Mälardalen University Press Licentiate Theses No. 253...
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Mälardalen University Press Licentiate Theses No. 253 GENERALIZED VANDERMONDE MATRICES AND DETERMINANTS IN ELECTROMAGNETIC COMPATIBILITY Karl Lundengård 2017 School of Education, Culture and Communication
Transcript of Mälardalen University Press Licentiate Theses No. 253...
Mälardalen University Press Licentiate ThesesNo. 253
GENERALIZED VANDERMONDE MATRICES ANDDETERMINANTS IN ELECTROMAGNETIC COMPATIBILITY
Karl Lundengård
2017
School of Education, Culture and Communication
Mälardalen University Press Licentiate ThesesNo. 253
GENERALIZED VANDERMONDE MATRICES ANDDETERMINANTS IN ELECTROMAGNETIC COMPATIBILITY
Karl Lundengård
2017
School of Education, Culture and Communication
Copyright © Karl Lundengård, 2017ISBN 978-91-7485-312-4ISSN 1651-9256Printed by E-Print AB, Stockholm, Sweden
Abstract
Matrices whose rows (or columns) consists of monomials of sequential powersare called Vandermonde matrices and can be used to describe several usefulconcepts and have properties that can be helpful for solving many kinds ofproblems. In this thesis we will discuss this matrix and some of its propertiesas well as a generalization of it and how it can be applied to curve fittingdischarge current for the purpose of ensuring electromagnetic compatibility.
In the first chapter the basic theory for later chapters is introduced. Thisincludes the Vandermonde matrix and some of its properties, history, appli-cations and generalizations, interpolation and regression problems, optimalexperiment design and modelling of electrostatic discharge currents with thepurpose to ensure electromagnetic compatibility.
The second chapter focuses on finding the extreme points for the deter-minant for the Vandermonde matrix on various surfaces including spheres,ellipsoids, cylinders and tori. The extreme points are analysed in threedimensions or more.
The third chapter discusses fitting a particular model called the p-peakedAnalytically Extended Function (AEF) to data taken either from a stan-dard for electromagnetic compatibility or experimental measurements. Morespecifically the AEF will be fitted to discharge currents from the IEC 62305-1and IEC 61000-4-2 standards for lightning protection and electrostatic dis-charge immunity as well as some experimentally measured data of similarphenomena.
1
Abstract
Matrices whose rows (or columns) consists of monomials of sequential powersare called Vandermonde matrices and can be used to describe several usefulconcepts and have properties that can be helpful for solving many kinds ofproblems. In this thesis we will discuss this matrix and some of its propertiesas well as a generalization of it and how it can be applied to curve fittingdischarge current for the purpose of ensuring electromagnetic compatibility.
In the first chapter the basic theory for later chapters is introduced. Thisincludes the Vandermonde matrix and some of its properties, history, appli-cations and generalizations, interpolation and regression problems, optimalexperiment design and modelling of electrostatic discharge currents with thepurpose to ensure electromagnetic compatibility.
The second chapter focuses on finding the extreme points for the deter-minant for the Vandermonde matrix on various surfaces including spheres,ellipsoids, cylinders and tori. The extreme points are analysed in threedimensions or more.
The third chapter discusses fitting a particular model called the p-peakedAnalytically Extended Function (AEF) to data taken either from a stan-dard for electromagnetic compatibility or experimental measurements. Morespecifically the AEF will be fitted to discharge currents from the IEC 62305-1and IEC 61000-4-2 standards for lightning protection and electrostatic dis-charge immunity as well as some experimentally measured data of similarphenomena.
1
Acknowledgements
First I am very grateful to my closest colleagues. My main supervisor Pro-fessor Sergei Silvestrov introduced the Vandermonde matrix to me and sug-gested the research pursued in this thesis. From him and my co-supervisorProfessor Anatoliy Malyarenko I have learned invaluable lessons about math-ematics research that will be very important in my future career. Optimisingthe Vandermonde determinant would have been less fruitful and engagingwithout fellow research student Jonas Osterberg whose skills and ideas com-plement my own very well. I am very glad to have worked with Dr. MilicaRancic whose conscientiousness, work ethic and patience when introducingme to the world of electromagnetic compatibility and developing and evalu-ating the ideas used in this thesis makes me consider her a true role modelfor an interdisciplinary researcher. The research related to electromagneticcompatibility owes a lot to the regular input from Assistant Professor VesnaJavor to ensure the relevance and quality of the work.
I am also grateful to the other research students at Malardalen Universitythat I have worked alongside with. I have found the environment at theSchool for Education, Culture and Communication at Malardalen Universityexcellent and full of friendly, helpful and skilled co-workers.
Last but not least a very heartfelt thank you to my family for all thesupport, encouragement and assistance you have given me. A special men-tion to my sister for help with translating from 18th century French. I amalso very sad that I will not get to spend time explaining the contents ofthis thesis to my father whose entire mathematics career consisted of unsuc-cessfully solving a single problem on the blackboard in 9th grade. On theother hand, my mother’s modest academic credentials have never stoppedher from engaging, discussing and enjoying my work, interests and othernew knowledge so I am sure she will keep me busy.
Without the ideas, requests, remarks, questions, encouragements andpatience of those around me this work would not have been completed.
Karl Lundengard Vasteras, March, 2017
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Acknowledgements
First I am very grateful to my closest colleagues. My main supervisor Pro-fessor Sergei Silvestrov introduced the Vandermonde matrix to me and sug-gested the research pursued in this thesis. From him and my co-supervisorProfessor Anatoliy Malyarenko I have learned invaluable lessons about math-ematics research that will be very important in my future career. Optimisingthe Vandermonde determinant would have been less fruitful and engagingwithout fellow research student Jonas Osterberg whose skills and ideas com-plement my own very well. I am very glad to have worked with Dr. MilicaRancic whose conscientiousness, work ethic and patience when introducingme to the world of electromagnetic compatibility and developing and evalu-ating the ideas used in this thesis makes me consider her a true role modelfor an interdisciplinary researcher. The research related to electromagneticcompatibility owes a lot to the regular input from Assistant Professor VesnaJavor to ensure the relevance and quality of the work.
I am also grateful to the other research students at Malardalen Universitythat I have worked alongside with. I have found the environment at theSchool for Education, Culture and Communication at Malardalen Universityexcellent and full of friendly, helpful and skilled co-workers.
Last but not least a very heartfelt thank you to my family for all thesupport, encouragement and assistance you have given me. A special men-tion to my sister for help with translating from 18th century French. I amalso very sad that I will not get to spend time explaining the contents ofthis thesis to my father whose entire mathematics career consisted of unsuc-cessfully solving a single problem on the blackboard in 9th grade. On theother hand, my mother’s modest academic credentials have never stoppedher from engaging, discussing and enjoying my work, interests and othernew knowledge so I am sure she will keep me busy.
Without the ideas, requests, remarks, questions, encouragements andpatience of those around me this work would not have been completed.
Karl Lundengard Vasteras, March, 2017
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Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Popularvetenskaplig sammanfattning
Denna licentiatuppsats behandlar tva olika amnen, optimering av determi-nanten av Vandermonde-matrisen over olika volymer i olika dimensioner ochhur en viss klass av funktioner kan anvandas for att approximera strommeni elektrostatiska urladdningar som anvands for att sakerstalla elektromag-netisk kompatibilitet. En exempel pa kopplingen mellan de tva omradenages i den sista delen av uppsatsen.
En Vandermonde-matris ar en matris dar raderna (eller kolonnerna) gesav stigande potenser och sadana matriser forekommer i manga olika sam-manhang, bade inom abstrakt matematik och tillampningar inom andraomraden. I uppsatsen ges en kort genomgang av Vandermonde-matrisenshistoria och tillampningar av den och nagra beslaktade matriser. Fokusligger pa interpolation or regression vilket kortfattat kan beskrivas sommetoder for att anpassa en matematisk beskrivning till given data, t.ex.fran experimentella matningar.
Determinanten av en matris ar ett tal som beraknas fran matrisens el-ement och som pa ett kompakt satt kan beskriva flera olika egenskaper avmatrisen eller systemet som matrisen beskriver. I denna uppsats diskuterashur man skall valja element i Vandermonde-matrisen for att maximera de-terminanten under forutsattningen att elementen i Vandermonde-matrisentolkas som en punkt i en volym (som kan ha fler an tre dimensioner). Fleravolymer undersoks, bland annat klot, kuber, ellipsoider och torus.
En motivering till varfor det ar anvandbart att veta hur Vandermonde-matrisens determinant kan maximeras ar att det kan anvandas till optimalexperiment design, det vill saga att avgora hur man skall valja matpunkterfor att kunna bygga en sa bra matematisk modell som mojligt. Ett exempelpa hur detta kan ga till ges i sista delen av uppsatsen.
Ett omrade dar det ar anvandbart att kunna bygga matematiska mod-eller fran experimentella data ar elektromagnetisk kompatibilitet. Dettaomrade handlar om att sakerstalla att system som innehaller elektronik intepaverkas for mycket av externa elektromagnetiska storningar eller stor an-dra system da de anvands. En viktig del av detta omrade ar att undersokahur systemet reagerar pa olika externa storningar sasom de beskrivs i olikakonstruktionsstandarder. I uppsatsen diskuterar vi hur man med hjalp aven specifik klass av funktioner kan konstruera matematiska modeller baser-ade pa specifikationer i standarder eller experimentella data. Valbekantafenomen sasom blixtnedslag och urladdningar av statisk elektricitet mellanmanniska och metallforemal diskuteras.
4
Popular-science summary
This licentiate thesis discusses two different topics, optimisation of the Van-dermonde determinant over different volumes in various dimensions and howa certain class of functions can be used to approximate the current in elec-trostatic discharges used in ensuring electromagnetic compatibility. An ex-ample of how the two topics can be connected is given in the final part ofthe thesis.
A Vandermonde matrix is a matrix with rows (or column) given byincreasing powers and such matrices appear in many different circumstances,both in abstract mathematics and various applications. In the thesis a briefhistory of the Vandermonde matrix is given as well as a discussion of someapplications of the Vandermonde matrix and some related matrices. Themain topics will be interpolation and regression which can be described asmethods for fitting a mathematical description to collected data from forexample experimental measurements.
The determinant of a matrix is a number calculated from the elements ofthe matrix in a particular way and it can describe properties of the matrixor the system it describes in a compact way. In this thesis it is discussedhow to choose the elements of the Vandermonde matrix to maximise thedeterminant under the constraint that the elements that define the Vander-monde determinant are interpreted as points in a certain volume (that canhave a dimension higher than three). Examined volumes include spheres,cubes, ellipsoids and tori.
One way to motivate the usefulness of knowing how to maximise theVandermonde determinant is that it can be used in optimal experiment de-sign, that is determining how to choose the data points in an experiment toconstruct the best possible mathematical model. An example of how to dothis can be found in the final section of the thesis.
One area where it is useful to construct mathematical model from ex-perimental data is electromagnetic compatibility. This is the study if how toensure that a system that contains electronics is not disturbed too much byexternal electromagnetic disturbances or disturbs other systems when it isused. An important aspect of this field is examining how systems respondto different external disturbances described in different construction stan-dards. In this thesis we discuss how a specific class of functions can be usedto construct mathematical models based on specifications in standards orexperimental data. Well-known phenomenon such as lightning strikes andelectrostatic discharges from a human being to a metal object are discussed.
5
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Popularvetenskaplig sammanfattning
Denna licentiatuppsats behandlar tva olika amnen, optimering av determi-nanten av Vandermonde-matrisen over olika volymer i olika dimensioner ochhur en viss klass av funktioner kan anvandas for att approximera strommeni elektrostatiska urladdningar som anvands for att sakerstalla elektromag-netisk kompatibilitet. En exempel pa kopplingen mellan de tva omradenages i den sista delen av uppsatsen.
En Vandermonde-matris ar en matris dar raderna (eller kolonnerna) gesav stigande potenser och sadana matriser forekommer i manga olika sam-manhang, bade inom abstrakt matematik och tillampningar inom andraomraden. I uppsatsen ges en kort genomgang av Vandermonde-matrisenshistoria och tillampningar av den och nagra beslaktade matriser. Fokusligger pa interpolation or regression vilket kortfattat kan beskrivas sommetoder for att anpassa en matematisk beskrivning till given data, t.ex.fran experimentella matningar.
Determinanten av en matris ar ett tal som beraknas fran matrisens el-ement och som pa ett kompakt satt kan beskriva flera olika egenskaper avmatrisen eller systemet som matrisen beskriver. I denna uppsats diskuterashur man skall valja element i Vandermonde-matrisen for att maximera de-terminanten under forutsattningen att elementen i Vandermonde-matrisentolkas som en punkt i en volym (som kan ha fler an tre dimensioner). Fleravolymer undersoks, bland annat klot, kuber, ellipsoider och torus.
En motivering till varfor det ar anvandbart att veta hur Vandermonde-matrisens determinant kan maximeras ar att det kan anvandas till optimalexperiment design, det vill saga att avgora hur man skall valja matpunkterfor att kunna bygga en sa bra matematisk modell som mojligt. Ett exempelpa hur detta kan ga till ges i sista delen av uppsatsen.
Ett omrade dar det ar anvandbart att kunna bygga matematiska mod-eller fran experimentella data ar elektromagnetisk kompatibilitet. Dettaomrade handlar om att sakerstalla att system som innehaller elektronik intepaverkas for mycket av externa elektromagnetiska storningar eller stor an-dra system da de anvands. En viktig del av detta omrade ar att undersokahur systemet reagerar pa olika externa storningar sasom de beskrivs i olikakonstruktionsstandarder. I uppsatsen diskuterar vi hur man med hjalp aven specifik klass av funktioner kan konstruera matematiska modeller baser-ade pa specifikationer i standarder eller experimentella data. Valbekantafenomen sasom blixtnedslag och urladdningar av statisk elektricitet mellanmanniska och metallforemal diskuteras.
4
Popular-science summary
This licentiate thesis discusses two different topics, optimisation of the Van-dermonde determinant over different volumes in various dimensions and howa certain class of functions can be used to approximate the current in elec-trostatic discharges used in ensuring electromagnetic compatibility. An ex-ample of how the two topics can be connected is given in the final part ofthe thesis.
A Vandermonde matrix is a matrix with rows (or column) given byincreasing powers and such matrices appear in many different circumstances,both in abstract mathematics and various applications. In the thesis a briefhistory of the Vandermonde matrix is given as well as a discussion of someapplications of the Vandermonde matrix and some related matrices. Themain topics will be interpolation and regression which can be described asmethods for fitting a mathematical description to collected data from forexample experimental measurements.
The determinant of a matrix is a number calculated from the elements ofthe matrix in a particular way and it can describe properties of the matrixor the system it describes in a compact way. In this thesis it is discussedhow to choose the elements of the Vandermonde matrix to maximise thedeterminant under the constraint that the elements that define the Vander-monde determinant are interpreted as points in a certain volume (that canhave a dimension higher than three). Examined volumes include spheres,cubes, ellipsoids and tori.
One way to motivate the usefulness of knowing how to maximise theVandermonde determinant is that it can be used in optimal experiment de-sign, that is determining how to choose the data points in an experiment toconstruct the best possible mathematical model. An example of how to dothis can be found in the final section of the thesis.
One area where it is useful to construct mathematical model from ex-perimental data is electromagnetic compatibility. This is the study if how toensure that a system that contains electronics is not disturbed too much byexternal electromagnetic disturbances or disturbs other systems when it isused. An important aspect of this field is examining how systems respondto different external disturbances described in different construction stan-dards. In this thesis we discuss how a specific class of functions can be usedto construct mathematical models based on specifications in standards orexperimental data. Well-known phenomenon such as lightning strikes andelectrostatic discharges from a human being to a metal object are discussed.
5
Notation
Matrix and vector notation
v, M - Bold, roman lower- and uppercase lettersdenote vectors and matrices respectively.
Mi,j - Element on the ith row and jth column of M.
M·,j , Mi,· - Column (row) vector containing all elementsfrom the jth column (ith row) of M.
[aij ]nmij - n×m matrix with element aij in
the ith row and jth column.
Vnm, Vn = Vnn - n×m Vandermonde matrix.
Gnm, Gn = Gnn - n×m generalized Vandermonde matrix.
Standard sets
Z, N, R, C - Sets of all integers, natural numbers (including 0),real numbers and complex numbers.
Snp , S
n = Sn2 - The n-dimensional sphere defined by the p - norm,
Snp (r) =
x ∈ Rn+1
∣∣∣∣n+1∑k=1
|xk|p = r
.
Ck[K] - All functions on K with continuous kth derivative.
Special functions
Definitions can be found in standard texts.Suggested sources use notation consistent with thesis.
Hn, C(α)n , P
(α,β)n - Hermite, Gegenbauer and Jacobi polynomials, see [2].
Γ(x), ψ(x) - The Gamma- and Digamma functions, see [2].
2F2(a, b; c;x) - The hypergeometric function, see [2].
Gm,np,q
(z
∣∣∣∣ab
)- The Meijer G-function, see [139].
Other
dfdx = f ′(x) - Derivative of the function f with respect to x.
dkfdxk = f (k)(x) - kth derivative of the function f with respect to x.
∂f∂x = f ′(x) - Partial derivative of the function f with respect to x.
ab - Rising factorial ab = a(a+ 1) · · · (a+ b− 1).
7
Notation
Matrix and vector notation
v, M - Bold, roman lower- and uppercase lettersdenote vectors and matrices respectively.
Mi,j - Element on the ith row and jth column of M.
M·,j , Mi,· - Column (row) vector containing all elementsfrom the jth column (ith row) of M.
[aij ]nmij - n×m matrix with element aij in
the ith row and jth column.
Vnm, Vn = Vnn - n×m Vandermonde matrix.
Gnm, Gn = Gnn - n×m generalized Vandermonde matrix.
Standard sets
Z, N, R, C - Sets of all integers, natural numbers (including 0),real numbers and complex numbers.
Snp , S
n = Sn2 - The n-dimensional sphere defined by the p - norm,
Snp (r) =
x ∈ Rn+1
∣∣∣∣n+1∑k=1
|xk|p = r
.
Ck[K] - All functions on K with continuous kth derivative.
Special functions
Definitions can be found in standard texts.Suggested sources use notation consistent with thesis.
Hn, C(α)n , P
(α,β)n - Hermite, Gegenbauer and Jacobi polynomials, see [2].
Γ(x), ψ(x) - The Gamma- and Digamma functions, see [2].
2F2(a, b; c;x) - The hypergeometric function, see [2].
Gm,np,q
(z
∣∣∣∣ab
)- The Meijer G-function, see [139].
Other
dfdx = f ′(x) - Derivative of the function f with respect to x.
dkfdxk = f (k)(x) - kth derivative of the function f with respect to x.
∂f∂x = f ′(x) - Partial derivative of the function f with respect to x.
ab - Rising factorial ab = a(a+ 1) · · · (a+ b− 1).
7
Contents
List of Papers 13
1 Introduction 15
1.1 The Vandermonde matrix . . . . . . . . . . . . . . . . . . . . 19
1.1.1 Who was Vandermonde? . . . . . . . . . . . . . . . . . 19
1.1.2 The Vandermonde determinant . . . . . . . . . . . . . 21
1.1.3 Inverse of the Vandermonde matrix . . . . . . . . . . . 26
1.1.4 The alternant matrix . . . . . . . . . . . . . . . . . . . 27
1.1.5 The generalized Vandermonde matrix . . . . . . . . . 30
1.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.2.1 Polynomial interpolation . . . . . . . . . . . . . . . . . 33
1.3 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.3.1 Linear regression models . . . . . . . . . . . . . . . . . 38
1.3.2 Non-linear regression models . . . . . . . . . . . . . . 39
1.3.3 The Marquardt least-squares method . . . . . . . . . . 39
1.3.4 D-optimal experiment design . . . . . . . . . . . . . . 43
1.4 Electromagnetic compatibility andelectrostatic discharge currents . . . . . . . . . . . . . . . . . 46
1.4.1 Electrostatic discharge modelling . . . . . . . . . . . . 48
1.5 Summaries of papers . . . . . . . . . . . . . . . . . . . . . . . 51
9
Contents
List of Papers 13
1 Introduction 15
1.1 The Vandermonde matrix . . . . . . . . . . . . . . . . . . . . 19
1.1.1 Who was Vandermonde? . . . . . . . . . . . . . . . . . 19
1.1.2 The Vandermonde determinant . . . . . . . . . . . . . 21
1.1.3 Inverse of the Vandermonde matrix . . . . . . . . . . . 26
1.1.4 The alternant matrix . . . . . . . . . . . . . . . . . . . 27
1.1.5 The generalized Vandermonde matrix . . . . . . . . . 30
1.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.2.1 Polynomial interpolation . . . . . . . . . . . . . . . . . 33
1.3 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.3.1 Linear regression models . . . . . . . . . . . . . . . . . 38
1.3.2 Non-linear regression models . . . . . . . . . . . . . . 39
1.3.3 The Marquardt least-squares method . . . . . . . . . . 39
1.3.4 D-optimal experiment design . . . . . . . . . . . . . . 43
1.4 Electromagnetic compatibility andelectrostatic discharge currents . . . . . . . . . . . . . . . . . 46
1.4.1 Electrostatic discharge modelling . . . . . . . . . . . . 48
1.5 Summaries of papers . . . . . . . . . . . . . . . . . . . . . . . 51
9
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
2 Extreme points of the Vandermonde determinant 53
2.1 Extreme points of the Vandermonde determinant and relateddeterminants on various surfaces in three dimensions . . . . . 55
2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions . . . . . . . . . . . . . . . 55
2.1.2 Extreme points of the Vandermonde determinant onthe three-dimensional unit sphere . . . . . . . . . . . . 59
2.1.3 Optimisation of the Vandermonde determinant on thethree-dimensional torus . . . . . . . . . . . . . . . . . 60
2.1.4 Optimisation using Grobner bases . . . . . . . . . . . 64
2.1.5 Extreme points on the ellipsoid in three dimensions . 66
2.1.6 Extreme points on the cylinder in three dimensions . . 68
2.1.7 Optimizing the Vandermonde determinant on a sur-face defined by a homogeneous polynomial . . . . . . . 70
2.2 Optimization of the Vandermondedeterminant on some n-dimensional surfaces . . . . . . . . . . 72
2.2.1 The extreme points on the sphere given by roots of apolynomial . . . . . . . . . . . . . . . . . . . . . . . . 73
2.2.2 Further visual exploration on the sphere . . . . . . . . 81
2.2.3 The extreme points on spheres defined by the p-normsgiven by roots of a polynomial . . . . . . . . . . . . . 88
2.2.4 The Vandermonde determinant on spheres defined bythe 4-norm . . . . . . . . . . . . . . . . . . . . . . . . 96
3 Approximation of electrostatic discharge currents using theanalytically extended function 99
3.1 The analytically extended function (AEF) . . . . . . . . . . . 101
3.1.1 The p-peak analytically extended function . . . . . . . 102
3.2 Approximation of lightning discharge current functions . . . . 109
3.2.1 Fitting the AEF . . . . . . . . . . . . . . . . . . . . . 110
10
CONTENTS
3.2.2 Estimating parameters for underdetermined systems . 111
3.2.3 Fitting with data points as well as charge flow andspecific energy conditions . . . . . . . . . . . . . . . . 112
3.2.4 Calculating the η-parameters from the β-parameters . 115
3.2.5 Explicit formulas for a single-peak AEF . . . . . . . . 116
3.2.6 Fitting to lightning discharge currents . . . . . . . . . 117
3.3 Approximation of electrostatic discharge currents . . . . . . . 121
3.3.1 IEC 61000-4-2 Standard current waveshape . . . . . . 122
3.3.2 D-Optimal approximation for exponents given by aclass of arithmetic sequences . . . . . . . . . . . . . . 125
3.3.3 Examples of models from applications and experiments 129
3.3.4 Summary of ESD modelling . . . . . . . . . . . . . . . 131
References 135
Index 151
List of Figures 153
List of Tables 156
List of Definitions 156
List of Theorems 157
List of Lemmas 157
11
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
2 Extreme points of the Vandermonde determinant 53
2.1 Extreme points of the Vandermonde determinant and relateddeterminants on various surfaces in three dimensions . . . . . 55
2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions . . . . . . . . . . . . . . . 55
2.1.2 Extreme points of the Vandermonde determinant onthe three-dimensional unit sphere . . . . . . . . . . . . 59
2.1.3 Optimisation of the Vandermonde determinant on thethree-dimensional torus . . . . . . . . . . . . . . . . . 60
2.1.4 Optimisation using Grobner bases . . . . . . . . . . . 64
2.1.5 Extreme points on the ellipsoid in three dimensions . 66
2.1.6 Extreme points on the cylinder in three dimensions . . 68
2.1.7 Optimizing the Vandermonde determinant on a sur-face defined by a homogeneous polynomial . . . . . . . 70
2.2 Optimization of the Vandermondedeterminant on some n-dimensional surfaces . . . . . . . . . . 72
2.2.1 The extreme points on the sphere given by roots of apolynomial . . . . . . . . . . . . . . . . . . . . . . . . 73
2.2.2 Further visual exploration on the sphere . . . . . . . . 81
2.2.3 The extreme points on spheres defined by the p-normsgiven by roots of a polynomial . . . . . . . . . . . . . 88
2.2.4 The Vandermonde determinant on spheres defined bythe 4-norm . . . . . . . . . . . . . . . . . . . . . . . . 96
3 Approximation of electrostatic discharge currents using theanalytically extended function 99
3.1 The analytically extended function (AEF) . . . . . . . . . . . 101
3.1.1 The p-peak analytically extended function . . . . . . . 102
3.2 Approximation of lightning discharge current functions . . . . 109
3.2.1 Fitting the AEF . . . . . . . . . . . . . . . . . . . . . 110
10
CONTENTS
3.2.2 Estimating parameters for underdetermined systems . 111
3.2.3 Fitting with data points as well as charge flow andspecific energy conditions . . . . . . . . . . . . . . . . 112
3.2.4 Calculating the η-parameters from the β-parameters . 115
3.2.5 Explicit formulas for a single-peak AEF . . . . . . . . 116
3.2.6 Fitting to lightning discharge currents . . . . . . . . . 117
3.3 Approximation of electrostatic discharge currents . . . . . . . 121
3.3.1 IEC 61000-4-2 Standard current waveshape . . . . . . 122
3.3.2 D-Optimal approximation for exponents given by aclass of arithmetic sequences . . . . . . . . . . . . . . 125
3.3.3 Examples of models from applications and experiments 129
3.3.4 Summary of ESD modelling . . . . . . . . . . . . . . . 131
References 135
Index 151
List of Figures 153
List of Tables 156
List of Definitions 156
List of Theorems 157
List of Lemmas 157
11
List of Papers
Paper A. Karl Lundengard, Jonas Osterberg and Sergei Silvestrov. Extreme points ofthe Vandermonde determinant on the sphere and some limits involving thegeneralized Vandermonde determinant.Preprint: arXiv:1312.6193 [math.ca], 2013.
Paper B. Karl Lundengard, Jonas Osterberg, and Sergei Silvestrov. Optimizationof the determinant of the Vandermonde matrix and related matrices. InAIP Conference Proceedings 1637, ICNPAA, Narvik, Norway, pages 627–636, 2014.
Paper C. Karl Lundengard, Jonas Osterberg, and Sergei Silvestrov. Optimization ofthe determinant of the Vandermonde matrix on the sphere and related sur-faces. In Christos H Skiadas, editor, ASMDA 2015 Proceedings: 16th AppliedStochastic Models and Data Analysis International Conference with 4th De-mographics 2015 Workshop, pages 637–648. ISAST: International Society forthe Advancement of Science and Technology, 2015.
Paper D. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov. On someproperties of the multi-peaked analytically extended function for approxima-tion of lightning discharge currents. Sergei Silvestrov and Milica Rancic,editors, Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Ma-terial Physics and Financial Engineering, volume 178 of Springer Proceedingsin Mathematics & Statistics. Springer International Publishing, 2016.
Paper E. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov. Estima-tion of parameters for the multi-peaked AEF current functions. Methodologyand Computing in Applied Probability, pages 1–15, 2016.
Paper F. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov. Electro-static discharge currents representation using the multi-peaked analyticallyextended function by interpolation on a D-optimal design.Preprint: arXiv:1701.03728 [physics.comp-ph], 2017.
13
List of Papers
Paper A. Karl Lundengard, Jonas Osterberg and Sergei Silvestrov. Extreme points ofthe Vandermonde determinant on the sphere and some limits involving thegeneralized Vandermonde determinant.Preprint: arXiv:1312.6193 [math.ca], 2013.
Paper B. Karl Lundengard, Jonas Osterberg, and Sergei Silvestrov. Optimizationof the determinant of the Vandermonde matrix and related matrices. InAIP Conference Proceedings 1637, ICNPAA, Narvik, Norway, pages 627–636, 2014.
Paper C. Karl Lundengard, Jonas Osterberg, and Sergei Silvestrov. Optimization ofthe determinant of the Vandermonde matrix on the sphere and related sur-faces. In Christos H Skiadas, editor, ASMDA 2015 Proceedings: 16th AppliedStochastic Models and Data Analysis International Conference with 4th De-mographics 2015 Workshop, pages 637–648. ISAST: International Society forthe Advancement of Science and Technology, 2015.
Paper D. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov. On someproperties of the multi-peaked analytically extended function for approxima-tion of lightning discharge currents. Sergei Silvestrov and Milica Rancic,editors, Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Ma-terial Physics and Financial Engineering, volume 178 of Springer Proceedingsin Mathematics & Statistics. Springer International Publishing, 2016.
Paper E. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov. Estima-tion of parameters for the multi-peaked AEF current functions. Methodologyand Computing in Applied Probability, pages 1–15, 2016.
Paper F. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov. Electro-static discharge currents representation using the multi-peaked analyticallyextended function by interpolation on a D-optimal design.Preprint: arXiv:1701.03728 [physics.comp-ph], 2017.
13
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Parts of the thesis have been presented at the following international conferences:
1. 10th International Conference on Mathematical Problems in Engineering,Aerospace and Sciences - ICNPAA 2014, Narvik, Norway, July 15-18, 2014.
2. 16th Applied Stochastic Models and Data Analysis International Conferencewith 4th Demographics 2015 Workshop - ASMDA 2015, Piraeus, Greece,June 30 - July 4, 2015.
Summaries of papers A-F with a brief description of the thesis authorscontributions to each paper can be found in Section 1.5.
14
Chapter 1
Introduction
This chapter is partially based on Paper D:
Paper D. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov.On some properties of the multi-peaked analytically extended functionfor approximation of lightning discharge currents. Sergei Silvestrovand Milica Rancic, editors, Engineering Mathematics I: Electromag-netics, Fluid Mechanics, Material Physics and Financial Engineer-ing, volume 178 of Springer Proceedings in Mathematics & Statistics.Springer International Publishing, 2016.
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Parts of the thesis have been presented at the following international conferences:
1. 10th International Conference on Mathematical Problems in Engineering,Aerospace and Sciences - ICNPAA 2014, Narvik, Norway, July 15-18, 2014.
2. 16th Applied Stochastic Models and Data Analysis International Conferencewith 4th Demographics 2015 Workshop - ASMDA 2015, Piraeus, Greece,June 30 - July 4, 2015.
Summaries of papers A-F with a brief description of the thesis authorscontributions to each paper can be found in Section 1.5.
14
Chapter 1
Introduction
This chapter is partially based on Paper D:
Paper D. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov.On some properties of the multi-peaked analytically extended functionfor approximation of lightning discharge currents. Sergei Silvestrovand Milica Rancic, editors, Engineering Mathematics I: Electromag-netics, Fluid Mechanics, Material Physics and Financial Engineer-ing, volume 178 of Springer Proceedings in Mathematics & Statistics.Springer International Publishing, 2016.
INTRODUCTION
The Vandermonde matrix is a well-known type of matrix that appears inmany different areas. In this thesis we will discuss this matrix and some ofits properties, specifically the extreme points of the determinant on varioussurfaces and we will use a generalized Vandermonde matrix for fitting acertain type of curve to data taken from sources that are important whenanalysing electromagnetic compatibility.
This thesis is based on the six papers listed on page 13. The contentshave been rearranged to clarify the relations between the material in thedifferent papers. If a section is based on a paper this is specified at thebeginning of the section and unless otherwise specified any subsection isfrom the same source. A section that is based on a paper consists of textfrom the paper unchanged except for modifications to correct misprints andpreserve consistency within the thesis. Parts of several papers have also beenomitted to avoid repetition and improve cohesion. The relations betweencontents of the the sections are of many kinds, common definitions anddependent results, conceptual connections as well as similarities in prooftechniques and problem formulations. This is illustrated in Figure 1.1. Areader only interested in a particular section or in a hurry can consult Figure1.2 to find a short route to the desired content.
In chapter 1 the basic theory for later chapters is introduced. The Van-dermonde matrix and some of its properties, history, applications and gen-eralizations are briefly introduced in Section 1.1. In Section 1.2 interpola-tion problems and their relations to alternant- and Vandermonde matricesis described. In Section 1.3 various regression models and the Marquardtleast-squares method for non-linear regression problems are discussed. Theoptimal design of experiments with respect to regression is discussed aswell. Section 1.4 introduces electromagnetic compatibility and certain curve-fitting problems that appear in modelling of electrostatic discharge currents.
Chapter 2 discusses the optimisation of the Vandermonde determinantover various surfaces. First the extreme points on the sphere in three dimen-sions are examined, Section 2.1.1 and 2.1.2. Further discussion includes thetorus, cylinder and ellipsoid, Section 2.1.3–2.1.7. In Section 2.2 the determi-nant is optimised on the sphere and related surfaces in higher dimensions.
Chapter 3 discusses fitting a piecewise non-linear regression model todata. The particular model is introduced in Section 3.1 and a general frame-work for fitting it to data using the Marquardt least-squares method is de-scribed in Section 3.2.1–3.2.5. The framework is then applied to lightningdischarge currents in Section 3.2.6. An alternate curve-fitting method basedon D-optimal interpolation (found analogously to the results in Section 2.2)is described and applied to electrostatic discharge currents in Section 3.3.
17
INTRODUCTION
The Vandermonde matrix is a well-known type of matrix that appears inmany different areas. In this thesis we will discuss this matrix and some ofits properties, specifically the extreme points of the determinant on varioussurfaces and we will use a generalized Vandermonde matrix for fitting acertain type of curve to data taken from sources that are important whenanalysing electromagnetic compatibility.
This thesis is based on the six papers listed on page 13. The contentshave been rearranged to clarify the relations between the material in thedifferent papers. If a section is based on a paper this is specified at thebeginning of the section and unless otherwise specified any subsection isfrom the same source. A section that is based on a paper consists of textfrom the paper unchanged except for modifications to correct misprints andpreserve consistency within the thesis. Parts of several papers have also beenomitted to avoid repetition and improve cohesion. The relations betweencontents of the the sections are of many kinds, common definitions anddependent results, conceptual connections as well as similarities in prooftechniques and problem formulations. This is illustrated in Figure 1.1. Areader only interested in a particular section or in a hurry can consult Figure1.2 to find a short route to the desired content.
In chapter 1 the basic theory for later chapters is introduced. The Van-dermonde matrix and some of its properties, history, applications and gen-eralizations are briefly introduced in Section 1.1. In Section 1.2 interpola-tion problems and their relations to alternant- and Vandermonde matricesis described. In Section 1.3 various regression models and the Marquardtleast-squares method for non-linear regression problems are discussed. Theoptimal design of experiments with respect to regression is discussed aswell. Section 1.4 introduces electromagnetic compatibility and certain curve-fitting problems that appear in modelling of electrostatic discharge currents.
Chapter 2 discusses the optimisation of the Vandermonde determinantover various surfaces. First the extreme points on the sphere in three dimen-sions are examined, Section 2.1.1 and 2.1.2. Further discussion includes thetorus, cylinder and ellipsoid, Section 2.1.3–2.1.7. In Section 2.2 the determi-nant is optimised on the sphere and related surfaces in higher dimensions.
Chapter 3 discusses fitting a piecewise non-linear regression model todata. The particular model is introduced in Section 3.1 and a general frame-work for fitting it to data using the Marquardt least-squares method is de-scribed in Section 3.2.1–3.2.5. The framework is then applied to lightningdischarge currents in Section 3.2.6. An alternate curve-fitting method basedon D-optimal interpolation (found analogously to the results in Section 2.2)is described and applied to electrostatic discharge currents in Section 3.3.
17
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Section 1.1 Section 1.1.1
Section 1.1.2
Section 1.1.3
Section 1.1.4 Section 1.1.5
Section 1.2
Section 1.2.1
Section 1.3 Section 1.3.1Section 1.3.2
Section 1.3.3Paper D
Section 1.3.4Section 1.4
Section 2.1.1Paper A
Sections 2.1.2-2.1.7Paper A–C
Section 2.2Paper A–C
Section 3.1Paper D
Section 3.2Paper E
Section 3.3Paper F
Figure 1.1: Relations between sections of the thesis. Arrows indicate that the targetsection uses some definition or theorem from the source section. Dashedlines indicates a tangential or conceptual relation.
Section 1.1
Section 1.3Section 1.1.2
Sections 1.1.4, 1.2.1
Section 1.1.5 Section 1.3.2
Sections 1.3.1, 1.3.4
Section 1.3.3Paper D
Section 2.1.1Paper A
Sections2.1.2-2.1.7, 2.2Paper A–C
Section 3.3Paper F
Sections 1.4, 3.1Paper D
Section 3.2Paper E
Figure 1.2: Reference that demonstrates short routes to the different chapters.
18
1.1. THE VANDERMONDE MATRIX
1.1 The Vandermonde matrix
The Vandermonde matrix is a well-known matrix with a very special formthat appears in many different circumstances, a few examples are poly-nomial interpolation (see Section 1.2.1), least square regression (see Sec-tion 1.3), optimal experiment design (see Section 1.3.4), construction oferror-detecting and error-correcting codes (see [18, 69, 143] as well as morerecent work such as [17]), determining if a market with a finite set oftraded assets is complete [32], calculation of the discrete Fourier trans-form [142] and related transforms such as the fractional discrete Fouriertransform [122], the quantum Fourier transform [36], and the Vandermondetransform [6, 7], solving systems of differential equations with constant co-efficients [120], various problems in mathematical- [168], nuclear- [29], andquantum physics [148,159] and describing properties of the Fisher informa-tion matrix of stationary stochastic processes [93].
In this section we will review some of the basic properties of the Van-dermonde matrix, starting with its definition.
Definition 1.1. A Vandermonde matrix is an n×m matrix of the form
Vmn(xn) =[xi−1j
]m,n
i,j=
1 1 · · · 1x1 x2 · · · xn...
.... . .
...
xm−11 xm−1
2 · · · xm−1n
(1)
where xi ∈ C, i = 1, . . . , n. If the matrix is square, n = m, the notationVn = Vnm will be used.
Remark 1.1. Note that in the literature the term Vandermonde matrix isoften used for the transpose of the matrix given in expression (1).
1.1.1 Who was Vandermonde?
The matrix is named after Alexandre Theophile Vandermonde (1735–1796)who had a varied career that began with law studies and some success asa concert violinist, transitioned into work in science and mathematics inthe beginning of the 1770s that gradually turned into administrative andleadership positions at various Parisian institutions as well as work in politicsand economics in the end of the 1780s [42]. His entire mathematical careerconsisted of four published papers, first presented to the French Academyof Sciences in 1770 and 1771 and published a few years later.
19
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Section 1.1 Section 1.1.1
Section 1.1.2
Section 1.1.3
Section 1.1.4 Section 1.1.5
Section 1.2
Section 1.2.1
Section 1.3 Section 1.3.1Section 1.3.2
Section 1.3.3Paper D
Section 1.3.4Section 1.4
Section 2.1.1Paper A
Sections 2.1.2-2.1.7Paper A–C
Section 2.2Paper A–C
Section 3.1Paper D
Section 3.2Paper E
Section 3.3Paper F
Figure 1.1: Relations between sections of the thesis. Arrows indicate that the targetsection uses some definition or theorem from the source section. Dashedlines indicates a tangential or conceptual relation.
Section 1.1
Section 1.3Section 1.1.2
Sections 1.1.4, 1.2.1
Section 1.1.5 Section 1.3.2
Sections 1.3.1, 1.3.4
Section 1.3.3Paper D
Section 2.1.1Paper A
Sections2.1.2-2.1.7, 2.2Paper A–C
Section 3.3Paper F
Sections 1.4, 3.1Paper D
Section 3.2Paper E
Figure 1.2: Reference that demonstrates short routes to the different chapters.
18
1.1. THE VANDERMONDE MATRIX
1.1 The Vandermonde matrix
The Vandermonde matrix is a well-known matrix with a very special formthat appears in many different circumstances, a few examples are poly-nomial interpolation (see Section 1.2.1), least square regression (see Sec-tion 1.3), optimal experiment design (see Section 1.3.4), construction oferror-detecting and error-correcting codes (see [18, 69, 143] as well as morerecent work such as [17]), determining if a market with a finite set oftraded assets is complete [32], calculation of the discrete Fourier trans-form [142] and related transforms such as the fractional discrete Fouriertransform [122], the quantum Fourier transform [36], and the Vandermondetransform [6, 7], solving systems of differential equations with constant co-efficients [120], various problems in mathematical- [168], nuclear- [29], andquantum physics [148,159] and describing properties of the Fisher informa-tion matrix of stationary stochastic processes [93].
In this section we will review some of the basic properties of the Van-dermonde matrix, starting with its definition.
Definition 1.1. A Vandermonde matrix is an n×m matrix of the form
Vmn(xn) =[xi−1j
]m,n
i,j=
1 1 · · · 1x1 x2 · · · xn...
.... . .
...
xm−11 xm−1
2 · · · xm−1n
(1)
where xi ∈ C, i = 1, . . . , n. If the matrix is square, n = m, the notationVn = Vnm will be used.
Remark 1.1. Note that in the literature the term Vandermonde matrix isoften used for the transpose of the matrix given in expression (1).
1.1.1 Who was Vandermonde?
The matrix is named after Alexandre Theophile Vandermonde (1735–1796)who had a varied career that began with law studies and some success asa concert violinist, transitioned into work in science and mathematics inthe beginning of the 1770s that gradually turned into administrative andleadership positions at various Parisian institutions as well as work in politicsand economics in the end of the 1780s [42]. His entire mathematical careerconsisted of four published papers, first presented to the French Academyof Sciences in 1770 and 1771 and published a few years later.
19
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The first paper, Memoire sur la resolution des equations [164], discussessome properties of the roots of polynomial equations, more specifically for-mulas for the sum of the roots and a sum of symmetric functions of the pow-ers of the roots. This paper has been mentioned as important since it con-tains some of the fundamental ideas of group theory (see for instance [99]),but generally this work is overshadowed by the works of the contempo-rary Joseph Louis Lagrange (1736–1813) [97]. He also notices the equalitya2b+b2c+ac2−a2c−ab2−bc2 = (a−b)(a−c)(b−c), which is a special caseof the formula for the determinant of the Vandermonde matrix. It seemsthat Vandermonde did not understand the significance of the expression.
The second paper, Remarques sur des problemes de situation [165], dis-cusses the problem of the knight’s tour (what sequence of moves allows aknight to visit all squares on a chessboard exactly once). This paper is con-sidered the first mathematical paper that uses the basic ideas of what is nowcalled knot theory [140].
The third paper, Memoire sur des irrationnelles de differents ordres avecune application au cercle [166], is a paper on combinatorics and the mostwell-known result from the paper is the Chu-Vandermonde identity,
n∑k=1
k∏j=1
r + 1− j
j
n−k∏j=1
s+ 1− j
j
=
n∏j=1
r + s+ 1− j
j
,
where r, s ∈ R and n ∈ Z. The identity was first found by Chu Shih-Chieh(ca 1260 – ca 1320, traditional chinese: 朱世傑
)in 1303 in The precious
mirror of the four elements(四元玉
)and was rediscovered (apparently
independently) by Vandermonde [4, 127].In the fourth paper Memoire sur l’elimination [167] Vandermonde dis-
cusses some ideas for what we today call determinants, which is functionsthat can tell us if a linear equation system has a unique solution or not.The paper predates the modern definitions of determinants but Vander-monde discusses a general method for solving linear equation systems usingalternating functions, which has strong relation to determinants. He alsonotices that exchanging exponents for indices in a class of expressions fromhis first paper will give a class of expressions that he discusses in his fourthpaper [179]. This relation is mirrored in the relationship between the deter-minant of the Vandermonde matrix and the determinant of a general matrixdescribed in Theorem 1.3.
While Vandermonde’s papers can be said to contain many importantideas they do not bring any of them to maturity and he is therefore usu-ally considered a minor scientist and mathematician, especially compared
20
1.1. THE VANDERMONDE MATRIX
to well-known mathematicians such as Etienne Bezout (1730 – 1783) andPierre-Simon de Laplace (1749 – 1827) as well as the chemist AntoineLavoisier (1743 – 1794) that he worked with for some time after his math-ematical career. The Vandermonde matrix does not appear in any of Van-dermonde’s published works, which is not surprising considering that themodern matrix concept did not really take shape until almost a hundredyears later in the works of Sylvester and Cayley [25, 157]. It is thereforestrange that the Vandermonde matrix was named after him, a thoroughdiscussion on this can be found in [179], but a possible reason is the simpleformula for the determinant that Vandermonde briefly discusses in his fourthpaper can be generalized to a Vandermonde matrix of any size. One of themain reasons that the Vandermonde matrix has become known is that ithas an exceptionally simple expression for its determinant that in turn hasa surprisingly fundamental relation to the determinant of a general matrix.We will be taking a closer look at the determinant of the Vandermonde ma-trix and related matrices several times in this thesis so the next section willintroduce it and some of its properties.
1.1.2 The Vandermonde determinant
Often it is not the Vandermonde matrix itself that is useful, instead it is themultivariate polynomial given by its determinant that is examined and used.The determinant of the Vandermonde matrix is usually called the Vander-monde determinant (or Vandermonde polynomial or Vandermondian [168])and can be written using an exceptionally simple formula. But before wediscuss the Vandermonde determinant we will disuss the general determi-nant.
Definition 1.2. The determinant is a function of square matrices over afield F to the field F, det : Mn×n(F) → F such that if we consider thedeterminant as a function of the columns
det(M) = det(M·,1,M·,2, . . . ,M·,n)
of the matrix the determinant must have the following properties
• The determinant must be multilinear
det(M·,1, . . . , aM·,k + bN·,k, . . . ,M·,n)
= a det(M·,1, . . . ,M·,k, . . . ,M·,n) + b det(M·,1, . . . ,N·,k, . . . ,M·,n).
• The determinant must be alternating, that is if M·,i = M·,j for somei = j then det(M) = 0.
21
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The first paper, Memoire sur la resolution des equations [164], discussessome properties of the roots of polynomial equations, more specifically for-mulas for the sum of the roots and a sum of symmetric functions of the pow-ers of the roots. This paper has been mentioned as important since it con-tains some of the fundamental ideas of group theory (see for instance [99]),but generally this work is overshadowed by the works of the contempo-rary Joseph Louis Lagrange (1736–1813) [97]. He also notices the equalitya2b+b2c+ac2−a2c−ab2−bc2 = (a−b)(a−c)(b−c), which is a special caseof the formula for the determinant of the Vandermonde matrix. It seemsthat Vandermonde did not understand the significance of the expression.
The second paper, Remarques sur des problemes de situation [165], dis-cusses the problem of the knight’s tour (what sequence of moves allows aknight to visit all squares on a chessboard exactly once). This paper is con-sidered the first mathematical paper that uses the basic ideas of what is nowcalled knot theory [140].
The third paper, Memoire sur des irrationnelles de differents ordres avecune application au cercle [166], is a paper on combinatorics and the mostwell-known result from the paper is the Chu-Vandermonde identity,
n∑k=1
k∏j=1
r + 1− j
j
n−k∏j=1
s+ 1− j
j
=
n∏j=1
r + s+ 1− j
j
,
where r, s ∈ R and n ∈ Z. The identity was first found by Chu Shih-Chieh(ca 1260 – ca 1320, traditional chinese: 朱世傑
)in 1303 in The precious
mirror of the four elements(四元玉
)and was rediscovered (apparently
independently) by Vandermonde [4, 127].In the fourth paper Memoire sur l’elimination [167] Vandermonde dis-
cusses some ideas for what we today call determinants, which is functionsthat can tell us if a linear equation system has a unique solution or not.The paper predates the modern definitions of determinants but Vander-monde discusses a general method for solving linear equation systems usingalternating functions, which has strong relation to determinants. He alsonotices that exchanging exponents for indices in a class of expressions fromhis first paper will give a class of expressions that he discusses in his fourthpaper [179]. This relation is mirrored in the relationship between the deter-minant of the Vandermonde matrix and the determinant of a general matrixdescribed in Theorem 1.3.
While Vandermonde’s papers can be said to contain many importantideas they do not bring any of them to maturity and he is therefore usu-ally considered a minor scientist and mathematician, especially compared
20
1.1. THE VANDERMONDE MATRIX
to well-known mathematicians such as Etienne Bezout (1730 – 1783) andPierre-Simon de Laplace (1749 – 1827) as well as the chemist AntoineLavoisier (1743 – 1794) that he worked with for some time after his math-ematical career. The Vandermonde matrix does not appear in any of Van-dermonde’s published works, which is not surprising considering that themodern matrix concept did not really take shape until almost a hundredyears later in the works of Sylvester and Cayley [25, 157]. It is thereforestrange that the Vandermonde matrix was named after him, a thoroughdiscussion on this can be found in [179], but a possible reason is the simpleformula for the determinant that Vandermonde briefly discusses in his fourthpaper can be generalized to a Vandermonde matrix of any size. One of themain reasons that the Vandermonde matrix has become known is that ithas an exceptionally simple expression for its determinant that in turn hasa surprisingly fundamental relation to the determinant of a general matrix.We will be taking a closer look at the determinant of the Vandermonde ma-trix and related matrices several times in this thesis so the next section willintroduce it and some of its properties.
1.1.2 The Vandermonde determinant
Often it is not the Vandermonde matrix itself that is useful, instead it is themultivariate polynomial given by its determinant that is examined and used.The determinant of the Vandermonde matrix is usually called the Vander-monde determinant (or Vandermonde polynomial or Vandermondian [168])and can be written using an exceptionally simple formula. But before wediscuss the Vandermonde determinant we will disuss the general determi-nant.
Definition 1.2. The determinant is a function of square matrices over afield F to the field F, det : Mn×n(F) → F such that if we consider thedeterminant as a function of the columns
det(M) = det(M·,1,M·,2, . . . ,M·,n)
of the matrix the determinant must have the following properties
• The determinant must be multilinear
det(M·,1, . . . , aM·,k + bN·,k, . . . ,M·,n)
= a det(M·,1, . . . ,M·,k, . . . ,M·,n) + b det(M·,1, . . . ,N·,k, . . . ,M·,n).
• The determinant must be alternating, that is if M·,i = M·,j for somei = j then det(M) = 0.
21
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
• If I is the identity matrix then det(I) = 1.
Remark 1.2. Defining the multilinear and alternating properties from therows of the matrix will give the same determinant. The name of the alter-nating property comes from the fact that it combined with multilinearityimplies that switching places between two columns changes the sign of thedeterminant.
This definition of the determinant is quite abstract but it is sufficient todefine a unique function.
Theorem 1.1 (Leibniz formula for determinants). A standard result fromlinear algebra says that the determinant is unique and that it is given by thefollowing formula
det(M) =∑σ∈Sn
(−1)I(σ)n∏
i=1
mi,σ(i) (2)
where Sn is the set of all permutations of the set 1, 2, . . . , n, that is all liststhat contain the numbers 1, 2, . . . , n exactly once, and if σ is a permutationthen σ(i) is the ith element of that permutation.
Remark 1.3. Often formula (2) is used immediately as the definition ofthe determinant of a matrix, see for instance [5]. The formula is usuallyattributed to Gottfried Wilhem Leibniz (1646–1716), probably due to aletter that he wrote to Guillaume de l’Hopital (1661–1704) in 1693 where hedescribes a method of solving linear equation systems that is closely relatedto Cramer’s rule [123], the particular letter was published in [100] and atranslation can be found in [153].
The determinant has several uses and interpretation, the two most well-known ones are
• If det(M) = 0 then the vectors corresponding to the columns (orrows) are linearly independent. Compare this to the properties of theWronskian matrix described on page 28.
• If the columns (or rows) of M are interpreted as sides defining ann-dimensional parallelepiped the absolute value of det(M) will give thevolume of this parallelepiped. Compare this to the interpretation ofD-optimality on page 43. The sign of the determinant is also importantwhen considering the orientation of the surface which is highly relevantin geometric algebra and integration over several variables, see forinstance [112,146] for examples.
22
1.1. THE VANDERMONDE MATRIX
We will now discuss the Vandermonde determinant specifically.
Theorem 1.2. The Vandermonde determinant, vn(x1, . . . , xn), is given by
vn(x1, . . . , xn) = det(Vn(x1, . . . , xn)) =∏
1≤i<j≤n
(xj − xi).
A simple way to phrase Theorem 1.2 is that the Vandermonde determi-nant is the product of all differences of the values that define the elements(note that this does not give the sign of the determinant).
There are several proofs of Theorem 1.2. Many are based on a com-bination of using elementary row or column operations together with in-duction [95, 117] but there are also several proofs using combinatorial [11]or graph-based techniques [60, 141]. Here we will present a simple proof,that dates back to some very early results on determinants [24] and has aninteresting connection to the general concept of a determinant.
Proof of Theorem 1.2. There are many versions of this proof, see for exam-ple [10,21,24,71], with focus on different aspects of the proof. Here we willprovide a fairly concise version that still makes all the steps of the proofclear. We start by only considering one of the variables xk, which gives asingle variable function vn(xk). From the general expression for the deter-minant (Theorem 1.1) it is clear that vn(xk) must be a polynomial of degreen in xk. We also know that if we let xk = xi for any 1 ≤ i ≤ n, i = k, thedeterminant will be equal to zero since the corresponding matrix will havetwo identical columns. Thus if vn(xi) = 0 we can write
vn(xk) = P (xk)
n∏i=1i =k
(xk − xi)
where P (xk) is a polynomial. If we repeat this argument for all the variables,and ensure that no roots appear twice in the factorization, we get
vn(x1, . . . , xn) = Pn(x1, . . . , xn)n−1∏i=1
(xn − xi)
= Pn−1(x1, . . . , xn)
n−2∏i=1
(xn−1 − xi)
n−1∏i=1
(xn − xi)
= P0(x1, . . . , xn)(x2 − x1)(x3 − x2)(x3 − x1) · · ·n−1∏i=1
(xn − xi)
23
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
• If I is the identity matrix then det(I) = 1.
Remark 1.2. Defining the multilinear and alternating properties from therows of the matrix will give the same determinant. The name of the alter-nating property comes from the fact that it combined with multilinearityimplies that switching places between two columns changes the sign of thedeterminant.
This definition of the determinant is quite abstract but it is sufficient todefine a unique function.
Theorem 1.1 (Leibniz formula for determinants). A standard result fromlinear algebra says that the determinant is unique and that it is given by thefollowing formula
det(M) =∑σ∈Sn
(−1)I(σ)n∏
i=1
mi,σ(i) (2)
where Sn is the set of all permutations of the set 1, 2, . . . , n, that is all liststhat contain the numbers 1, 2, . . . , n exactly once, and if σ is a permutationthen σ(i) is the ith element of that permutation.
Remark 1.3. Often formula (2) is used immediately as the definition ofthe determinant of a matrix, see for instance [5]. The formula is usuallyattributed to Gottfried Wilhem Leibniz (1646–1716), probably due to aletter that he wrote to Guillaume de l’Hopital (1661–1704) in 1693 where hedescribes a method of solving linear equation systems that is closely relatedto Cramer’s rule [123], the particular letter was published in [100] and atranslation can be found in [153].
The determinant has several uses and interpretation, the two most well-known ones are
• If det(M) = 0 then the vectors corresponding to the columns (orrows) are linearly independent. Compare this to the properties of theWronskian matrix described on page 28.
• If the columns (or rows) of M are interpreted as sides defining ann-dimensional parallelepiped the absolute value of det(M) will give thevolume of this parallelepiped. Compare this to the interpretation ofD-optimality on page 43. The sign of the determinant is also importantwhen considering the orientation of the surface which is highly relevantin geometric algebra and integration over several variables, see forinstance [112,146] for examples.
22
1.1. THE VANDERMONDE MATRIX
We will now discuss the Vandermonde determinant specifically.
Theorem 1.2. The Vandermonde determinant, vn(x1, . . . , xn), is given by
vn(x1, . . . , xn) = det(Vn(x1, . . . , xn)) =∏
1≤i<j≤n
(xj − xi).
A simple way to phrase Theorem 1.2 is that the Vandermonde determi-nant is the product of all differences of the values that define the elements(note that this does not give the sign of the determinant).
There are several proofs of Theorem 1.2. Many are based on a com-bination of using elementary row or column operations together with in-duction [95, 117] but there are also several proofs using combinatorial [11]or graph-based techniques [60, 141]. Here we will present a simple proof,that dates back to some very early results on determinants [24] and has aninteresting connection to the general concept of a determinant.
Proof of Theorem 1.2. There are many versions of this proof, see for exam-ple [10,21,24,71], with focus on different aspects of the proof. Here we willprovide a fairly concise version that still makes all the steps of the proofclear. We start by only considering one of the variables xk, which gives asingle variable function vn(xk). From the general expression for the deter-minant (Theorem 1.1) it is clear that vn(xk) must be a polynomial of degreen in xk. We also know that if we let xk = xi for any 1 ≤ i ≤ n, i = k, thedeterminant will be equal to zero since the corresponding matrix will havetwo identical columns. Thus if vn(xi) = 0 we can write
vn(xk) = P (xk)
n∏i=1i =k
(xk − xi)
where P (xk) is a polynomial. If we repeat this argument for all the variables,and ensure that no roots appear twice in the factorization, we get
vn(x1, . . . , xn) = Pn(x1, . . . , xn)n−1∏i=1
(xn − xi)
= Pn−1(x1, . . . , xn)
n−2∏i=1
(xn−1 − xi)
n−1∏i=1
(xn − xi)
= P0(x1, . . . , xn)(x2 − x1)(x3 − x2)(x3 − x1) · · ·n−1∏i=1
(xn − xi)
23
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and since this factorization has each xk appear as a root n times we canconclude that
vn(x1, . . . , xn) = C det(Vn(x1, . . . , xn)) =∏
1≤i<j≤n
(xj − xi)
where C is a constant. From Leibniz formula for the determinant, equation(2), we can see that the coefficient in front of any term of the form xnk mustbe 1, thus C = 1, which concludes the proof.
Theorem 1.3. There is a relationship between the exponents of the expandedVandermonde determinant and the indices in the expression for a generaldeterminant, more specifically
(n∏
i=1
xi
)vn(x1, . . . , xn) =
(n∏
i=1
xi
) ∏1≤i<j≤n
(xj − xi)
=∑σ∈Sn
(−1)I(σ)n∏
i=1
xiσ(i). (3)
Clearly replacing xji with xi,j in equation (3) gives equation (2).
Proof. We will prove this theorem by showing that replacing exponents withindices will give a function that by Definition 1.2 is a determinant. InDefinition 1.2 we interpreted the determinant as a function of the columns ofthe matrix, for the Vandermonde determinant this corresponds to a functionof the xi since they define the columns. Here we will interpret each part ofDefinition 1.2 as a statement about the xi and then show how it is impliedby the Vandermonde determinant.
• Alternating: The alternating property is easy to interpret in termsof the xi since if xi = xj for some i = j then we have two identicalcolumns. Consider the product form of the Vandermonde determinantgiven in Theorem 1.2. Switching places between xi and xj with i < j inthe Vandermonde determinant is equal to switching sign in all factorsthat contain either xi or xj as well as xk with i ≤ k ≤ j. There willbe j − i− 1 factors that contain xi and satisfy i < k ≤ j and j − i− 1factors that contain xj satisfy i ≤ k < j and one factor (xi−xj). Thismeans that in total we will change sign in 2(j − i) − 1 factors whichmeans the sign of the whole product will change.
24
1.1. THE VANDERMONDE MATRIX
• Multilinearity: If we denote the left hand side in (3) with w
w =
(n∏
k=1
xk
)vn(x1, . . . , xn)
then multiplying the kth column by a scalar can be interpreted asfollows
M·,k → aM·,k ⇔ w =n∑
i=1
xikci →n∑
i=1
axikci
and addition of columns as
M·,k → M·,k +N·,k ⇔ w =
n∑i=1
xikci →n∑
i=1
(xik + yik)ci
and multilinearity follows immediately from this.
• det(I) = 1: For the identity matrix we have
xi,j =
1 i = j
0 i = j
which for the expanded Vandermonde determinant corresponds to thetransformation
xji →
1 i = j
0 i = j
when expanding the Vandermonde determinant we get
vn(x1, . . . , xn) = vn−1(x1, . . . , xn−1)
n−1∏i=1
(xn − xi)
= xn−1n vn−1(x1, . . . , xn−1) + P (n)
= xn−1n vn−2(x1, . . . , xn−2)
n−2∏i=1
(xn−1 − xi) + P (n)
= xn−1n xn−2
n−1vn−2(x1, . . . , xn−2) + P (n, n− 1)
=n∏
k=1
xk−1k + P (n, n− 1, . . . , 1)
25
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and since this factorization has each xk appear as a root n times we canconclude that
vn(x1, . . . , xn) = C det(Vn(x1, . . . , xn)) =∏
1≤i<j≤n
(xj − xi)
where C is a constant. From Leibniz formula for the determinant, equation(2), we can see that the coefficient in front of any term of the form xnk mustbe 1, thus C = 1, which concludes the proof.
Theorem 1.3. There is a relationship between the exponents of the expandedVandermonde determinant and the indices in the expression for a generaldeterminant, more specifically
(n∏
i=1
xi
)vn(x1, . . . , xn) =
(n∏
i=1
xi
) ∏1≤i<j≤n
(xj − xi)
=∑σ∈Sn
(−1)I(σ)n∏
i=1
xiσ(i). (3)
Clearly replacing xji with xi,j in equation (3) gives equation (2).
Proof. We will prove this theorem by showing that replacing exponents withindices will give a function that by Definition 1.2 is a determinant. InDefinition 1.2 we interpreted the determinant as a function of the columns ofthe matrix, for the Vandermonde determinant this corresponds to a functionof the xi since they define the columns. Here we will interpret each part ofDefinition 1.2 as a statement about the xi and then show how it is impliedby the Vandermonde determinant.
• Alternating: The alternating property is easy to interpret in termsof the xi since if xi = xj for some i = j then we have two identicalcolumns. Consider the product form of the Vandermonde determinantgiven in Theorem 1.2. Switching places between xi and xj with i < j inthe Vandermonde determinant is equal to switching sign in all factorsthat contain either xi or xj as well as xk with i ≤ k ≤ j. There willbe j − i− 1 factors that contain xi and satisfy i < k ≤ j and j − i− 1factors that contain xj satisfy i ≤ k < j and one factor (xi−xj). Thismeans that in total we will change sign in 2(j − i) − 1 factors whichmeans the sign of the whole product will change.
24
1.1. THE VANDERMONDE MATRIX
• Multilinearity: If we denote the left hand side in (3) with w
w =
(n∏
k=1
xk
)vn(x1, . . . , xn)
then multiplying the kth column by a scalar can be interpreted asfollows
M·,k → aM·,k ⇔ w =n∑
i=1
xikci →n∑
i=1
axikci
and addition of columns as
M·,k → M·,k +N·,k ⇔ w =
n∑i=1
xikci →n∑
i=1
(xik + yik)ci
and multilinearity follows immediately from this.
• det(I) = 1: For the identity matrix we have
xi,j =
1 i = j
0 i = j
which for the expanded Vandermonde determinant corresponds to thetransformation
xji →
1 i = j
0 i = j
when expanding the Vandermonde determinant we get
vn(x1, . . . , xn) = vn−1(x1, . . . , xn−1)
n−1∏i=1
(xn − xi)
= xn−1n vn−1(x1, . . . , xn−1) + P (n)
= xn−1n vn−2(x1, . . . , xn−2)
n−2∏i=1
(xn−1 − xi) + P (n)
= xn−1n xn−2
n−1vn−2(x1, . . . , xn−2) + P (n, n− 1)
=n∏
k=1
xk−1k + P (n, n− 1, . . . , 1)
25
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
where P (I), I ⊂ Z>0, does not contain any terms of the form xk−1k
for all k ∈ I. Thus applying the transformation corresponding to theidentity matrix we get
(n∏
i=1
xi
)vn(x1, . . . , xn) =
n∏k=1
xkk + P (n, . . . , 1) →n∏
k=1
1 + 0 = 1.
Thus if we take the right hand side in equation (3) and exchange exponentsfor indices we get a determinant be Definition 1.2 and since the determinantis unique by Theorem 1.1 and xi,j = xji in the Vandermonde matrix thismust be equal to
(n∏
i=1
xi
)vn(x1, . . . , xn) =
∑σ∈Sn
(−1)I(σ)n∏
i=1
xiσ(i).
1.1.3 Inverse of the Vandermonde matrix
The inverse for the Vandermonde matrix has been known for a long time, es-pecially since the solution to a Lagrange interpolation problems (see Section1.2.1) gives the inverse indirectly. Here we will only give a short overviewof the work on expressing the inverse as an explicit matrix.
An explicit expression for the inverse matrix has been known since atleast the end of the 1950s, see [113].
Theorem 1.4. The elements of the inverse of an n-dimensional Vander-monde matrix V can be calculated by
(V−1
n
)ij=
(−1)j−1σn−j,in∏
k=1k =i
(xk − xi)
(4)
where σj,i is the j:th elementary symmetric polynomial with variable xi setto zero.
σj,i =∑
1≤m1<m2<... <mj≤n
j∏k=1
xmk(1− δmk,i) , δa,b =
1 , a = b0 , a = b
(5)
We will not give the proof of this theorem here, but the general outlineof a proof will be given in Section 1.2.1.
26
1.1. THE VANDERMONDE MATRIX
In the literature there are many cases where the inverse is instead writtenas a product of several simpler matrices, usually triangular or diagonal [121,129,130,162]. There is also a lot of literature that takes a more algorithmicapproach and tries to find fast ways of computing the elements, classicalexamples include the Parker-Traub algorithm [160] and the Bjorck-Pereyraalgorithm [13], and more recent results can be found in [40].
1.1.4 The alternant matrix
Many generalizations of the Vandermonde matrix have been proposed andstudied in the literature. An early generalization is the alternant matrixwhich is a matrix that exchanges the powers in the Vandermonde matrixwith other functions [124].
Definition 1.3. An alternant matrix is a matrix of the form
Amn(fm;xn) = [fi(xj)]m,ni,j =
f1(x1) f1(x2) · · · f1(xn)f2(x1) f2(x2) · · · f2(xn)
......
. . ....
fm(x1) fm(x2) · · · fm(xn)
(6)
where fi : F → F where F is a field.
Remark 1.4. In some literature the alternant matrix is used as an al-ternative name for the Vandermonde matrix or the Vandermonde matrixmultiplied by a diagonal matrix [161].
There are several special cases of alternant matrices that are useful orinteresting in various mathematical fields:
Interpolation and regression
Just like the Vandermonde matrix can be used for polynomial interpolationthe alternant matrix can be used to describe interpolation with other setsof function, see Section 1.2 and 1.3.
Alternant codes
As mentioned on page 19 there are several different error-detecting anderror-correcting codes that can be described using the Vandermonde matrix.These and some related codes can also be categorized as alternant codes, aterm introduced in [68]. For a survey on these codes see [176].
27
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
where P (I), I ⊂ Z>0, does not contain any terms of the form xk−1k
for all k ∈ I. Thus applying the transformation corresponding to theidentity matrix we get
(n∏
i=1
xi
)vn(x1, . . . , xn) =
n∏k=1
xkk + P (n, . . . , 1) →n∏
k=1
1 + 0 = 1.
Thus if we take the right hand side in equation (3) and exchange exponentsfor indices we get a determinant be Definition 1.2 and since the determinantis unique by Theorem 1.1 and xi,j = xji in the Vandermonde matrix thismust be equal to
(n∏
i=1
xi
)vn(x1, . . . , xn) =
∑σ∈Sn
(−1)I(σ)n∏
i=1
xiσ(i).
1.1.3 Inverse of the Vandermonde matrix
The inverse for the Vandermonde matrix has been known for a long time, es-pecially since the solution to a Lagrange interpolation problems (see Section1.2.1) gives the inverse indirectly. Here we will only give a short overviewof the work on expressing the inverse as an explicit matrix.
An explicit expression for the inverse matrix has been known since atleast the end of the 1950s, see [113].
Theorem 1.4. The elements of the inverse of an n-dimensional Vander-monde matrix V can be calculated by
(V−1
n
)ij=
(−1)j−1σn−j,in∏
k=1k =i
(xk − xi)
(4)
where σj,i is the j:th elementary symmetric polynomial with variable xi setto zero.
σj,i =∑
1≤m1<m2<... <mj≤n
j∏k=1
xmk(1− δmk,i) , δa,b =
1 , a = b0 , a = b
(5)
We will not give the proof of this theorem here, but the general outlineof a proof will be given in Section 1.2.1.
26
1.1. THE VANDERMONDE MATRIX
In the literature there are many cases where the inverse is instead writtenas a product of several simpler matrices, usually triangular or diagonal [121,129,130,162]. There is also a lot of literature that takes a more algorithmicapproach and tries to find fast ways of computing the elements, classicalexamples include the Parker-Traub algorithm [160] and the Bjorck-Pereyraalgorithm [13], and more recent results can be found in [40].
1.1.4 The alternant matrix
Many generalizations of the Vandermonde matrix have been proposed andstudied in the literature. An early generalization is the alternant matrixwhich is a matrix that exchanges the powers in the Vandermonde matrixwith other functions [124].
Definition 1.3. An alternant matrix is a matrix of the form
Amn(fm;xn) = [fi(xj)]m,ni,j =
f1(x1) f1(x2) · · · f1(xn)f2(x1) f2(x2) · · · f2(xn)
......
. . ....
fm(x1) fm(x2) · · · fm(xn)
(6)
where fi : F → F where F is a field.
Remark 1.4. In some literature the alternant matrix is used as an al-ternative name for the Vandermonde matrix or the Vandermonde matrixmultiplied by a diagonal matrix [161].
There are several special cases of alternant matrices that are useful orinteresting in various mathematical fields:
Interpolation and regression
Just like the Vandermonde matrix can be used for polynomial interpolationthe alternant matrix can be used to describe interpolation with other setsof function, see Section 1.2 and 1.3.
Alternant codes
As mentioned on page 19 there are several different error-detecting anderror-correcting codes that can be described using the Vandermonde matrix.These and some related codes can also be categorized as alternant codes, aterm introduced in [68]. For a survey on these codes see [176].
27
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Jacobian matrix
One of the most well-known examples of an alternant matrix is the Jaco-bian matrix. Let f : Fn → Fn be a vector-valued function that is n timesdifferentiable with respect to each variable, then the Jacobian matrix is thematrix J, given by
∂y1∂x1
∂y2∂x1
· · · ∂yn∂x1
∂y1∂x2
∂y2∂x2
· · · ∂yn∂x2
......
. . ....
∂y1∂xn
∂y2∂xn
· · · ∂yn∂xn
where y = f(x). The most common application of the Jacobian matrix isto use its determinant to describe how volume elements are deformed whenchanging variables in multivariate calculus [146]. The numerous applicationsand generalizations that follow from this alone are too numerous to listso here we will only note that it holds a central role in many methodsfor multivariate optimizations, such as the Marquardt least-square methoddescribed in Section 1.3.3.
Wronskian matrix
If fi =di−1
dxi−1 and gi ∈ Cn−1[C], i = 1, . . . , n the alternant matrix An(fn;gn)will be the Wronskian matrix . The Wronskian matrix has a long history [70]and is commonly used to test if a set of functions gi are linearly indepen-dent as well as finding solutions to ordinary differential equations [52]. Ifthe determinant of the Wronskian matrix is non-zero then the functions arelinearly independent, see [16, 19], but proving linear dependence requiresfurther conditions, see [14, 15,133,134,174].
A classical application of the Wronskian is confirming that a set of so-lutions to a linear differential equation are linearly independent or if n − 1linearly independent solutions are know construct the remaining linearly in-dependent solution using Abel’s identity (for n = 2) or a generalisation ofit [20].
If Li is a linear partial differential operator of order i then the alter-nant matrix An(Ln;gn), where Ln = (L1, . . . , Ln), is the generalized Wron-skian matrix [131], that has been used in for example diophantine geome-try [39, 145] and for solving Korteweg-de Vries equations, see [111] and thereferences therein. The generalized Wronskian matrix has similar propertieswith respect to the linear dependence of the functions it is created from asthe standard Wronskian [175].
28
1.1. THE VANDERMONDE MATRIX
Both the Wronskian and generalized Wronskian is also useful in algebraicgeometry, see [52] for several examples.
Bell matrix
Alternating matrices can be used to convert function composition into ma-trix multiplication. By letting Di =
di−1
dxi−1 and gj(x) = (f(x))j , where f isinfinitely differentiable, the alternant matrix B[f ] = An(Dn,gn) is calleda Bell matrix (its transpose is known as the Carleman matrix ). Some au-thors, for instance [94], refer to Bell matrices as Jabotinsky matrices due toa special case of Bell matrices considered in [77].
That Bell matrices converts function composition into matrix multiplica-tion can be seen by noting that the power series expansion of the jth powerof f can be written as
(f(x))j =∞∑i=1
B[f ]ijxi
and from this equality follows that B[f g] = B[g]B[f ]. This is the basicproperty behind a popular technique called Carleman linearisation or Carle-man embedding that has seen wide use in the theory of non-linear dynamicalsystems. The literature on the subject is vast but a systematic introductionis offered in [96].
Moore matrix
When working in a finite field with prime characteristic p an analogue ofthe Vandermonde and Wronskian matrix can be constructed by taking analternant matrix where the rows are given by power of the Frobenius au-tomorphism, F (ω) = ωp. This matrix is called the Moore matrix and isnamed after its originator E. H. Moore who also calculated its determinant,
∣∣∣∣∣∣∣∣∣
ω1 · · · ωn
ωp1 · · · ωp
n...
. . ....
ωpn−1
1 · · · ωpn−1
n
∣∣∣∣∣∣∣∣∣=
n−1∏i=1
p−1∏ki−1
· · ·p−1∏k1=0
(ωi + ki−1ωi−1 + . . .+ k1ω1)(mod p),
and showed that if this determinant was not equal to zero then ω1, . . . , ωn
are linearly independent [118]. There are several uses for the determinant ofthe Moore matrix in function field arithmetic, see for instance [62], a classicalexample is finding the modular invariants of the general linear group over
29
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Jacobian matrix
One of the most well-known examples of an alternant matrix is the Jaco-bian matrix. Let f : Fn → Fn be a vector-valued function that is n timesdifferentiable with respect to each variable, then the Jacobian matrix is thematrix J, given by
∂y1∂x1
∂y2∂x1
· · · ∂yn∂x1
∂y1∂x2
∂y2∂x2
· · · ∂yn∂x2
......
. . ....
∂y1∂xn
∂y2∂xn
· · · ∂yn∂xn
where y = f(x). The most common application of the Jacobian matrix isto use its determinant to describe how volume elements are deformed whenchanging variables in multivariate calculus [146]. The numerous applicationsand generalizations that follow from this alone are too numerous to listso here we will only note that it holds a central role in many methodsfor multivariate optimizations, such as the Marquardt least-square methoddescribed in Section 1.3.3.
Wronskian matrix
If fi =di−1
dxi−1 and gi ∈ Cn−1[C], i = 1, . . . , n the alternant matrix An(fn;gn)will be the Wronskian matrix . The Wronskian matrix has a long history [70]and is commonly used to test if a set of functions gi are linearly indepen-dent as well as finding solutions to ordinary differential equations [52]. Ifthe determinant of the Wronskian matrix is non-zero then the functions arelinearly independent, see [16, 19], but proving linear dependence requiresfurther conditions, see [14, 15,133,134,174].
A classical application of the Wronskian is confirming that a set of so-lutions to a linear differential equation are linearly independent or if n − 1linearly independent solutions are know construct the remaining linearly in-dependent solution using Abel’s identity (for n = 2) or a generalisation ofit [20].
If Li is a linear partial differential operator of order i then the alter-nant matrix An(Ln;gn), where Ln = (L1, . . . , Ln), is the generalized Wron-skian matrix [131], that has been used in for example diophantine geome-try [39, 145] and for solving Korteweg-de Vries equations, see [111] and thereferences therein. The generalized Wronskian matrix has similar propertieswith respect to the linear dependence of the functions it is created from asthe standard Wronskian [175].
28
1.1. THE VANDERMONDE MATRIX
Both the Wronskian and generalized Wronskian is also useful in algebraicgeometry, see [52] for several examples.
Bell matrix
Alternating matrices can be used to convert function composition into ma-trix multiplication. By letting Di =
di−1
dxi−1 and gj(x) = (f(x))j , where f isinfinitely differentiable, the alternant matrix B[f ] = An(Dn,gn) is calleda Bell matrix (its transpose is known as the Carleman matrix ). Some au-thors, for instance [94], refer to Bell matrices as Jabotinsky matrices due toa special case of Bell matrices considered in [77].
That Bell matrices converts function composition into matrix multiplica-tion can be seen by noting that the power series expansion of the jth powerof f can be written as
(f(x))j =∞∑i=1
B[f ]ijxi
and from this equality follows that B[f g] = B[g]B[f ]. This is the basicproperty behind a popular technique called Carleman linearisation or Carle-man embedding that has seen wide use in the theory of non-linear dynamicalsystems. The literature on the subject is vast but a systematic introductionis offered in [96].
Moore matrix
When working in a finite field with prime characteristic p an analogue ofthe Vandermonde and Wronskian matrix can be constructed by taking analternant matrix where the rows are given by power of the Frobenius au-tomorphism, F (ω) = ωp. This matrix is called the Moore matrix and isnamed after its originator E. H. Moore who also calculated its determinant,
∣∣∣∣∣∣∣∣∣
ω1 · · · ωn
ωp1 · · · ωp
n...
. . ....
ωpn−1
1 · · · ωpn−1
n
∣∣∣∣∣∣∣∣∣=
n−1∏i=1
p−1∏ki−1
· · ·p−1∏k1=0
(ωi + ki−1ωi−1 + . . .+ k1ω1)(mod p),
and showed that if this determinant was not equal to zero then ω1, . . . , ωn
are linearly independent [118]. There are several uses for the determinant ofthe Moore matrix in function field arithmetic, see for instance [62], a classicalexample is finding the modular invariants of the general linear group over
29
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
a finite field [37, 128]. The determinant also plays an important role in thetheory of Drinfeld modules [125].
1.1.5 The generalized Vandermonde matrix
There are several types of matrices (or determinants) that have been re-ferred to as generalized Vandermonde matrices, for example the confluentVandermonde matrix is sometimes referred to as the generalized Vander-monde matrix [85,86,102,109,155], this matrix and its role in interpolationproblems is briefly described on page 36. Other examples include modifiedversions of confluent Vandermonde matrices [45], as well as matrices withelements given by multivariate monomials of increasing multidegree [23], orsimilarly over the algebraic closure of a field [31], matrices with elementsgiven by multivariate polynomials with univariate terms [168].
In this thesis we call the alternant matrix Amn(xα1 , . . . , xαn ;x1, . . . , xn)
the generalized Vandermonde matrix.
Definition 1.4. A generalized Vandermonde matrix is an n×m matrix ofthe form
Gmn(xn) =[xαij
]m,n
i,j=
xα11 xα1
2 · · · xα1n
xα21 xα2
2 · · · xα2n
......
. . ....
xαm1 xαm
2 · · · xαmn
(7)
where xi ∈ C, αi ∈ C, i = 1, . . . , n.
This name has been used for quite some time, see [67] for instance.The main reason to study this matrix seems to be its connection to Schurpolynomials, see below, and thus the research on the matrix is primarilyfocused on its determinant. Many of the results are algorithmic in nature[28,33–35,92] but there are also more algebraic examinations [41,49,149,177].
There are a couple of examples where the determinant of generalizedVandermonde matrices are interesting or useful:
Schur polynomials
If λ = (λ1, . . . , λn) is an integer partition, that is 0 < λ1 ≤ λ2 ≤ . . . λn andeach λi ∈ N, then
a(λ1+n−1,λ2+n−2,...,λn)(x1, . . . , xn)
= det(Gn(λ1 + n− 1, λ2 + n− 2, . . . , λn;x1, . . . , xn))
30
1.1. THE VANDERMONDE MATRIX
are the Schur functions that were introduced by Cauchy [24] but namedafter Issai Schur (1875 – 1941) that showed that they were highly useful ininvariant theory and representation theory. For instance they can be usedto determine the character of conjugacy classes of representations of thesymmetric group [50]. They have also been used in other areas, for instanceto describe the generating function of many classes of plane partitions, seefor instance [21] for several examples. The literature on Schur polynomialsis vast and so are the applications so there will be no attempt to summarisethem here.
Integration of an exponential function over a unitary group
If we let U(n) be the n-dimensional unitary group and dU a Haar mea-sure normalised to 1 then the Harish-Chandra–Itzykson-Zuber integral for-mula [64,76], says that if A and B are Hermitian matrices with eigenvaluesλ1(A) ≤ . . . ≤ λn(A) and λ1(B) ≤ . . . ≤ λn(B) then
∫
U(n)et tr(AUBU∗) dU =
det([exp(tλj(A)λk(B))]nnj,k
)
tn(n−1)
2 vn(λ(A))vn(λ(B))
n−1∏i=1
i! (8)
where vn is the determinant of the Vandermonde matrix. If t = 1 and Aand B are chosen as diagonal matrices
Aij =
ai if i = j,
0 if i = j,Bij =
bi if i = j,
0 if i = j,
then formula (8) reduces to an expression involving determinants of a gen-eralized Vandermonde matrix and two Vandermonde matrices,
∫
U(n)etr(AUBU∗) dU =
∣∣∣∣∣∣∣∣∣
ea1b1 ea1b2 . . . ea1bn
ea2b1 ea2b2 . . . ea2bn
......
. . ....
eanb1 eanb2 . . . eanbn
∣∣∣∣∣∣∣∣∣vn(a1, . . . , an)vn(b1, . . . , bn)
.
31
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
a finite field [37, 128]. The determinant also plays an important role in thetheory of Drinfeld modules [125].
1.1.5 The generalized Vandermonde matrix
There are several types of matrices (or determinants) that have been re-ferred to as generalized Vandermonde matrices, for example the confluentVandermonde matrix is sometimes referred to as the generalized Vander-monde matrix [85,86,102,109,155], this matrix and its role in interpolationproblems is briefly described on page 36. Other examples include modifiedversions of confluent Vandermonde matrices [45], as well as matrices withelements given by multivariate monomials of increasing multidegree [23], orsimilarly over the algebraic closure of a field [31], matrices with elementsgiven by multivariate polynomials with univariate terms [168].
In this thesis we call the alternant matrix Amn(xα1 , . . . , xαn ;x1, . . . , xn)
the generalized Vandermonde matrix.
Definition 1.4. A generalized Vandermonde matrix is an n×m matrix ofthe form
Gmn(xn) =[xαij
]m,n
i,j=
xα11 xα1
2 · · · xα1n
xα21 xα2
2 · · · xα2n
......
. . ....
xαm1 xαm
2 · · · xαmn
(7)
where xi ∈ C, αi ∈ C, i = 1, . . . , n.
This name has been used for quite some time, see [67] for instance.The main reason to study this matrix seems to be its connection to Schurpolynomials, see below, and thus the research on the matrix is primarilyfocused on its determinant. Many of the results are algorithmic in nature[28,33–35,92] but there are also more algebraic examinations [41,49,149,177].
There are a couple of examples where the determinant of generalizedVandermonde matrices are interesting or useful:
Schur polynomials
If λ = (λ1, . . . , λn) is an integer partition, that is 0 < λ1 ≤ λ2 ≤ . . . λn andeach λi ∈ N, then
a(λ1+n−1,λ2+n−2,...,λn)(x1, . . . , xn)
= det(Gn(λ1 + n− 1, λ2 + n− 2, . . . , λn;x1, . . . , xn))
30
1.1. THE VANDERMONDE MATRIX
are the Schur functions that were introduced by Cauchy [24] but namedafter Issai Schur (1875 – 1941) that showed that they were highly useful ininvariant theory and representation theory. For instance they can be usedto determine the character of conjugacy classes of representations of thesymmetric group [50]. They have also been used in other areas, for instanceto describe the generating function of many classes of plane partitions, seefor instance [21] for several examples. The literature on Schur polynomialsis vast and so are the applications so there will be no attempt to summarisethem here.
Integration of an exponential function over a unitary group
If we let U(n) be the n-dimensional unitary group and dU a Haar mea-sure normalised to 1 then the Harish-Chandra–Itzykson-Zuber integral for-mula [64,76], says that if A and B are Hermitian matrices with eigenvaluesλ1(A) ≤ . . . ≤ λn(A) and λ1(B) ≤ . . . ≤ λn(B) then
∫
U(n)et tr(AUBU∗) dU =
det([exp(tλj(A)λk(B))]nnj,k
)
tn(n−1)
2 vn(λ(A))vn(λ(B))
n−1∏i=1
i! (8)
where vn is the determinant of the Vandermonde matrix. If t = 1 and Aand B are chosen as diagonal matrices
Aij =
ai if i = j,
0 if i = j,Bij =
bi if i = j,
0 if i = j,
then formula (8) reduces to an expression involving determinants of a gen-eralized Vandermonde matrix and two Vandermonde matrices,
∫
U(n)etr(AUBU∗) dU =
∣∣∣∣∣∣∣∣∣
ea1b1 ea1b2 . . . ea1bn
ea2b1 ea2b2 . . . ea2bn
......
. . ....
eanb1 eanb2 . . . eanbn
∣∣∣∣∣∣∣∣∣vn(a1, . . . , an)vn(b1, . . . , bn)
.
31
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
1.2 Interpolation
The problem of finding a function that generates a given set of points isusually referred to as an interpolation problem and the function generatingthe points is called an interpolating function. A common type of inter-polation problem is to find a continuous function, f , such that the givenset of points (x1, y1), (x2, y2), . . . can be generated by calculating the set(x1, f(x1)), (x2, f(x2)), . . .. Often the interpolating function is also a lin-ear combination of elementary functions, but interpolation can also be donein other ways, for instance with fractals (the classical texts on this is [8, 9])or parametrised curves. For some examples, see Figure 1.3.
Figure 1.3: Some examples of different interpolating curves. The set of red pointsare interpolated by a polynomial (left), a self-affine fractal (middle) anda Lissajous curve (right).
In the case of the interpolating function being a linear combination ofother functions and the interpolation is achieved by changing the coefficientsof the linear combination this is said to be a linear model (not to be confusedwith linear interpolation that is interpolation with piecewise straight lines).
For linear models the interpolation problem can be described using al-ternant matrices. Suppose we want to find a function
f(x) =
m∑i=1
aigi(x) (9)
that fits as well as possible to the data points (xi, yi), i = 1, . . . , n. Wethen get an interpolation problem described by the linear equation systemXa = y where a are the coefficients of f , y are the data values and A is the
32
1.2. INTERPOLATION
appropriate alternant matrix,
A =
g1(x1) g2(x1) . . . gm(x1)g1(x2) g2(x2) . . . gm(x2)
......
. . ....
g1(xn) g2(xn) . . . gm(xn)
, a =
a1a2...an
, y =
y1y2...yn
.
1.2.1 Polynomial interpolation
A classical form of interpolation is polynomial interpolation where n datapoints are interpolated by a polynomial of at most degree n− 1.
The Vandermonde matrix can be used to describe this type of interpola-tion problem simply by rewriting the equation system given by p(xk) = ykas a matrix equation
1 x1 · · · xn−11
1 x2 · · · xn−12
......
. . ....
1 xn · · · xn−1n
a1a2...an
=
y1y2...yn
.
That the polynomial is unique (if it exists) is easy to see when consideringthe determinant of the Vandermonde matrix
det(Vn(x1, . . . , xn)) =∏
1≤i<j≤n
(xj − xi).
Clearly this determinant is non-zero whenever all xi are disctinct whichmeans that he matrix is invertible whenever all xi are distinct. If not all xiare distinct there is no function of the x coordinate that can interpolate allthe points.
There are several ways to construct the interpolating polynomial withoutexplicitly inverting the Vandermonde matrix. The most straight-forward isprobably Lagrange interpolation, named after Joseph-Louis Lagrange (1736– 1813) [98] who independently discovered it a few years after Edward War-ing (1736 – 1798) [172].
The idea behind Lagrange interpolation is simple, construct a set of npolynomials p1, p2, . . . , pn such that
pi(xj) =
0, i = j
1, i = j
33
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
1.2 Interpolation
The problem of finding a function that generates a given set of points isusually referred to as an interpolation problem and the function generatingthe points is called an interpolating function. A common type of inter-polation problem is to find a continuous function, f , such that the givenset of points (x1, y1), (x2, y2), . . . can be generated by calculating the set(x1, f(x1)), (x2, f(x2)), . . .. Often the interpolating function is also a lin-ear combination of elementary functions, but interpolation can also be donein other ways, for instance with fractals (the classical texts on this is [8, 9])or parametrised curves. For some examples, see Figure 1.3.
Figure 1.3: Some examples of different interpolating curves. The set of red pointsare interpolated by a polynomial (left), a self-affine fractal (middle) anda Lissajous curve (right).
In the case of the interpolating function being a linear combination ofother functions and the interpolation is achieved by changing the coefficientsof the linear combination this is said to be a linear model (not to be confusedwith linear interpolation that is interpolation with piecewise straight lines).
For linear models the interpolation problem can be described using al-ternant matrices. Suppose we want to find a function
f(x) =
m∑i=1
aigi(x) (9)
that fits as well as possible to the data points (xi, yi), i = 1, . . . , n. Wethen get an interpolation problem described by the linear equation systemXa = y where a are the coefficients of f , y are the data values and A is the
32
1.2. INTERPOLATION
appropriate alternant matrix,
A =
g1(x1) g2(x1) . . . gm(x1)g1(x2) g2(x2) . . . gm(x2)
......
. . ....
g1(xn) g2(xn) . . . gm(xn)
, a =
a1a2...an
, y =
y1y2...yn
.
1.2.1 Polynomial interpolation
A classical form of interpolation is polynomial interpolation where n datapoints are interpolated by a polynomial of at most degree n− 1.
The Vandermonde matrix can be used to describe this type of interpola-tion problem simply by rewriting the equation system given by p(xk) = ykas a matrix equation
1 x1 · · · xn−11
1 x2 · · · xn−12
......
. . ....
1 xn · · · xn−1n
a1a2...an
=
y1y2...yn
.
That the polynomial is unique (if it exists) is easy to see when consideringthe determinant of the Vandermonde matrix
det(Vn(x1, . . . , xn)) =∏
1≤i<j≤n
(xj − xi).
Clearly this determinant is non-zero whenever all xi are disctinct whichmeans that he matrix is invertible whenever all xi are distinct. If not all xiare distinct there is no function of the x coordinate that can interpolate allthe points.
There are several ways to construct the interpolating polynomial withoutexplicitly inverting the Vandermonde matrix. The most straight-forward isprobably Lagrange interpolation, named after Joseph-Louis Lagrange (1736– 1813) [98] who independently discovered it a few years after Edward War-ing (1736 – 1798) [172].
The idea behind Lagrange interpolation is simple, construct a set of npolynomials p1, p2, . . . , pn such that
pi(xj) =
0, i = j
1, i = j
33
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and then construct the final interpolating polynomial by the sum of thesepi weighted by the corresponding yi.
The pi polynomials are called Lagrange basis polynomials and can easilybe constructed by placing the roots appropriately and then normalizing theresult such that pi(xi) = 1, this gives the expression
pi(x) =(x− x1) · · · (x− xi−1)(x− xi+1) · · · (x− xn)
(xi − x1) · · · (xi − xk−1)(xi − xi+1) · · · (xi − xn).
0 1 2 3 4 5 6 7 8
−2
0
2
p1(x) p2(x) p3(x) p4(x) p(x) (x, y)
Figure 1.4: Illustration of Lagrange interpolation of 4 data points. The red dots are
the data set and p(x) =
4∑k=1
ykp(xk) is the interpolating polynomial.
The explicit formula for the full interpolating polynomial is
p(x) =
n∑k=1
yk(x− x1) · · · (x− xk−1)(x− xk+1) · · · (x− xn)
(xk − x1) · · · (xk − xk−1)(xk − xk+1) · · · (xk − xn)
and from this formula the expression for the inverse of the Vandermondematrix can be found by noting that the jth row of the inverse will consistof the coefficients of pj , the resulting expression for the elements is given inTheorem 1.4.
Polynomial interpolation is mostly used when the data set we wish tointerpolate is small. The main reason for this is the instability of the inter-polation method. One example of this is Runge’s phenomenon that showsthat when certain functions are approximated by polynomial interpolation
34
1.2. INTERPOLATION
−40−20 0 20 400
0.5
1
1.5
−40−20 0 20 400
0.5
1
1.5
−40−20 0 20 400
0.5
1
1.5
Figure 1.5: Illustration of Runge’s phenomenon. Here we attempt to approximate afunction (dashed line) by polynomial interpolation (solid line). With 7equidistant sample points (left figure) the approximation is poor near theedges of the interval and increasing the number of sample points to 14(center) and 19 (right) clearly reduces accuracy at the edges further.
fitted to equidistantly sampled points will sometimes lose precision when thenumber of interpolating points is increased, see Figure 1.5 for an example.
One way to predict this instability of polynomial interpolation is thatthe conditional number of the Vandermonde matrix can be very large forequidistant points [59].
There are different ways to mitigate the issue of stability, for examplechoosing data points that minimize the conditional number of the relevantmatrix [57, 59] or by choosing a polynomial basis that is more stable forthe given set of data points such as Bernstein polynomials in the case ofequidistant points [126]. Other polynomial schemes can also be considered,for instance by interpolating with different basis functions in different inter-vals, for example using polynomial splines.
Naturally another choice is to not choose polynomials as basis functionsbut instead choose some other functions that are more suitable. For anexample of this see Section 3.3.
While the instability of polynomial interpolation does not prevent it frombeing useful for analytical examinations it is generally considered imprac-tical when there is noise present or when calculations are performed withlimited precision. Often interpolating polynomials are not constructed byinverting the Vandermonde matrix or calculating the Lagrange basis poly-nomials, instead a more computationally efficient method such as Newtoninterpolation or Neville’s algorithm are used [138]. There are some variantsof Lagrange interpolation, such as barycentric Lagrange interpolation, that
35
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and then construct the final interpolating polynomial by the sum of thesepi weighted by the corresponding yi.
The pi polynomials are called Lagrange basis polynomials and can easilybe constructed by placing the roots appropriately and then normalizing theresult such that pi(xi) = 1, this gives the expression
pi(x) =(x− x1) · · · (x− xi−1)(x− xi+1) · · · (x− xn)
(xi − x1) · · · (xi − xk−1)(xi − xi+1) · · · (xi − xn).
0 1 2 3 4 5 6 7 8
−2
0
2
p1(x) p2(x) p3(x) p4(x) p(x) (x, y)
Figure 1.4: Illustration of Lagrange interpolation of 4 data points. The red dots are
the data set and p(x) =
4∑k=1
ykp(xk) is the interpolating polynomial.
The explicit formula for the full interpolating polynomial is
p(x) =
n∑k=1
yk(x− x1) · · · (x− xk−1)(x− xk+1) · · · (x− xn)
(xk − x1) · · · (xk − xk−1)(xk − xk+1) · · · (xk − xn)
and from this formula the expression for the inverse of the Vandermondematrix can be found by noting that the jth row of the inverse will consistof the coefficients of pj , the resulting expression for the elements is given inTheorem 1.4.
Polynomial interpolation is mostly used when the data set we wish tointerpolate is small. The main reason for this is the instability of the inter-polation method. One example of this is Runge’s phenomenon that showsthat when certain functions are approximated by polynomial interpolation
34
1.2. INTERPOLATION
−40−20 0 20 400
0.5
1
1.5
−40−20 0 20 400
0.5
1
1.5
−40−20 0 20 400
0.5
1
1.5
Figure 1.5: Illustration of Runge’s phenomenon. Here we attempt to approximate afunction (dashed line) by polynomial interpolation (solid line). With 7equidistant sample points (left figure) the approximation is poor near theedges of the interval and increasing the number of sample points to 14(center) and 19 (right) clearly reduces accuracy at the edges further.
fitted to equidistantly sampled points will sometimes lose precision when thenumber of interpolating points is increased, see Figure 1.5 for an example.
One way to predict this instability of polynomial interpolation is thatthe conditional number of the Vandermonde matrix can be very large forequidistant points [59].
There are different ways to mitigate the issue of stability, for examplechoosing data points that minimize the conditional number of the relevantmatrix [57, 59] or by choosing a polynomial basis that is more stable forthe given set of data points such as Bernstein polynomials in the case ofequidistant points [126]. Other polynomial schemes can also be considered,for instance by interpolating with different basis functions in different inter-vals, for example using polynomial splines.
Naturally another choice is to not choose polynomials as basis functionsbut instead choose some other functions that are more suitable. For anexample of this see Section 3.3.
While the instability of polynomial interpolation does not prevent it frombeing useful for analytical examinations it is generally considered imprac-tical when there is noise present or when calculations are performed withlimited precision. Often interpolating polynomials are not constructed byinverting the Vandermonde matrix or calculating the Lagrange basis poly-nomials, instead a more computationally efficient method such as Newtoninterpolation or Neville’s algorithm are used [138]. There are some variantsof Lagrange interpolation, such as barycentric Lagrange interpolation, that
35
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
have good computational performance [12].
In applications where the data is noisy it is often suitable to use regres-sion instead of interpolation, which will be discussed in the next section.
Before moving on to regression we will discuss an interesting and im-portant (but for the rest of the thesis irrelevant) form of polynomial in-terpolation called Hermite interpolation where it is not only required thatp(xk) = yk but also that the derivatives up to a certain order (sometimesallowed vary per point) are also given. This requires a higher degree poly-nomial that can be found by solving the equation system
p(xk) = yk0
p′(xk) = yk1...
p(i)(xk) = yki
for all k = 1, 2, . . . , n where ki are integers that defines the order of thederivative that needs to match at the point given by xk.
When this equation system is written as a matrix equation the resulting
matrix, C, will have dimension m×m with m =n∑
i=1
ki with rows given by
Ca,b =
0, b ≤ kj(b−1)!
(b−c−1)!xb−c−1k , b > kj
with c = a−j∑
i=1
ki and c < a ≤ c+ kj+1.
The matrix C is called a confluent Vandermonde matrix and has beenstudied extensively since Hermite interpolation is important both for nu-merical and analytical purposes. For example the confluent Vandermondematrix also has a very elegant formula for the determinant [3]
det(C) =∏
1≤i<j≤n
(xj − xi)(ki+1)(kj+1).
There are also many results related to its inverse and numerical proper-ties, classical examples are [55,56,58], some further examples are mentionedon page 30 but this is a vanishingly small part of the total literature on thesubject.
36
1.3. REGRESSION
1.3 Regression
Regression is similar to interpolation except that the presence of noise inthe data is taken into consideration. The typical regression problem assumesthat the data points (xi, yi), i = 1, . . . , n can be described by
yi = f(β;xi) + εi
where f(β;x) is a given function with a fixed number of undetermined pa-rameters β ∈ B and εi for i = 1, . . . , n are random variables with expectedvalue zero, usually referred to as the errors or the noise for the data set.
There are many different classes of regression problems defined by thetype of function f(β1, . . . , βm;x) and the probability distribution of the errorvariables.
Here we will only consider the situation when the εi variables are in-dependent and normally distributed with identical variance and that theparameter space B is a compact subset of Rk and that for all xi the functionf(β;xi) is a continuous function of β ∈ B. It is well known in statistics thatunder these conditions the parameter set β∗ that minimizes the sum of thesquares of the residuals
S(β) =n∑
i=1
(yi − f(β;xi))2
is the maximum-likelihood estimator of the true parameter set β [150].When choosing the parameters β∗ the function is usually said to be thebest possible fit the data in the least-square sense.
The most wide-spread form of regression is linear regression where, anal-ogously to linear interpolation, the function f(β;x) depends linearly on β.This is a common type of regression that has a unique solution that is simpleto find. It is commonly known as the least-squares method and we describethis method in the next section.
With a non-linear f(β;x) it is usually much more difficult to solve theregression problem and numerical methods are commonly used, in Sec-tion 1.3.3 we describe one such method called the Marquardt least-squaresmethod.
In Section 3.2 we present a scheme for approximating electrostaticaldischarges to ensure electromagnetic compatibility (see Section 1.4) thatuses both the least-squares method and the Marquard least-squares method.
37
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
have good computational performance [12].
In applications where the data is noisy it is often suitable to use regres-sion instead of interpolation, which will be discussed in the next section.
Before moving on to regression we will discuss an interesting and im-portant (but for the rest of the thesis irrelevant) form of polynomial in-terpolation called Hermite interpolation where it is not only required thatp(xk) = yk but also that the derivatives up to a certain order (sometimesallowed vary per point) are also given. This requires a higher degree poly-nomial that can be found by solving the equation system
p(xk) = yk0
p′(xk) = yk1...
p(i)(xk) = yki
for all k = 1, 2, . . . , n where ki are integers that defines the order of thederivative that needs to match at the point given by xk.
When this equation system is written as a matrix equation the resulting
matrix, C, will have dimension m×m with m =n∑
i=1
ki with rows given by
Ca,b =
0, b ≤ kj(b−1)!
(b−c−1)!xb−c−1k , b > kj
with c = a−j∑
i=1
ki and c < a ≤ c+ kj+1.
The matrix C is called a confluent Vandermonde matrix and has beenstudied extensively since Hermite interpolation is important both for nu-merical and analytical purposes. For example the confluent Vandermondematrix also has a very elegant formula for the determinant [3]
det(C) =∏
1≤i<j≤n
(xj − xi)(ki+1)(kj+1).
There are also many results related to its inverse and numerical proper-ties, classical examples are [55,56,58], some further examples are mentionedon page 30 but this is a vanishingly small part of the total literature on thesubject.
36
1.3. REGRESSION
1.3 Regression
Regression is similar to interpolation except that the presence of noise inthe data is taken into consideration. The typical regression problem assumesthat the data points (xi, yi), i = 1, . . . , n can be described by
yi = f(β;xi) + εi
where f(β;x) is a given function with a fixed number of undetermined pa-rameters β ∈ B and εi for i = 1, . . . , n are random variables with expectedvalue zero, usually referred to as the errors or the noise for the data set.
There are many different classes of regression problems defined by thetype of function f(β1, . . . , βm;x) and the probability distribution of the errorvariables.
Here we will only consider the situation when the εi variables are in-dependent and normally distributed with identical variance and that theparameter space B is a compact subset of Rk and that for all xi the functionf(β;xi) is a continuous function of β ∈ B. It is well known in statistics thatunder these conditions the parameter set β∗ that minimizes the sum of thesquares of the residuals
S(β) =n∑
i=1
(yi − f(β;xi))2
is the maximum-likelihood estimator of the true parameter set β [150].When choosing the parameters β∗ the function is usually said to be thebest possible fit the data in the least-square sense.
The most wide-spread form of regression is linear regression where, anal-ogously to linear interpolation, the function f(β;x) depends linearly on β.This is a common type of regression that has a unique solution that is simpleto find. It is commonly known as the least-squares method and we describethis method in the next section.
With a non-linear f(β;x) it is usually much more difficult to solve theregression problem and numerical methods are commonly used, in Sec-tion 1.3.3 we describe one such method called the Marquardt least-squaresmethod.
In Section 3.2 we present a scheme for approximating electrostaticaldischarges to ensure electromagnetic compatibility (see Section 1.4) thatuses both the least-squares method and the Marquard least-squares method.
37
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
1.3.1 Linear regression models
Suppose we want to find a function
f(x) =m∑i=1
βigi(x) (10)
that fits as well as possible in the least-square sense to the data points(xi, yi), i = 1, . . . , n, n > m. We then get a regression problem described bythe linear equation system Aβ = y where β are the coefficients of f , y isthe vector of data values and A is the appropriate alternant matrix,
A =
g1(x1) g2(x1) . . . gm(x1)g1(x2) g2(x2) . . . gm(x2)
......
. . ....
g1(xn) g2(xn) . . . gm(xn)
, β =
β1β2...βn
, y =
y1y2...yn
.
This is an overdetermined version of the linear interpolation problemdescribed in Section 1.2.
How can we actually find the coefficients that minimize the sum of thesquares of the residuals? First we can define the square of the length of theresidual vector, e = Aβ − y, as a function
s(e) = ee =n∑
i=1
|ei|2 = (Aβ − y)(Aβ − y)
This kind of function is a positive second degree polynomial with no mixedterms and thus has a global minima where ∂s
∂ei= 0 for all 1 ≤ i ≤ n. We
can find the global minima by looking at the derivative of the function, eiis determined by βi and
∂ei∂βj
= Ai,j
thus
∂s
∂βi=
n∑i=1
2ei∂ei∂βj
=
n∑i=1
2(Ai,·β − yi)Ai,j = 0 ⇔ AAβ = Ay
This givesAAβ = Ay ⇔ β = (AA)−1Ay
and by the Gauss-Markov theorem ( [53, 54, 114], see for instance [116] fora more modern description), if (AA)−1 exists then (10) gives the linear,
38
1.3. REGRESSION
unbiased estimator that gives the lowest variance possible for any linear,unbiased estimator. The matrix given by (AA)−1A is often referred toas the Moore-Penrose pseudoinverse of A.
Clearly a linear regression model with gi(x) = xi−1 gives a regressionmodel described by a rectangular Vandermonde matrix.
1.3.2 Non-linear regression models
So far we have only considered models that are linear with respect to theparameters that specify them. If we relax the linearity condition and simplyconsider fitting a function with m parameters, f(β1, . . . , βm;x), to n datapoints it is usually referred to as a non-linear regression model.
There is no general analogue to the Gauss-Markov theorem for non-linearregression models and therefore finding the appropriate estimator requiresmore knowledge about the specifics of the model. In practice non-linearregression problems are often solved using some numerical method for non-linear optimization of which there are many (see for instance [147] for anoverview).
In this thesis we will use a standard method called the Marquardt least-squares method that the next section will give an overview of. In Section3.2.2 we will use a combination of the Marquardt least-squares method andmethods for fitting linear regression models to fit a non-linear regressionmodel described by
G (β; t)η = i
where β, η are vectors of parameters to be fitted, i is the data we wish tofit the model to and G (β; t) is the generalized Vandermonde matrix
G (β; t) =
(t1e1−t1)β1 (t1e
1−t1)β2 · · · (t1e1−t1)βn
(t2e1−t2)β1 (t2e
1−t2)β2 · · · (t2e1−t2)βn
......
. . ....
(tme1−tm)β1 (tme1−tm)β2 · · · (tme1−tm)βn
.
1.3.3 The Marquardt least-squares method
This section is based on Section 3.1 of Paper D
TheMarquardt least-squares method , also known as the Levenberg-Marquardtalgorithm or damped least-squares, is an efficient method for least-squaresestimation for functions with non-linear parameters that was developed inthe middle of the 20th century (see [101], [115]).
39
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
1.3.1 Linear regression models
Suppose we want to find a function
f(x) =m∑i=1
βigi(x) (10)
that fits as well as possible in the least-square sense to the data points(xi, yi), i = 1, . . . , n, n > m. We then get a regression problem described bythe linear equation system Aβ = y where β are the coefficients of f , y isthe vector of data values and A is the appropriate alternant matrix,
A =
g1(x1) g2(x1) . . . gm(x1)g1(x2) g2(x2) . . . gm(x2)
......
. . ....
g1(xn) g2(xn) . . . gm(xn)
, β =
β1β2...βn
, y =
y1y2...yn
.
This is an overdetermined version of the linear interpolation problemdescribed in Section 1.2.
How can we actually find the coefficients that minimize the sum of thesquares of the residuals? First we can define the square of the length of theresidual vector, e = Aβ − y, as a function
s(e) = ee =n∑
i=1
|ei|2 = (Aβ − y)(Aβ − y)
This kind of function is a positive second degree polynomial with no mixedterms and thus has a global minima where ∂s
∂ei= 0 for all 1 ≤ i ≤ n. We
can find the global minima by looking at the derivative of the function, eiis determined by βi and
∂ei∂βj
= Ai,j
thus
∂s
∂βi=
n∑i=1
2ei∂ei∂βj
=
n∑i=1
2(Ai,·β − yi)Ai,j = 0 ⇔ AAβ = Ay
This givesAAβ = Ay ⇔ β = (AA)−1Ay
and by the Gauss-Markov theorem ( [53, 54, 114], see for instance [116] fora more modern description), if (AA)−1 exists then (10) gives the linear,
38
1.3. REGRESSION
unbiased estimator that gives the lowest variance possible for any linear,unbiased estimator. The matrix given by (AA)−1A is often referred toas the Moore-Penrose pseudoinverse of A.
Clearly a linear regression model with gi(x) = xi−1 gives a regressionmodel described by a rectangular Vandermonde matrix.
1.3.2 Non-linear regression models
So far we have only considered models that are linear with respect to theparameters that specify them. If we relax the linearity condition and simplyconsider fitting a function with m parameters, f(β1, . . . , βm;x), to n datapoints it is usually referred to as a non-linear regression model.
There is no general analogue to the Gauss-Markov theorem for non-linearregression models and therefore finding the appropriate estimator requiresmore knowledge about the specifics of the model. In practice non-linearregression problems are often solved using some numerical method for non-linear optimization of which there are many (see for instance [147] for anoverview).
In this thesis we will use a standard method called the Marquardt least-squares method that the next section will give an overview of. In Section3.2.2 we will use a combination of the Marquardt least-squares method andmethods for fitting linear regression models to fit a non-linear regressionmodel described by
G (β; t)η = i
where β, η are vectors of parameters to be fitted, i is the data we wish tofit the model to and G (β; t) is the generalized Vandermonde matrix
G (β; t) =
(t1e1−t1)β1 (t1e
1−t1)β2 · · · (t1e1−t1)βn
(t2e1−t2)β1 (t2e
1−t2)β2 · · · (t2e1−t2)βn
......
. . ....
(tme1−tm)β1 (tme1−tm)β2 · · · (tme1−tm)βn
.
1.3.3 The Marquardt least-squares method
This section is based on Section 3.1 of Paper D
TheMarquardt least-squares method , also known as the Levenberg-Marquardtalgorithm or damped least-squares, is an efficient method for least-squaresestimation for functions with non-linear parameters that was developed inthe middle of the 20th century (see [101], [115]).
39
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The least-squares estimation problem for functions with non-linear pa-rameters arises when a function of m independent variables and describedby k unknown parameters needs to be fitted to a set of n data points suchthat the sum of squares of residuals is minimized.
The vector containing the independent variables is x = (x1, · · · , xn), thevector containing the parameters β = (β1, · · · , βk) and the data points
(Yi, X1i, X2i, · · · , Xmi) = (Yi,Xi) , i = 1, 2, · · · , n.
Let the residuals be denoted by Ei = f(Xi;β) − Yi and the sum ofsquares of Ei is then written as
S =n∑
i=1
(f(Xi;β)− Yi)2 ,
which is the function to be minimized with respect to β.The Marquardt least-square method is an iterative method that gives ap-
proximate values of β by combining the Gauss-Newton method (also knownas the inverse Hessian method) and the steepest descent (also known as thegradient) method to minimize S. The method is based around solving thelinear equation system
(A∗(r) + λ(r)I
)δ∗(r) = g∗(r), (11)
where A∗(r) is a modified Hessian matrix of E(b) (or f(Xi;b)), g∗(r) is a
rescaled version of the gradient of S, r is the number of the current iterationof the method, and λ is a real positive number sometimes referred to as thefudge factor [138]. The Hessian, the gradient and their modifications aredefined as follows:
A = JJ,
Jij =∂fi∂bj
=∂Ei
∂bj, i = 1, 2, · · · ,m; j = 1, 2, · · · , k,
and(A∗)ij =
aij√aii
√ajj
,
whileg = J(Y − f0), f0i = f(Xi,b, c), g∗
i =giaii
.
Solving (11) gives a vector which, after some scaling, describes how theparameters b should be changed in order to get a new approximation of β,
b(r+1) = b(r) + δ(r), δ(r) =δ∗(r)i√aii
. (12)
40
1.3. REGRESSION
It is obvious from (11) that δ(r) depends on the value of the fudge factorλ. Note that if λ = 0, then (11) reduces to the regular Gauss-Newtonmethod [115], and if λ → ∞ the method will converge towards the steepestdescent method [115]. The reason that the two methods are combined is thatthe Gauss-Newton method often has faster convergence than the steepestdescent method, but is also an unstable method [115]. Therefore, λ must bechosen appropriately in each step. In the Marquardt least-squares methodthis amounts to increasing λ with a chosen factor v whenever an iterationincreases S, and if an iteration reduces S then λ is reduced by a factor v asmany times as possible. Below follows a detailed description of the methodusing the following notation:
S(r) =n∑
i=1
(Yi − f(Xi,b
(r), c))2
, (13)
S(λ(r)
)=
n∑i=1
(Yi − f(Xi,b
(r) + δ(r), c))2
. (14)
The iteration step of the Marquardt least-squares method can be de-scribed as follows:
• Input: v > 1 and b(r), λ(r).
Compute S(λ(r)
).
• If λ(r) 1 then compute S(λ(r)
v
), else go to .
• If S(λ(r)
v
)≤ S(r) let λ(r+1) = λ(r)
v .
If S(λ(r)
)≤ S(r) let λ(r+1) = λ(r).
• If S(λ(r)
)> S(r) then find the smallest integer ω > 0 such that
S(λ(r)vω
)≤ S(r), and set λ(r+1) = λ(r)vω.
• Output: b(r+1) = b(r) + δ(r), δ(r).
This iteration procedure is also described in figure 1.6. Naturally, somecondition for what constitutes an acceptable fit for the function must alsobe chosen. If this condition is not satisfied the new values for b(r+1) andλ(r+1) will be used as input for the next iteration and if the condition issatisfied the algorithm terminates. The quality of the fitting, in other words
41
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The least-squares estimation problem for functions with non-linear pa-rameters arises when a function of m independent variables and describedby k unknown parameters needs to be fitted to a set of n data points suchthat the sum of squares of residuals is minimized.
The vector containing the independent variables is x = (x1, · · · , xn), thevector containing the parameters β = (β1, · · · , βk) and the data points
(Yi, X1i, X2i, · · · , Xmi) = (Yi,Xi) , i = 1, 2, · · · , n.
Let the residuals be denoted by Ei = f(Xi;β) − Yi and the sum ofsquares of Ei is then written as
S =n∑
i=1
(f(Xi;β)− Yi)2 ,
which is the function to be minimized with respect to β.The Marquardt least-square method is an iterative method that gives ap-
proximate values of β by combining the Gauss-Newton method (also knownas the inverse Hessian method) and the steepest descent (also known as thegradient) method to minimize S. The method is based around solving thelinear equation system
(A∗(r) + λ(r)I
)δ∗(r) = g∗(r), (11)
where A∗(r) is a modified Hessian matrix of E(b) (or f(Xi;b)), g∗(r) is a
rescaled version of the gradient of S, r is the number of the current iterationof the method, and λ is a real positive number sometimes referred to as thefudge factor [138]. The Hessian, the gradient and their modifications aredefined as follows:
A = JJ,
Jij =∂fi∂bj
=∂Ei
∂bj, i = 1, 2, · · · ,m; j = 1, 2, · · · , k,
and(A∗)ij =
aij√aii
√ajj
,
whileg = J(Y − f0), f0i = f(Xi,b, c), g∗
i =giaii
.
Solving (11) gives a vector which, after some scaling, describes how theparameters b should be changed in order to get a new approximation of β,
b(r+1) = b(r) + δ(r), δ(r) =δ∗(r)i√aii
. (12)
40
1.3. REGRESSION
It is obvious from (11) that δ(r) depends on the value of the fudge factorλ. Note that if λ = 0, then (11) reduces to the regular Gauss-Newtonmethod [115], and if λ → ∞ the method will converge towards the steepestdescent method [115]. The reason that the two methods are combined is thatthe Gauss-Newton method often has faster convergence than the steepestdescent method, but is also an unstable method [115]. Therefore, λ must bechosen appropriately in each step. In the Marquardt least-squares methodthis amounts to increasing λ with a chosen factor v whenever an iterationincreases S, and if an iteration reduces S then λ is reduced by a factor v asmany times as possible. Below follows a detailed description of the methodusing the following notation:
S(r) =n∑
i=1
(Yi − f(Xi,b
(r), c))2
, (13)
S(λ(r)
)=
n∑i=1
(Yi − f(Xi,b
(r) + δ(r), c))2
. (14)
The iteration step of the Marquardt least-squares method can be de-scribed as follows:
• Input: v > 1 and b(r), λ(r).
Compute S(λ(r)
).
• If λ(r) 1 then compute S(λ(r)
v
), else go to .
• If S(λ(r)
v
)≤ S(r) let λ(r+1) = λ(r)
v .
If S(λ(r)
)≤ S(r) let λ(r+1) = λ(r).
• If S(λ(r)
)> S(r) then find the smallest integer ω > 0 such that
S(λ(r)vω
)≤ S(r), and set λ(r+1) = λ(r)vω.
• Output: b(r+1) = b(r) + δ(r), δ(r).
This iteration procedure is also described in figure 1.6. Naturally, somecondition for what constitutes an acceptable fit for the function must alsobe chosen. If this condition is not satisfied the new values for b(r+1) andλ(r+1) will be used as input for the next iteration and if the condition issatisfied the algorithm terminates. The quality of the fitting, in other words
41
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
the value of S, is determined by the stopping condition and the initial valuesfor b(0). The initial value of λ(0) affects the performance of the algorithmto some extent since after the first iteration λ(r) will be self-regulating.Suitable values for b(0) are challenging to find for many functions f andthey are often, together with λ(0), found using heuristic methods.
Input:b(r), λ(r) and v > 1
Compute S(λ(r)
)
ω = ω + 1 λ(r) 1 Compute S(λ(r)
v
)
S(λ(r)vω
)≤ S(r)
ω = 1
S(λ(r)
)≤ S(r) S
(λ(r)
v
)≤ S(r)
λ(r+1) = λ(r)vω λ(r+1) = λ(r) λ(r+1) = λ(r)
v
Output:b(r+1) = b(r) + δ(r), δ(r)
YES
YES
NO
NO
YES
NONO
YES
Figure 1.6: The basic iteration step of the Marquardt least-squares method, defini-tions of computed quantities are given in (12), (13) and (14).
In Section 3.2 the Marquardt least-squares method will be used for re-gression with power-exponential functions.
42
1.3. REGRESSION
1.3.4 D-optimal experiment design
For the class of linear non-weighted regression problems described in Sec-tion 1.3.1 minimizing the square of the sum of residuals gives the maximum-likelihood estimation of the parameters that specify the fitted function. Thisestimation naturally has a variance as well and minimizing this variance canbe interpreted as improving the reliability of the fitted function by minimiz-ing its sensitivity to noise in measurements. This minimization is usuallydone by choosing where to sample the data carefully, in other words, giventhe regression problem defined by
yi = f(β;xi) + εi
for i = 1, . . . , n with the same conditions on f(β;x) and εi as in Section 1.3we want to choose a design xi, i = 1, . . . , n that minimizes the variance ofthe values predicted by the regression model. This is usually referred to asG-optimality.
To give a proper definition of G-optimality we will need the concept ofthe Fischer information matrix.
Definition 1.5. For a finite design x ∈ X ⊆ Rn the Fischer informationmatrix is the matrix defined by
M(x) =
n∑i=1
f(xi)f(xi)
where f(x) =[f1(x) f2(x) · · · fn(x)
].
Definition 1.6 (The G-optimality criterion). A design ξ is said to beG-optimal if it minimizes the maximum variance of any predicted value
Var(y(ξ)) = minxi, i=1,2,...,n
maxx∈X
Var(y(x)) = minz∈X
maxx∈X
f(x)M(z)f(x).
The G-optimality condition was first introduced in [154] (the name G-optimality comes from later work by Kiefer and Wolfowitz where they de-scribe several different types of optimal design using alphabetical letters[89], [90]) and is an example of a minimax criterion, since it minimizes themaximum variance of the values given by the regression model [116].
There are many kinds of optimality conditions related to G-optimality.One which is suitable for us to consider is D-optimality. This type of opti-mality was first introduced in [169] and instead of focusing on the varianceof the predicted values of the model it instead minimizes the volume of theconfidence ellipsoid for the parameters (for a given confidence level).
43
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
the value of S, is determined by the stopping condition and the initial valuesfor b(0). The initial value of λ(0) affects the performance of the algorithmto some extent since after the first iteration λ(r) will be self-regulating.Suitable values for b(0) are challenging to find for many functions f andthey are often, together with λ(0), found using heuristic methods.
Input:b(r), λ(r) and v > 1
Compute S(λ(r)
)
ω = ω + 1 λ(r) 1 Compute S(λ(r)
v
)
S(λ(r)vω
)≤ S(r)
ω = 1
S(λ(r)
)≤ S(r) S
(λ(r)
v
)≤ S(r)
λ(r+1) = λ(r)vω λ(r+1) = λ(r) λ(r+1) = λ(r)
v
Output:b(r+1) = b(r) + δ(r), δ(r)
YES
YES
NO
NO
YES
NONO
YES
Figure 1.6: The basic iteration step of the Marquardt least-squares method, defini-tions of computed quantities are given in (12), (13) and (14).
In Section 3.2 the Marquardt least-squares method will be used for re-gression with power-exponential functions.
42
1.3. REGRESSION
1.3.4 D-optimal experiment design
For the class of linear non-weighted regression problems described in Sec-tion 1.3.1 minimizing the square of the sum of residuals gives the maximum-likelihood estimation of the parameters that specify the fitted function. Thisestimation naturally has a variance as well and minimizing this variance canbe interpreted as improving the reliability of the fitted function by minimiz-ing its sensitivity to noise in measurements. This minimization is usuallydone by choosing where to sample the data carefully, in other words, giventhe regression problem defined by
yi = f(β;xi) + εi
for i = 1, . . . , n with the same conditions on f(β;x) and εi as in Section 1.3we want to choose a design xi, i = 1, . . . , n that minimizes the variance ofthe values predicted by the regression model. This is usually referred to asG-optimality.
To give a proper definition of G-optimality we will need the concept ofthe Fischer information matrix.
Definition 1.5. For a finite design x ∈ X ⊆ Rn the Fischer informationmatrix is the matrix defined by
M(x) =
n∑i=1
f(xi)f(xi)
where f(x) =[f1(x) f2(x) · · · fn(x)
].
Definition 1.6 (The G-optimality criterion). A design ξ is said to beG-optimal if it minimizes the maximum variance of any predicted value
Var(y(ξ)) = minxi, i=1,2,...,n
maxx∈X
Var(y(x)) = minz∈X
maxx∈X
f(x)M(z)f(x).
The G-optimality condition was first introduced in [154] (the name G-optimality comes from later work by Kiefer and Wolfowitz where they de-scribe several different types of optimal design using alphabetical letters[89], [90]) and is an example of a minimax criterion, since it minimizes themaximum variance of the values given by the regression model [116].
There are many kinds of optimality conditions related to G-optimality.One which is suitable for us to consider is D-optimality. This type of opti-mality was first introduced in [169] and instead of focusing on the varianceof the predicted values of the model it instead minimizes the volume of theconfidence ellipsoid for the parameters (for a given confidence level).
43
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Definition 1.7 (The D-optimality criterion). A design ξ is said to beD-optimal if it maximizes the determinant of the Fischer information matrix
det(M(ξ)) = maxx∈X
det(M(x)).
The D-optimal designs are often good design with respect to othertypes of criterion (see for example [61] for a brief discussion on this) andis often practical to consider due to being invariant with respect to lin-ear transformations of the design matrix. A well-known theorem called theKiefer-Wolfowitz equivalence theorem shows that under certain conditionsG-optimality is equivalent to D-optimality.
Theorem 1.5 (Kiefer-Wolfowitz equivalence theorem). For any linear re-gression model with independent, uncorrelated errors and continuous andlinearly independent basis functions fi(x) defined on a fixed compact topo-logical space X there exists a D-optimal design and any D-optimal design isalso G-optimal.
This equivalence theorem was originally proven in [91] but the formula-tion above is taken from [116].
Thus maximizing the determinant of the Fischer information matrix cor-responds to minimizing the variance of the estimated β. Interpolation canbe considered a special case of regression when the sum of the square of theresiduals can be reduced to zero. Thus we can speak of D-optimal designfor interpolation as well, in fact optimal experiment design is often usedto find the minimum number of points needed for a certain model. For alinear interpolation problem defined by the alternant matrix A(f ;x) the Fis-cher information matrix is M(x) = A(f ;x)A(f ;x) and since A(f ;x) is ann × n matrix det(M(x)) = det(A(f ;x)) det(A(f ;x)) = det(f ;x))2. Thusthe maximization of the determinant of the Fischer information matrix isequivalent to finding the extreme points of the determinant of an alternantmatrix in some volume given by the set of possible designs.
A standard case of this is polynomial interpolation where the x-valuesare in a limited interval, for instance −1 ≤ xi ≤ 1 for i = 1, 2, . . . , n. In thiscase the regression problem can be written as Vn(x)
β = y where Vn(x)is a Vandermonde matrix as defined in equation (1) and the constraints onthe elements of β means that the volume we want to optimize over is acube in n dimensions. There is a number of classical results that describehow to find the D-optimal designs for weighted univariate polynomials withvarious efficiency functions, see for instance [43], and in Section 2.2.3 we
44
1.3. REGRESSION
will demonstrate one way to optimize the Vandermonde determinant over acube.
The shape of the volume to optimize the determinant in is given by con-straints on the data points. For example, if there is a cost associated witheach data point that increases quadratically with x and there is a total bud-get, C, for the experiment that cannot be exceeded the constraint on thex-values becomes x21 + x22 + . . . + x2n ≤ C and the determinant needs to beoptimized in a ball. In Chapter 2 we examine the optimization of the Van-dermonde determinant over several different surfaces in several dimensions.
In Section 3.3 we use a D-optimal design to improve the stability of aninterpolation problem as an alternative to the non-linear regression donein Section 3.2. Note that while choosing a D-optimal design can give anapproximation method that is more stable since it minimizes the varianceof the parameters the function used in the approximation can still be highlysensitive to changes in parameters (the variance of the predicted values canbe minimized but still high) so it does necessarily maximize stability orstop instability phenomenons similar to Runge’s phenomenon for polynomialinterpolation.
45
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Definition 1.7 (The D-optimality criterion). A design ξ is said to beD-optimal if it maximizes the determinant of the Fischer information matrix
det(M(ξ)) = maxx∈X
det(M(x)).
The D-optimal designs are often good design with respect to othertypes of criterion (see for example [61] for a brief discussion on this) andis often practical to consider due to being invariant with respect to lin-ear transformations of the design matrix. A well-known theorem called theKiefer-Wolfowitz equivalence theorem shows that under certain conditionsG-optimality is equivalent to D-optimality.
Theorem 1.5 (Kiefer-Wolfowitz equivalence theorem). For any linear re-gression model with independent, uncorrelated errors and continuous andlinearly independent basis functions fi(x) defined on a fixed compact topo-logical space X there exists a D-optimal design and any D-optimal design isalso G-optimal.
This equivalence theorem was originally proven in [91] but the formula-tion above is taken from [116].
Thus maximizing the determinant of the Fischer information matrix cor-responds to minimizing the variance of the estimated β. Interpolation canbe considered a special case of regression when the sum of the square of theresiduals can be reduced to zero. Thus we can speak of D-optimal designfor interpolation as well, in fact optimal experiment design is often usedto find the minimum number of points needed for a certain model. For alinear interpolation problem defined by the alternant matrix A(f ;x) the Fis-cher information matrix is M(x) = A(f ;x)A(f ;x) and since A(f ;x) is ann × n matrix det(M(x)) = det(A(f ;x)) det(A(f ;x)) = det(f ;x))2. Thusthe maximization of the determinant of the Fischer information matrix isequivalent to finding the extreme points of the determinant of an alternantmatrix in some volume given by the set of possible designs.
A standard case of this is polynomial interpolation where the x-valuesare in a limited interval, for instance −1 ≤ xi ≤ 1 for i = 1, 2, . . . , n. In thiscase the regression problem can be written as Vn(x)
β = y where Vn(x)is a Vandermonde matrix as defined in equation (1) and the constraints onthe elements of β means that the volume we want to optimize over is acube in n dimensions. There is a number of classical results that describehow to find the D-optimal designs for weighted univariate polynomials withvarious efficiency functions, see for instance [43], and in Section 2.2.3 we
44
1.3. REGRESSION
will demonstrate one way to optimize the Vandermonde determinant over acube.
The shape of the volume to optimize the determinant in is given by con-straints on the data points. For example, if there is a cost associated witheach data point that increases quadratically with x and there is a total bud-get, C, for the experiment that cannot be exceeded the constraint on thex-values becomes x21 + x22 + . . . + x2n ≤ C and the determinant needs to beoptimized in a ball. In Chapter 2 we examine the optimization of the Van-dermonde determinant over several different surfaces in several dimensions.
In Section 3.3 we use a D-optimal design to improve the stability of aninterpolation problem as an alternative to the non-linear regression donein Section 3.2. Note that while choosing a D-optimal design can give anapproximation method that is more stable since it minimizes the varianceof the parameters the function used in the approximation can still be highlysensitive to changes in parameters (the variance of the predicted values canbe minimized but still high) so it does necessarily maximize stability orstop instability phenomenons similar to Runge’s phenomenon for polynomialinterpolation.
45
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
1.4 Electromagnetic compatibility andelectrostatic discharge currents
There are many examples of electromagnetic phenomena that involve twoobjects influencing each other without touching. Almost everyone is familiarwith magnets that attract or repel other object, sparks that bridge physicalgaps and radio waves that send messages across the globe. While this action-at-a-distance can be very useful it can also cause unintended interactionsbetween different systems. This is usually referred to as electromagneticdisturbance or electromagnetic interference and the field of electromagneticcompatibility (EMC) is the study and design of systems that are not sus-ceptible to disturbances from other systems and does not cause interferencewith other systems or themselves [132,173].
There are many possible causes of electromagnetic disturbance includinga multitude of sources. Some examples are man-made sources such as broad-casting and receiving devices, power generators and converters, power con-version and ignition systems for combustion engines, manufacturing equip-ment like ovens, saws, mills, welders, blenders and mixers, other equipmentsuch as fans, heaters, coolers, lights, computers, instruments for measure-ments and control, examples of natural sources are atmospheric-, solar- andcosmic noise, static discharges and lightning [119].
Mathematical modelling is an important tool for EMC [119]. Using com-puters for electromagnetic analysis have been done since the 1950s [63] andit rapidly became more and more useful and important over time [137]. Inpractice many different types of models and methods are used, all with theirown advantages and disadvantages, and the design process often involvesa combination of analytical and numerical techniques [46]. The sources ofelectromagnetic disturbances are not always well understood or cannot bewell described which means that it is not always feasible to derive all partsof the model using first principles, especially since the behaviour of manysystems contain some degree of randomness which means that it is some-times most reasonable to use models that are based on typical behavioursbased on statistical data [27,84].
Requirements for a product or system to be considered electromagneti-cally compatible can be found in standards such as the IEC 61000-4-2 [73]and IEC 62305-1 [74]. In several of these standard approximations of typicalcurrents for various phenomena are given and electromagnetic compatibilityrequirements are based on the effects of the system being exposed to thesecurrents, such as the radiated electromagnetical fields. Ideally the descrip-
46
1.4. ELECTROMAGNETIC COMPATIBILITY ANDELECTROSTATIC DISCHARGE CURRENTS
tions of these currents should give an accurate description of the observedbehaviour that the standard is based on as well being computationally effi-cient (since computer simulations replacing construction of prototypes cansave both time and resources) and be compatible with the mathematicaltools that are commonly used in electromagnetic calculations, for instanceLaplace and Fourier transforms.
In this thesis we will discuss approximations of electrostatic dischargecurrents, either from a standard or based on experimental data. In Section1.4.1 a review of models in the literature can be found and in Chapter3 we propose a new function, the analytically extended function (AEF),for modelling these currents that has some advantages compared to thecommonly used models and can be applied to many different cases, typicallyat the cost of some extra manual work in fitting the model.
Electrostatic discharge (ESD) is a common phenomenon where a sud-den flow of electricity between two charged object occurs, examples includesparks and lightning strikes. The main mechanism behind is usually saidto be contact electrification, this phenomena is due to all materials occa-sionally emitting electrons, usually at a higher rate when they are heated.Typically the emission and absorption balances out but since the rate ofemission varies between different materials an imbalance can occur whentwo materials come sufficiently close to each other. When the materials areseparated this charge imbalance might remain for some time, it can be re-stored by the charged objects slowly emitting electrons to the surroundingobjects but in the right conditions, for example if the charged object comesnear a conductive material with an opposite charge, the restoration of thecharge balance can be very rapid resulting in an electrostatic discharge. Thereader is likely to be familiar with the case of two materials rubbing againsteach other building up a charge imbalance and one of the objects generatinga spark when moved close to a metal object. This case is common since fric-tion between objects typically means a larger contact area where charges cantransfer and movement is necessary for charge separation. For this reasonthis mechanism is often referred to as friction charging or the triboelectriceffect. Contact charging can happen between any material, including liq-uids and gases, and can also be affected by many other types of phenomena,such as ion transfer or energetic charged particles colliding with other ob-jects [51]. Therefore the exact mechanisms behind electrostatic dischargescan be difficult to understand and describe, even when the circumstanceswhere the electrostatic discharge are likely are well known [110].
In this thesis we focus on two types of electrostatic discharge, lightningdischarge and human-to-object (human-to-metal or human-to-human).
47
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
1.4 Electromagnetic compatibility andelectrostatic discharge currents
There are many examples of electromagnetic phenomena that involve twoobjects influencing each other without touching. Almost everyone is familiarwith magnets that attract or repel other object, sparks that bridge physicalgaps and radio waves that send messages across the globe. While this action-at-a-distance can be very useful it can also cause unintended interactionsbetween different systems. This is usually referred to as electromagneticdisturbance or electromagnetic interference and the field of electromagneticcompatibility (EMC) is the study and design of systems that are not sus-ceptible to disturbances from other systems and does not cause interferencewith other systems or themselves [132,173].
There are many possible causes of electromagnetic disturbance includinga multitude of sources. Some examples are man-made sources such as broad-casting and receiving devices, power generators and converters, power con-version and ignition systems for combustion engines, manufacturing equip-ment like ovens, saws, mills, welders, blenders and mixers, other equipmentsuch as fans, heaters, coolers, lights, computers, instruments for measure-ments and control, examples of natural sources are atmospheric-, solar- andcosmic noise, static discharges and lightning [119].
Mathematical modelling is an important tool for EMC [119]. Using com-puters for electromagnetic analysis have been done since the 1950s [63] andit rapidly became more and more useful and important over time [137]. Inpractice many different types of models and methods are used, all with theirown advantages and disadvantages, and the design process often involvesa combination of analytical and numerical techniques [46]. The sources ofelectromagnetic disturbances are not always well understood or cannot bewell described which means that it is not always feasible to derive all partsof the model using first principles, especially since the behaviour of manysystems contain some degree of randomness which means that it is some-times most reasonable to use models that are based on typical behavioursbased on statistical data [27,84].
Requirements for a product or system to be considered electromagneti-cally compatible can be found in standards such as the IEC 61000-4-2 [73]and IEC 62305-1 [74]. In several of these standard approximations of typicalcurrents for various phenomena are given and electromagnetic compatibilityrequirements are based on the effects of the system being exposed to thesecurrents, such as the radiated electromagnetical fields. Ideally the descrip-
46
1.4. ELECTROMAGNETIC COMPATIBILITY ANDELECTROSTATIC DISCHARGE CURRENTS
tions of these currents should give an accurate description of the observedbehaviour that the standard is based on as well being computationally effi-cient (since computer simulations replacing construction of prototypes cansave both time and resources) and be compatible with the mathematicaltools that are commonly used in electromagnetic calculations, for instanceLaplace and Fourier transforms.
In this thesis we will discuss approximations of electrostatic dischargecurrents, either from a standard or based on experimental data. In Section1.4.1 a review of models in the literature can be found and in Chapter3 we propose a new function, the analytically extended function (AEF),for modelling these currents that has some advantages compared to thecommonly used models and can be applied to many different cases, typicallyat the cost of some extra manual work in fitting the model.
Electrostatic discharge (ESD) is a common phenomenon where a sud-den flow of electricity between two charged object occurs, examples includesparks and lightning strikes. The main mechanism behind is usually saidto be contact electrification, this phenomena is due to all materials occa-sionally emitting electrons, usually at a higher rate when they are heated.Typically the emission and absorption balances out but since the rate ofemission varies between different materials an imbalance can occur whentwo materials come sufficiently close to each other. When the materials areseparated this charge imbalance might remain for some time, it can be re-stored by the charged objects slowly emitting electrons to the surroundingobjects but in the right conditions, for example if the charged object comesnear a conductive material with an opposite charge, the restoration of thecharge balance can be very rapid resulting in an electrostatic discharge. Thereader is likely to be familiar with the case of two materials rubbing againsteach other building up a charge imbalance and one of the objects generatinga spark when moved close to a metal object. This case is common since fric-tion between objects typically means a larger contact area where charges cantransfer and movement is necessary for charge separation. For this reasonthis mechanism is often referred to as friction charging or the triboelectriceffect. Contact charging can happen between any material, including liq-uids and gases, and can also be affected by many other types of phenomena,such as ion transfer or energetic charged particles colliding with other ob-jects [51]. Therefore the exact mechanisms behind electrostatic dischargescan be difficult to understand and describe, even when the circumstanceswhere the electrostatic discharge are likely are well known [110].
In this thesis we focus on two types of electrostatic discharge, lightningdischarge and human-to-object (human-to-metal or human-to-human).
47
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Lightning discharges can cause electromagnetic disturbances in threeways, by passing through an system directly, by passing through a nearbyobject which then radiates electrical fields that disturbs the system, or byindirectly inducing transient currents in systems when the electrical fieldassociated with a thundercloud disappears when the lightning discharge re-moves the charge imbalance between cloud and ground [27]. We discussmodelling of some lightning discharges from standards and experimentaldata in Section 3.2.
Electrostatical discharges from humans are very common and are typ-ically just a nuisance, but they can damage sensitive electronics and cancause severe accidents, either by the shock from the discharge causing ahuman error or by directly causing gas or dust explosions [84,110]. We dis-cuss modelling of a simulated human-to-object electrostatical discharge inSection 3.3.
1.4.1 Electrostatic discharge modelling
Well-defined representation of real electrostatic discharge currents is neededin order to establish realistic requirements for ESD generators used in testingthe equipment and devices, as well as to provide and improve the repeata-bility of tests. It should be able to approximate the current for varioustest levels, test set-ups and procedures, and also for various ESD conditionssuch as approach speeds, types of electrodes, relative arc length, humidity,etc. A mathematical function is necessary for computer simulation of suchphenomena, for verification of test generators and for improving standardwaveshape definitions.
A number of current functions, mostly based on exponential functions,have been proposed in the literature to model the ESD currents, [26,47,48,80, 82, 88, 156, 163, 170, 171, 180, 181]. Here we will give a brief presentationof some of them and in Section 3.1 we will propose an alternative functionand a scheme for fitting it to a waveshape.
A number of mathematical expressions have been introduced in the liter-ature for the purpose of representation of the ESD currents, either the IEC61000-4-2 Standard one [73], or experimentally measured ones, e.g. [47]. Inthis section we give an overview of most commonly applied ESD currentapproximations.
A double-exponential function has been proposed by Cerri et al. [26] forrepresentation of ESD currents for commercial simulators in the form
i(t) = I1e− t
τ1 − I2e− t
τ2 ,
48
1.4. ELECTROMAGNETIC COMPATIBILITY ANDELECTROSTATIC DISCHARGE CURRENTS
this type of function is also applied in other types of engineering, see sec-tion 3.1 for some examples.
This model was also extended with a four-exponential version by Keenanand Rossi [88]:
i(t) = I1
(e− t
τ1 − e− t
τ2
)− I2
(e− t
τ3 − e− t
τ4
). (15)
The Pulse function was proposed in [44],
i(t) = I0
(1− e
− tτ1
)pe− t
τ2 ,
and has been used for representation of lightning discharge currents both inits single term form [105] as well as linear combinations of two [156], threeor four Pulse functions [180].
The Heidler function [65] is one of the most commonly used functionsfor lightning discharge modelling
i(t) =I0η
(tτ1
)n
1 +(
tτ1
)n e− t
τ2 ,
Wang et al. [170] proposed an ESD model in the form of a sum of two Heidlerfunctions:
i(t) =I1η1
(tτ1
)n
1 +(
tτ1
)n e− t
τ2 +I2η2
(tτ3
)n
1 +(
tτ3
)n e− t
τ4 , (16)
with η1 = exp
(− τ1
τ2
(nτ2τ1
)1/n)
and η2 = exp
(− τ3
τ4
(nτ4τ3
)1/n)
being the
peak correction factors. The function has been used to fit different electro-static discharge currents using different methods [47,170,181].
Berghe and Zutter [163] proposed an ESD current model constructed asa sum of two Gaussian functions in the form:
i(t) = A exp
(−(t− τ1σ1
)2)
+Bt exp
(−(t− τ2σ2
)2). (17)
The following approximation using exponential polynomials is presentedin [171] by Wang et al.
i(t) = Ate−Ct +Bte−Dt, (18)
49
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Lightning discharges can cause electromagnetic disturbances in threeways, by passing through an system directly, by passing through a nearbyobject which then radiates electrical fields that disturbs the system, or byindirectly inducing transient currents in systems when the electrical fieldassociated with a thundercloud disappears when the lightning discharge re-moves the charge imbalance between cloud and ground [27]. We discussmodelling of some lightning discharges from standards and experimentaldata in Section 3.2.
Electrostatical discharges from humans are very common and are typ-ically just a nuisance, but they can damage sensitive electronics and cancause severe accidents, either by the shock from the discharge causing ahuman error or by directly causing gas or dust explosions [84,110]. We dis-cuss modelling of a simulated human-to-object electrostatical discharge inSection 3.3.
1.4.1 Electrostatic discharge modelling
Well-defined representation of real electrostatic discharge currents is neededin order to establish realistic requirements for ESD generators used in testingthe equipment and devices, as well as to provide and improve the repeata-bility of tests. It should be able to approximate the current for varioustest levels, test set-ups and procedures, and also for various ESD conditionssuch as approach speeds, types of electrodes, relative arc length, humidity,etc. A mathematical function is necessary for computer simulation of suchphenomena, for verification of test generators and for improving standardwaveshape definitions.
A number of current functions, mostly based on exponential functions,have been proposed in the literature to model the ESD currents, [26,47,48,80, 82, 88, 156, 163, 170, 171, 180, 181]. Here we will give a brief presentationof some of them and in Section 3.1 we will propose an alternative functionand a scheme for fitting it to a waveshape.
A number of mathematical expressions have been introduced in the liter-ature for the purpose of representation of the ESD currents, either the IEC61000-4-2 Standard one [73], or experimentally measured ones, e.g. [47]. Inthis section we give an overview of most commonly applied ESD currentapproximations.
A double-exponential function has been proposed by Cerri et al. [26] forrepresentation of ESD currents for commercial simulators in the form
i(t) = I1e− t
τ1 − I2e− t
τ2 ,
48
1.4. ELECTROMAGNETIC COMPATIBILITY ANDELECTROSTATIC DISCHARGE CURRENTS
this type of function is also applied in other types of engineering, see sec-tion 3.1 for some examples.
This model was also extended with a four-exponential version by Keenanand Rossi [88]:
i(t) = I1
(e− t
τ1 − e− t
τ2
)− I2
(e− t
τ3 − e− t
τ4
). (15)
The Pulse function was proposed in [44],
i(t) = I0
(1− e
− tτ1
)pe− t
τ2 ,
and has been used for representation of lightning discharge currents both inits single term form [105] as well as linear combinations of two [156], threeor four Pulse functions [180].
The Heidler function [65] is one of the most commonly used functionsfor lightning discharge modelling
i(t) =I0η
(tτ1
)n
1 +(
tτ1
)n e− t
τ2 ,
Wang et al. [170] proposed an ESD model in the form of a sum of two Heidlerfunctions:
i(t) =I1η1
(tτ1
)n
1 +(
tτ1
)n e− t
τ2 +I2η2
(tτ3
)n
1 +(
tτ3
)n e− t
τ4 , (16)
with η1 = exp
(− τ1
τ2
(nτ2τ1
)1/n)
and η2 = exp
(− τ3
τ4
(nτ4τ3
)1/n)
being the
peak correction factors. The function has been used to fit different electro-static discharge currents using different methods [47,170,181].
Berghe and Zutter [163] proposed an ESD current model constructed asa sum of two Gaussian functions in the form:
i(t) = A exp
(−(t− τ1σ1
)2)
+Bt exp
(−(t− τ2σ2
)2). (17)
The following approximation using exponential polynomials is presentedin [171] by Wang et al.
i(t) = Ate−Ct +Bte−Dt, (18)
49
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and has been used for design of simple electric circuits which can be usedto simulate ESD currents.
One of the most commonly used ESD standard currents is the IEC 61000-4-2 current that represent a typical electrostatic discharge generated by thehuman body [73]. In the IEC 61000-4-2 standard [73] this current is given bya graphical representation, see Figure 3.10, together with some constraints,see page 122. In Figure 1.7 the models discussed in this section have beenfitted to the graph given in the standard. The data from the standard is notincluded in this figure since some features, notably the initial delay visiblein the standard is not reproduced in either model. The different modelsalso give quite different quantitative behaviour in the region 2.5− 25 ns. InSection 3.1 we propose a new scheme for modelling this type of functionsand in Section 3.3 we fit this model to the IEC 61000-4-2 standard currentand some experimental data.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
·10−8
0
2
4
6
8
10
12
14
t [s]
i(t)
[A]
Two Heidler, [47] Two Heidler, [181]
Pulse binomial, [156] Exponential polynomial, [171]
Two Gaussians, [163] Four exponential, [88]
Figure 1.7: Functions representing the Standard ESD current waveshape for 4kV.
The model given in Section 3.1 is also fitted to both lightning dischargecurrent from the standard and from measured data in Section 3.2.
50
1.5. SUMMARIES OF PAPERS
1.5 Summaries of papers
Paper A
In this paper we examine the extreme points of the Vandermonde determi-nant in three or more dimensions. The paper discusses the three-dimensionalcase, see Section 2.1.2, and gives a more detailed description of the methodused to solve the n-dimensional problem from [158], see Section 2.2.1. Theextreme points are given in terms of roots of rescaled Hermite polynomialsand explicit expressions are given for dimensions three to seven. The resultsare also visualized in three to seven dimensions by using symmetries of theanswers to project all the extreme points onto a two-dimensional plane, seeSection 2.2.2. There is also a brief discussion on optimising the generalisedVandermonde determinant in three dimensions, see Section 2.1.1. The thesisauthor contributed primarily to the derivation of some of the recursive prop-erties of the Vandermonde determinant and its derivatives and to a lesserextent to the visualisation aspects of the problem.
Paper B
Here the Vandermonde determinant is optimised over the three-dimensionaltorus, see Section 2.1.3, and the sphere defined by the p - norm in n di-mensions, see Section 2.2.3. Main focus is on optimisation over the cubethat corresponds to p = ∞. The thesis author contributed primarily to theexamination of the torus.
Paper C
The value of the Vandermonde determinant is optimized over the ellipsoidand cylinder in three dimensions, see Section 2.1.5 and 2.1.6. Lagrange mul-tipliers are used to find a system of polynomial equations which give the localextreme points as its solutions. Using Grobner basis and other techniquesthe extreme points are given either explicitly or as roots of polynomials inone variable. The behaviour of the Vandermonde determinant is also pre-sented visually in some interesting cases. The method is also extended tosurfaces defined by homogeneous polynomials, see Section 2.1.7. Finally thepaper discusses the extreme points on sphere defined by the p - norm (pri-marily p = 4). The thesis author primarily contributed to the examinationof the ellipsoid, cylinder and surfaces defined by homogenous polynomials.
51
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and has been used for design of simple electric circuits which can be usedto simulate ESD currents.
One of the most commonly used ESD standard currents is the IEC 61000-4-2 current that represent a typical electrostatic discharge generated by thehuman body [73]. In the IEC 61000-4-2 standard [73] this current is given bya graphical representation, see Figure 3.10, together with some constraints,see page 122. In Figure 1.7 the models discussed in this section have beenfitted to the graph given in the standard. The data from the standard is notincluded in this figure since some features, notably the initial delay visiblein the standard is not reproduced in either model. The different modelsalso give quite different quantitative behaviour in the region 2.5− 25 ns. InSection 3.1 we propose a new scheme for modelling this type of functionsand in Section 3.3 we fit this model to the IEC 61000-4-2 standard currentand some experimental data.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
·10−8
0
2
4
6
8
10
12
14
t [s]
i(t)
[A]
Two Heidler, [47] Two Heidler, [181]
Pulse binomial, [156] Exponential polynomial, [171]
Two Gaussians, [163] Four exponential, [88]
Figure 1.7: Functions representing the Standard ESD current waveshape for 4kV.
The model given in Section 3.1 is also fitted to both lightning dischargecurrent from the standard and from measured data in Section 3.2.
50
1.5. SUMMARIES OF PAPERS
1.5 Summaries of papers
Paper A
In this paper we examine the extreme points of the Vandermonde determi-nant in three or more dimensions. The paper discusses the three-dimensionalcase, see Section 2.1.2, and gives a more detailed description of the methodused to solve the n-dimensional problem from [158], see Section 2.2.1. Theextreme points are given in terms of roots of rescaled Hermite polynomialsand explicit expressions are given for dimensions three to seven. The resultsare also visualized in three to seven dimensions by using symmetries of theanswers to project all the extreme points onto a two-dimensional plane, seeSection 2.2.2. There is also a brief discussion on optimising the generalisedVandermonde determinant in three dimensions, see Section 2.1.1. The thesisauthor contributed primarily to the derivation of some of the recursive prop-erties of the Vandermonde determinant and its derivatives and to a lesserextent to the visualisation aspects of the problem.
Paper B
Here the Vandermonde determinant is optimised over the three-dimensionaltorus, see Section 2.1.3, and the sphere defined by the p - norm in n di-mensions, see Section 2.2.3. Main focus is on optimisation over the cubethat corresponds to p = ∞. The thesis author contributed primarily to theexamination of the torus.
Paper C
The value of the Vandermonde determinant is optimized over the ellipsoidand cylinder in three dimensions, see Section 2.1.5 and 2.1.6. Lagrange mul-tipliers are used to find a system of polynomial equations which give the localextreme points as its solutions. Using Grobner basis and other techniquesthe extreme points are given either explicitly or as roots of polynomials inone variable. The behaviour of the Vandermonde determinant is also pre-sented visually in some interesting cases. The method is also extended tosurfaces defined by homogeneous polynomials, see Section 2.1.7. Finally thepaper discusses the extreme points on sphere defined by the p - norm (pri-marily p = 4). The thesis author primarily contributed to the examinationof the ellipsoid, cylinder and surfaces defined by homogenous polynomials.
51
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Paper D
This paper is a detailed description and derivation of some properties ofthe analytically extended function (AEF) and a scheme for how it can beused in approximation of lightning discharge currents, see sections 3.1.1 and3.2.2. Lightning discharge currents are classified in the IEC 62305-1 Stan-dard into waveshapes representing important observed phenomena. Thesewaveshapes are approximated with mathematical functions in order to beused in lightning discharge models for ensuring electromagnetic compatibil-ity. A general framework for estimating the parameters of the AEF usingthe Marquardt least-squares method (MLSM) for a waveform with an arbi-trary (finite) number of peaks as well as for the given charge transfer andspecific energy is described, see sections 1.3.3, 3.2 and 3.2.3. This frameworkis used to find parameters for some single-peak waveshapes and advantagesand disadvantages of the approach are discussed, see Section 3.2.6. The the-sis author contributed with the p-peak formulation of the AEF, modificationto the MLSM and basic software for fitting the AEF to data.
Paper E
In this paper is an examination of how the analytically extended function(AEF) can be used to approximate multi-peaked lightning current wave-forms. A general framework for estimating the parameters of the AEF usingthe Marquardt least-squares method (MLSM) for a waveform with an arbi-trary (finite) number of peaks is presented, see Section 3.2. This frameworkis used to find parameters for some waveforms, such as lightning currentsfrom the IEC 62305-1 Standard and recorded lightning current data, see Sec-tion 3.2.6. The thesis author contributed with improved software for fittingthe AEF to the more complicated waveforms (compared to Paper D).
Paper F
The multi-peaked analytically extended function (AEF) is used in this pa-per for representation of electrostatic discharge (ESD) currents. In order tominimize unstable behaviour and the number of free parameters the expo-nents of the AEF are chosen from an arithmetic sequence. The function isfitted by interpolating data chosen according to a D-optimal design. ESDcurrent modelling is illustrated through two examples: an approximation ofthe IEC Standard 61000-4-2 waveshape, and a representation of some mea-sured ESD current. The contents of this paper is in Section 3.3. The thesisauthor contributed with the derivation of the D-optimal design, motivatingits use as well as software for fitting the AEF according to the design.
52
Chapter 2
Extreme points of theVandermonde determinant
This chapter is based on Papers A, B and C:
Paper A. Karl Lundengard, Jonas Osterberg and Sergei Silvestrov. Extremepoints of the Vandermonde determinant on the sphere and some limitsinvolving the generalized Vandermonde determinant.Preprint: arXiv:1312.6193 [math.ca], 2013.
Paper B. Karl Lundengard, Jonas Osterberg, and Sergei Silvestrov. Optimiza-tion of the determinant of the Vandermonde matrix and related matri-ces. In AIP Conference Proceedings 1637, ICNPAA, Narvik, Norway,pages 627–636, 2014.
Paper C. Karl Lundengard, Jonas Osterberg, and Sergei Silvestrov. Optimiza-tion of the determinant of the Vandermonde matrix on the sphereand related surfaces. In Christos H Skiadas, editor, ASMDA 2015Proceedings: 16th Applied Stochastic Models and Data Analysis In-ternational Conference with 4th Demographics 2015 Workshop, pages637–648. ISAST: International Society for the Advancement of Scienceand Technology, 2015.
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Paper D
This paper is a detailed description and derivation of some properties ofthe analytically extended function (AEF) and a scheme for how it can beused in approximation of lightning discharge currents, see sections 3.1.1 and3.2.2. Lightning discharge currents are classified in the IEC 62305-1 Stan-dard into waveshapes representing important observed phenomena. Thesewaveshapes are approximated with mathematical functions in order to beused in lightning discharge models for ensuring electromagnetic compatibil-ity. A general framework for estimating the parameters of the AEF usingthe Marquardt least-squares method (MLSM) for a waveform with an arbi-trary (finite) number of peaks as well as for the given charge transfer andspecific energy is described, see sections 1.3.3, 3.2 and 3.2.3. This frameworkis used to find parameters for some single-peak waveshapes and advantagesand disadvantages of the approach are discussed, see Section 3.2.6. The the-sis author contributed with the p-peak formulation of the AEF, modificationto the MLSM and basic software for fitting the AEF to data.
Paper E
In this paper is an examination of how the analytically extended function(AEF) can be used to approximate multi-peaked lightning current wave-forms. A general framework for estimating the parameters of the AEF usingthe Marquardt least-squares method (MLSM) for a waveform with an arbi-trary (finite) number of peaks is presented, see Section 3.2. This frameworkis used to find parameters for some waveforms, such as lightning currentsfrom the IEC 62305-1 Standard and recorded lightning current data, see Sec-tion 3.2.6. The thesis author contributed with improved software for fittingthe AEF to the more complicated waveforms (compared to Paper D).
Paper F
The multi-peaked analytically extended function (AEF) is used in this pa-per for representation of electrostatic discharge (ESD) currents. In order tominimize unstable behaviour and the number of free parameters the expo-nents of the AEF are chosen from an arithmetic sequence. The function isfitted by interpolating data chosen according to a D-optimal design. ESDcurrent modelling is illustrated through two examples: an approximation ofthe IEC Standard 61000-4-2 waveshape, and a representation of some mea-sured ESD current. The contents of this paper is in Section 3.3. The thesisauthor contributed with the derivation of the D-optimal design, motivatingits use as well as software for fitting the AEF according to the design.
52
Chapter 2
Extreme points of theVandermonde determinant
This chapter is based on Papers A, B and C:
Paper A. Karl Lundengard, Jonas Osterberg and Sergei Silvestrov. Extremepoints of the Vandermonde determinant on the sphere and some limitsinvolving the generalized Vandermonde determinant.Preprint: arXiv:1312.6193 [math.ca], 2013.
Paper B. Karl Lundengard, Jonas Osterberg, and Sergei Silvestrov. Optimiza-tion of the determinant of the Vandermonde matrix and related matri-ces. In AIP Conference Proceedings 1637, ICNPAA, Narvik, Norway,pages 627–636, 2014.
Paper C. Karl Lundengard, Jonas Osterberg, and Sergei Silvestrov. Optimiza-tion of the determinant of the Vandermonde matrix on the sphereand related surfaces. In Christos H Skiadas, editor, ASMDA 2015Proceedings: 16th Applied Stochastic Models and Data Analysis In-ternational Conference with 4th Demographics 2015 Workshop, pages637–648. ISAST: International Society for the Advancement of Scienceand Technology, 2015.
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
2.1 Extreme points of the Vandermonde determi-nant and related determinants on various sur-faces in three dimensions
In this chapter we will discuss how to optimize the determinant of the Van-dermonde matrix and some related determinants over various surfaces inthree dimensions and the results will be visualized.
2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions
This section is based on Section 1.1 of Paper A
In this section we plot the values of the determinant
v3(x3) = (x3 − x2)(x3 − x1)(x2 − x1),
and also the generalized Vandermonde determinant g3(x3,a3) for three dif-ferent choices of a3 over the unit sphere x
21+x22+x23 = 1 in R3. Our plots are
over the unit sphere but the determinant exhibits the same general behaviorover centered spheres of any radius. This follows directly from (1.4) andthat exactly one element from each row appears in the determinant. Forany scalar c we get
gn(cxn,an) =
(n∏
i=1
cai
)gn(xn,an),
which for vn becomes
vn(cxn) = cn(n−1)
2 vn(xn), (19)
and so the values over different radii differ only by a constant factor.
In Figure 2.1 value of v3(x3) has been plotted over the unit sphere andthe curves where the determinant vanishes are traced as black lines. Thecoordinates in Figure 2.1 (b) are related to x3 by
x3 =
2 0 1−1 1 1−1 −1 1
1/√6 0 0
0 1/√2 0
0 0 1/√3
t, (20)
55
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
2.1 Extreme points of the Vandermonde determi-nant and related determinants on various sur-faces in three dimensions
In this chapter we will discuss how to optimize the determinant of the Van-dermonde matrix and some related determinants over various surfaces inthree dimensions and the results will be visualized.
2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions
This section is based on Section 1.1 of Paper A
In this section we plot the values of the determinant
v3(x3) = (x3 − x2)(x3 − x1)(x2 − x1),
and also the generalized Vandermonde determinant g3(x3,a3) for three dif-ferent choices of a3 over the unit sphere x
21+x22+x23 = 1 in R3. Our plots are
over the unit sphere but the determinant exhibits the same general behaviorover centered spheres of any radius. This follows directly from (1.4) andthat exactly one element from each row appears in the determinant. Forany scalar c we get
gn(cxn,an) =
(n∏
i=1
cai
)gn(xn,an),
which for vn becomes
vn(cxn) = cn(n−1)
2 vn(xn), (19)
and so the values over different radii differ only by a constant factor.
In Figure 2.1 value of v3(x3) has been plotted over the unit sphere andthe curves where the determinant vanishes are traced as black lines. Thecoordinates in Figure 2.1 (b) are related to x3 by
x3 =
2 0 1−1 1 1−1 −1 1
1/√6 0 0
0 1/√2 0
0 0 1/√3
t, (20)
55
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
(a) Plot with respect tothe regular x-basis.
(b) Plot with respect tothe t-basis, see (20).
(c) Plot with respectto parametrization(21).
Figure 2.1: Plot of v3(x3) over the unit sphere.
where the columns in the product of the two matrices are the basis vectors inR3. The unit sphere in R3 can also be described using spherical coordinates.In Figure 2.1 (c) the following parametrization was used.
t(θ, φ) =
cos(φ) sin(θ)
sin(φ)cos(φ) cos(θ)
. (21)
We will use this t-basis and spherical parametrization throughout this sec-tion.
From the plots in Figure 2.1 it can be seen that the number of extremepoints for v3 over the unit sphere seem to be 6 = 3!. It can also been seenthat all extreme points seem to lie in the plane through the origin thatis orthogonal to an apparent symmetry axis in the direction (1, 1, 1), thedirection of t3. We will see later that the extreme points for vn indeed lie in
the hyperplanen∑
i=1
xi = 0 for all n, see Theorem 2.2, and the total number
of extreme points for vn equals n!, see Remark 2.1.
The black lines where v3(x3) vanishes are actually the intersections be-tween the sphere and the three planes x3 − x1 = 0, x3 − x2 = 0 andx2 − x1 = 0, as these differences appear as factors in v3(x3).
We will see later on that the extreme points are the six points acquiredfrom permuting the coordinates in
x3 =1√2(−1, 0, 1) .
56
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
For reasons that will become clear in Section 2.2.1 it is also useful to thinkabout these coordinates as the roots of the polynomial
P3(x) = x3 − 1
2x.
So far we have only considered the behavior of v3(x3), that is g3(x3,a3)with a3 = (0, 1, 2). We now consider three generalized Vandermonde de-terminants, namely g3 with a3 = (0, 1, 3), a3 = (0, 2, 3) and a3 = (1, 2, 3).These three determinants show increasingly more structure and they all havea neat formula in terms of v3 and the elementary symmetric polynomials
ekn = ek(x1, · · · , xn) =∑
1≤i1<i2<···<ik≤n
xi1xi2 · · ·xik ,
where we will simply use ek whenever n is clear from the context.
(a) Plot with respect tothe regular x-basis.
(b) Plot with respect tothe t-basis, see (20).
(c) Plot with respect toangles given in (21).
Figure 2.2: Plot of g3(x3, (0, 1, 3)) over the unit sphere.
In Figure 2.2 we see the determinant
g3(x3, (0, 1, 3)) =
∣∣∣∣∣∣1 1 1x1 x2 x3x31 x32 x33
∣∣∣∣∣∣= v3(x3)e1,
plotted over the unit sphere. The expression v3(x3)e1 is easy to derive, thev3(x3) is there since the determinant must vanish whenever any two columnsare equal, which is exactly what the Vandermonde determinant expresses.The e1 follows by a simple polynomial division. As can be seen in the plotswe have an extra black circle where the determinant vanishes compared toFigure 2.1. This circle lies in the plane e1 = x1 + x2 + x3 = 0 where we
57
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
(a) Plot with respect tothe regular x-basis.
(b) Plot with respect tothe t-basis, see (20).
(c) Plot with respectto parametrization(21).
Figure 2.1: Plot of v3(x3) over the unit sphere.
where the columns in the product of the two matrices are the basis vectors inR3. The unit sphere in R3 can also be described using spherical coordinates.In Figure 2.1 (c) the following parametrization was used.
t(θ, φ) =
cos(φ) sin(θ)
sin(φ)cos(φ) cos(θ)
. (21)
We will use this t-basis and spherical parametrization throughout this sec-tion.
From the plots in Figure 2.1 it can be seen that the number of extremepoints for v3 over the unit sphere seem to be 6 = 3!. It can also been seenthat all extreme points seem to lie in the plane through the origin thatis orthogonal to an apparent symmetry axis in the direction (1, 1, 1), thedirection of t3. We will see later that the extreme points for vn indeed lie in
the hyperplanen∑
i=1
xi = 0 for all n, see Theorem 2.2, and the total number
of extreme points for vn equals n!, see Remark 2.1.
The black lines where v3(x3) vanishes are actually the intersections be-tween the sphere and the three planes x3 − x1 = 0, x3 − x2 = 0 andx2 − x1 = 0, as these differences appear as factors in v3(x3).
We will see later on that the extreme points are the six points acquiredfrom permuting the coordinates in
x3 =1√2(−1, 0, 1) .
56
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
For reasons that will become clear in Section 2.2.1 it is also useful to thinkabout these coordinates as the roots of the polynomial
P3(x) = x3 − 1
2x.
So far we have only considered the behavior of v3(x3), that is g3(x3,a3)with a3 = (0, 1, 2). We now consider three generalized Vandermonde de-terminants, namely g3 with a3 = (0, 1, 3), a3 = (0, 2, 3) and a3 = (1, 2, 3).These three determinants show increasingly more structure and they all havea neat formula in terms of v3 and the elementary symmetric polynomials
ekn = ek(x1, · · · , xn) =∑
1≤i1<i2<···<ik≤n
xi1xi2 · · ·xik ,
where we will simply use ek whenever n is clear from the context.
(a) Plot with respect tothe regular x-basis.
(b) Plot with respect tothe t-basis, see (20).
(c) Plot with respect toangles given in (21).
Figure 2.2: Plot of g3(x3, (0, 1, 3)) over the unit sphere.
In Figure 2.2 we see the determinant
g3(x3, (0, 1, 3)) =
∣∣∣∣∣∣1 1 1x1 x2 x3x31 x32 x33
∣∣∣∣∣∣= v3(x3)e1,
plotted over the unit sphere. The expression v3(x3)e1 is easy to derive, thev3(x3) is there since the determinant must vanish whenever any two columnsare equal, which is exactly what the Vandermonde determinant expresses.The e1 follows by a simple polynomial division. As can be seen in the plotswe have an extra black circle where the determinant vanishes compared toFigure 2.1. This circle lies in the plane e1 = x1 + x2 + x3 = 0 where we
57
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
previously found the extreme points of v3(x3) and thus doubles the numberof extreme points to 2 · 3!.
A similar treatment can be made of the remaining two generalized de-terminants that we are interested in, plotted in the following two figures.
(a) Plot with respect tothe regular x-basis.
(b) Plot with respect tothe t-basis, see (20).
(c) Plot with respect toangles given in (21).
Figure 2.3: Plot of g3(x3, (0, 2, 3)) over the unit sphere.
(a) Plot with respect tothe regular x-basis.
(b) Plot with respect tothe t-basis, see (20).
(c) Plot with respect toangles given in (21).
Figure 2.4: Plot of g3(x3, (1, 2, 3)) over the unit sphere.
The four determinants treated so far are collected in Table 2.1. Deriva-tion of these determinants is straight forward. We note that all but one ofthem vanish on a set of planes through the origin. For a = (0, 2, 3) we havethe usual Vandermonde planes but the intersection of e2 = 0 and the unitsphere occur at two circles.
x1x2 + x1x3 + x2x3 =1
2
((x1 + x2 + x3)
2 − (x21 + x22 + x23))
=1
2
((x1 + x2 + x3)
2 − 1)=
1
2(x1 + x2 + x3 + 1) (x1 + x2 + x3 − 1) ,
58
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
a3 g3(x3,a3)
(0, 1, 2) v3(x3)e0 = (x3 − x2)(x3 − x1)(x2 − x1)(0, 1, 3) v3(x3)e1 = (x3 − x2)(x3 − x1)(x2 − x1)(x1 + x2 + x3)(0, 2, 3) v3(x3)e2 = (x3 − x2)(x3 − x1)(x2 − x1)(x1x2 + x1x3 + x2x3)(1, 2, 3) v3(x3)e3 = (x3 − x2)(x3 − x1)(x2 − x1)x1x2x3
Table 2.1: Table of some determinants of generalized Vandermonde matrices.
and so g3(x3, (0, 2, 3)) vanish on the sphere on two circles lying on the planesx1 + x2 + x3 + 1 = 0 and x1 + x2 + x3 − 1 = 0. These can be seen in Figure2.3 as the two black circles perpendicular to the direction (1, 1, 1).
Note also that while v3 and g3(x3, (0, 1, 3)) have the same absolute valueon all their respective local extreme points (by symmetry) we have that bothg3(x3, (0, 2, 3)) and g3(x3, (1, 2, 3)) have different absolute values for some oftheir respective extreme points.
2.1.2 Extreme points of the Vandermonde determinant onthe three-dimensional unit sphere
This section is based on Section 2.2 of Paper A
It is fairly simple to describe v3(x3) on the circle that is formed by theintersection of the unit sphere and the plane x1 + x2 + x3 = 0. UsingRodrigues’ rotation formula to rotate a point, x, around the axis 1√
3(1, 1, 1)
with the angle θ will give the rotation matrix
Rθ =1
3
2 cos(θ) + 1 1− cos(θ)−√3 sin(θ) 1− cos(θ)+
√3 sin(θ)
1− cos(θ)+√3 sin(θ) 2 cos(θ) + 1 1− cos(θ)−
√3 sin(θ)
1− cos(θ)−√3 sin(θ) 1− cos(θ)+
√3 sin(θ) 2 cos(θ) + 1
.
A point which already lies on S2 can then be rotated to any other pointon S2 by letting Rθ act on the point. Choosing the point x = 1√
2(−1, 0, 1)
gives the Vandermonde determinant a convenient form on the circle since:
Rθx =1√6
−√3 cos(θ)− sin(θ)−2 sin(θ)√
3 cos(θ) + sin(θ)
,
59
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
previously found the extreme points of v3(x3) and thus doubles the numberof extreme points to 2 · 3!.
A similar treatment can be made of the remaining two generalized de-terminants that we are interested in, plotted in the following two figures.
(a) Plot with respect tothe regular x-basis.
(b) Plot with respect tothe t-basis, see (20).
(c) Plot with respect toangles given in (21).
Figure 2.3: Plot of g3(x3, (0, 2, 3)) over the unit sphere.
(a) Plot with respect tothe regular x-basis.
(b) Plot with respect tothe t-basis, see (20).
(c) Plot with respect toangles given in (21).
Figure 2.4: Plot of g3(x3, (1, 2, 3)) over the unit sphere.
The four determinants treated so far are collected in Table 2.1. Deriva-tion of these determinants is straight forward. We note that all but one ofthem vanish on a set of planes through the origin. For a = (0, 2, 3) we havethe usual Vandermonde planes but the intersection of e2 = 0 and the unitsphere occur at two circles.
x1x2 + x1x3 + x2x3 =1
2
((x1 + x2 + x3)
2 − (x21 + x22 + x23))
=1
2
((x1 + x2 + x3)
2 − 1)=
1
2(x1 + x2 + x3 + 1) (x1 + x2 + x3 − 1) ,
58
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
a3 g3(x3,a3)
(0, 1, 2) v3(x3)e0 = (x3 − x2)(x3 − x1)(x2 − x1)(0, 1, 3) v3(x3)e1 = (x3 − x2)(x3 − x1)(x2 − x1)(x1 + x2 + x3)(0, 2, 3) v3(x3)e2 = (x3 − x2)(x3 − x1)(x2 − x1)(x1x2 + x1x3 + x2x3)(1, 2, 3) v3(x3)e3 = (x3 − x2)(x3 − x1)(x2 − x1)x1x2x3
Table 2.1: Table of some determinants of generalized Vandermonde matrices.
and so g3(x3, (0, 2, 3)) vanish on the sphere on two circles lying on the planesx1 + x2 + x3 + 1 = 0 and x1 + x2 + x3 − 1 = 0. These can be seen in Figure2.3 as the two black circles perpendicular to the direction (1, 1, 1).
Note also that while v3 and g3(x3, (0, 1, 3)) have the same absolute valueon all their respective local extreme points (by symmetry) we have that bothg3(x3, (0, 2, 3)) and g3(x3, (1, 2, 3)) have different absolute values for some oftheir respective extreme points.
2.1.2 Extreme points of the Vandermonde determinant onthe three-dimensional unit sphere
This section is based on Section 2.2 of Paper A
It is fairly simple to describe v3(x3) on the circle that is formed by theintersection of the unit sphere and the plane x1 + x2 + x3 = 0. UsingRodrigues’ rotation formula to rotate a point, x, around the axis 1√
3(1, 1, 1)
with the angle θ will give the rotation matrix
Rθ =1
3
2 cos(θ) + 1 1− cos(θ)−√3 sin(θ) 1− cos(θ)+
√3 sin(θ)
1− cos(θ)+√3 sin(θ) 2 cos(θ) + 1 1− cos(θ)−
√3 sin(θ)
1− cos(θ)−√3 sin(θ) 1− cos(θ)+
√3 sin(θ) 2 cos(θ) + 1
.
A point which already lies on S2 can then be rotated to any other pointon S2 by letting Rθ act on the point. Choosing the point x = 1√
2(−1, 0, 1)
gives the Vandermonde determinant a convenient form on the circle since:
Rθx =1√6
−√3 cos(θ)− sin(θ)−2 sin(θ)√
3 cos(θ) + sin(θ)
,
59
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
which gives
2v3(Rθx) = 2(√
3 cos(θ) + sin(θ))
(√3 cos(θ) + sin(θ) + 2 sin(θ)
)(−2 sin(θ) +
√3 cos(θ) + sin(θ)
)
=1√2
(4 cos(θ)3 − 3 cos(θ)
)=
1√2cos(3θ).
Note that the final equality follows from cos(nθ) = Tn(cos(θ)) where Tn isthe nth Chebyshev polynomial of the first kind. From formula (48) if followsthat P3(x) = T3(x) but for higher dimensions the relationship between theChebyshev polynomials and Pn is not as simple.Finding the maximum points for v3(x3) on this form is simple. The Van-dermonde determinant will be maximal when 3θ = 2nπ where n is someinteger. This gives three local maxima corresponding to θ1 = 0, θ2 = 2π
3and θ3 = 4π
3 . These points correspond to cyclic permutation of the coordi-nates of x = 1√
2(−1, 0, 1). Analogously the minimas for v3(x3) can be shown
to be a transposition followed by cyclic permutation of the coordinates of x.Thus any permutation of the coordinates of x correspond to a local extremepoint just like it was stated on page 56.
2.1.3 Optimisation of the Vandermonde determinant on thethree-dimensional torus
This section is based on page 627–630 in Paper B
There are two equivalent conditions that describe the three-dimensionaltorus with radii r2 and r1, T
2 = S1(r1)× S1(r2):
T2 =
x ∈ R3
∣∣∣∣g(x) =(r2 −
√x2 + y2
)2+ z2 − r21 = 0
, (22)
T2 =
x ∈ R3
∣∣∣∣h(x) =(x2 + y2 + z2 + r22 − r21
)2+ 4r2(x
2 + y2) = 0
.
(23)
The surface of the torus can also be parametrised as follows:
x = (r2 + r1 cos(φ)) cos(θ)
y = (r2 + r1 cos(φ)) sin(θ)
z = r1 sin(θ)
(24)
60
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
Lemma 2.1. For any x(φ, θ) such that
f(φ, θ) =∂h
∂x1
∣∣∣∣x=x(φ,θ)
+∂h
∂x2
∣∣∣∣x=x(φ,θ)
+∂h
∂x3
∣∣∣∣x=x(φ,θ)
= 0,
where h is the previously given implicit equation for T2,
h(x) =(x21 + x22 + x23 + r22 − r21
)2+ 4r2
(x21 + x22
),
then f(φ+ π, θ) = 0 as well unless x1 = x2 = 0.
Proof. Calculate and parametrise the partial derivatives of h:
∂h
∂x1= 8r2r1(r2 + r1 cos(φ)) cos(φ) cos(θ),
∂h
∂x2= 8r2r1(r2 + r1 cos(φ)) cos(φ) sin(θ),
∂h
∂x3= 8r2r1(r2 + r1 cos(φ)) sin(φ).
This gives
f(φ, θ) = 8r2r1(r2 + r1 cos(φ))(cos(φ)(cos(θ) + sin(θ)) + sin(φ))
unless cos(φ) = 0 or r2 + r1 cos(φ) = 0 the condition f(φ, θ) = 0 can berewritten as
cos(θ) + sin(θ) = − sin(φ)
cos(φ). (25)
Substituting (25) into the explicit expression for f(φ+ π, θ) gives
f(φ+ π, θ) = 8r2r1(r2 − r1 cos(φ))
(− cos(φ)
(− sin(φ)
cos(φ)
)− sin(φ)
)= 0.
If cos(θ) = 0 then sin(θ) = ±1 which gives f(φ, θ) = ±8r2r1 thus f(φ, θ) = 0if cos(θ) = 0. It remains to see what happens when r2+r1 cos(φ) = 0. Fromthe parametrization (24) it is clear that this corresponds to x1 = x2 = 0which means that v3 at any such point will be zero.
Using the method of Lagrange multipliers and the constraint g(x) = 0 in(23) gives the following conditions on stationary points for the Vandermonde
61
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
which gives
2v3(Rθx) = 2(√
3 cos(θ) + sin(θ))
(√3 cos(θ) + sin(θ) + 2 sin(θ)
)(−2 sin(θ) +
√3 cos(θ) + sin(θ)
)
=1√2
(4 cos(θ)3 − 3 cos(θ)
)=
1√2cos(3θ).
Note that the final equality follows from cos(nθ) = Tn(cos(θ)) where Tn isthe nth Chebyshev polynomial of the first kind. From formula (48) if followsthat P3(x) = T3(x) but for higher dimensions the relationship between theChebyshev polynomials and Pn is not as simple.Finding the maximum points for v3(x3) on this form is simple. The Van-dermonde determinant will be maximal when 3θ = 2nπ where n is someinteger. This gives three local maxima corresponding to θ1 = 0, θ2 = 2π
3and θ3 = 4π
3 . These points correspond to cyclic permutation of the coordi-nates of x = 1√
2(−1, 0, 1). Analogously the minimas for v3(x3) can be shown
to be a transposition followed by cyclic permutation of the coordinates of x.Thus any permutation of the coordinates of x correspond to a local extremepoint just like it was stated on page 56.
2.1.3 Optimisation of the Vandermonde determinant on thethree-dimensional torus
This section is based on page 627–630 in Paper B
There are two equivalent conditions that describe the three-dimensionaltorus with radii r2 and r1, T
2 = S1(r1)× S1(r2):
T2 =
x ∈ R3
∣∣∣∣g(x) =(r2 −
√x2 + y2
)2+ z2 − r21 = 0
, (22)
T2 =
x ∈ R3
∣∣∣∣h(x) =(x2 + y2 + z2 + r22 − r21
)2+ 4r2(x
2 + y2) = 0
.
(23)
The surface of the torus can also be parametrised as follows:
x = (r2 + r1 cos(φ)) cos(θ)
y = (r2 + r1 cos(φ)) sin(θ)
z = r1 sin(θ)
(24)
60
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
Lemma 2.1. For any x(φ, θ) such that
f(φ, θ) =∂h
∂x1
∣∣∣∣x=x(φ,θ)
+∂h
∂x2
∣∣∣∣x=x(φ,θ)
+∂h
∂x3
∣∣∣∣x=x(φ,θ)
= 0,
where h is the previously given implicit equation for T2,
h(x) =(x21 + x22 + x23 + r22 − r21
)2+ 4r2
(x21 + x22
),
then f(φ+ π, θ) = 0 as well unless x1 = x2 = 0.
Proof. Calculate and parametrise the partial derivatives of h:
∂h
∂x1= 8r2r1(r2 + r1 cos(φ)) cos(φ) cos(θ),
∂h
∂x2= 8r2r1(r2 + r1 cos(φ)) cos(φ) sin(θ),
∂h
∂x3= 8r2r1(r2 + r1 cos(φ)) sin(φ).
This gives
f(φ, θ) = 8r2r1(r2 + r1 cos(φ))(cos(φ)(cos(θ) + sin(θ)) + sin(φ))
unless cos(φ) = 0 or r2 + r1 cos(φ) = 0 the condition f(φ, θ) = 0 can berewritten as
cos(θ) + sin(θ) = − sin(φ)
cos(φ). (25)
Substituting (25) into the explicit expression for f(φ+ π, θ) gives
f(φ+ π, θ) = 8r2r1(r2 − r1 cos(φ))
(− cos(φ)
(− sin(φ)
cos(φ)
)− sin(φ)
)= 0.
If cos(θ) = 0 then sin(θ) = ±1 which gives f(φ, θ) = ±8r2r1 thus f(φ, θ) = 0if cos(θ) = 0. It remains to see what happens when r2+r1 cos(φ) = 0. Fromthe parametrization (24) it is clear that this corresponds to x1 = x2 = 0which means that v3 at any such point will be zero.
Using the method of Lagrange multipliers and the constraint g(x) = 0 in(23) gives the following conditions on stationary points for the Vandermonde
61
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
determinant on the surface of the three-dimensional torus
∂v3∂x
= λ∂g
∂x= λ
(r2 −
√x2 + y2
)√x2 + y2
x = λ 2 r1 cos(φ) cos(θ),
∂v3∂y
= λ∂g
∂y= λ
(r2 −
√x2 + y2
)√x2 + y2
y = λ 2 r1 cos(φ) sin(θ),
∂v3∂z
= λ∂g
∂z= λ 2 z = λ 2 r1 sin(φ),
(26)
where g(x) is defined by (22).
From equation (26) it is clear that if the partial derivatives of v3 at aparticular point on T2 corresponds to the coordinates a point on S2(2r1).It is also known (see Section 2.1.2) that the sum of all partial derivatives ofv3 vanishes,
∂v3∂x
+∂v3∂y
+∂v3∂z
= 0, (x, y, z) ∈ R3. (27)
Thus all points where the condition given in (26) is satisfied for λ = 0 canbe found in the intersection between S2(2r1) and the plane given by (27). Asimple way of finding this intersection is to finds some points that belongsto the intersection and rotate the point around the planes normal vector.
Using Rodrigues’ rotation formula to rotate a point, x, around the axis1√3(1, 1, 1) with the angle θ will give the rotation matrix
Rα=1
3
2 cos(α) + 1 1−cos(α)−√3 sin(α) 1−cos(α)+
√3 sin(α)
1−cos(α)+√3 sin(α) 2 cos(α) + 1 1−cos(α)−
√3 sin(α)
1−cos(α)−√3 sin(α) 1−cos(α)+
√3 sin(α) 2 cos(α) + 1
.
A point which already lies on S2 can then be rotated to any otherpoint on S2 by letting Rθ act on the point. Rotating the point x
(0, π4
)=
r1+r2√2
(1,−1, 0) gives the new point:
x(α) = Rαx(0,
π
4
)=
√2r13
3 cos(α) +√3 sin(α)
−3 cos(α) +√3 sin(α)
2√3 sin(α)
.
A rotated point, x(α) = (x, y, z), will correspond to two points on the torus,
62
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
x±, given by
x± =x
2
(1± r2
r1cos
(arcsin
(z
2r1
))−1),
y± =y
2
(1± r2
r1cos
(arcsin
(z
2r1
))−1),
z± =z
2.
Figures 2.5-2.7 illustrates this result for a regular, horn and spindle torus.
Figure 2.5: Plot of v3(x3) over a proper torus (r1 = 1, r2 = 3), 3D-plot with curvemarked (left), parametrised plot with curve marked (center), values ofv3(x(α)) along the curve (right).
Figure 2.6: Plot of v3(x3) over a horn torus (r1 = 1, r2 = 1), 3D-plot with curvemarked (left), parametrised plot with curve marked (center), values ofv3(x(α)) along the curve (right).
63
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
determinant on the surface of the three-dimensional torus
∂v3∂x
= λ∂g
∂x= λ
(r2 −
√x2 + y2
)√x2 + y2
x = λ 2 r1 cos(φ) cos(θ),
∂v3∂y
= λ∂g
∂y= λ
(r2 −
√x2 + y2
)√x2 + y2
y = λ 2 r1 cos(φ) sin(θ),
∂v3∂z
= λ∂g
∂z= λ 2 z = λ 2 r1 sin(φ),
(26)
where g(x) is defined by (22).
From equation (26) it is clear that if the partial derivatives of v3 at aparticular point on T2 corresponds to the coordinates a point on S2(2r1).It is also known (see Section 2.1.2) that the sum of all partial derivatives ofv3 vanishes,
∂v3∂x
+∂v3∂y
+∂v3∂z
= 0, (x, y, z) ∈ R3. (27)
Thus all points where the condition given in (26) is satisfied for λ = 0 canbe found in the intersection between S2(2r1) and the plane given by (27). Asimple way of finding this intersection is to finds some points that belongsto the intersection and rotate the point around the planes normal vector.
Using Rodrigues’ rotation formula to rotate a point, x, around the axis1√3(1, 1, 1) with the angle θ will give the rotation matrix
Rα=1
3
2 cos(α) + 1 1−cos(α)−√3 sin(α) 1−cos(α)+
√3 sin(α)
1−cos(α)+√3 sin(α) 2 cos(α) + 1 1−cos(α)−
√3 sin(α)
1−cos(α)−√3 sin(α) 1−cos(α)+
√3 sin(α) 2 cos(α) + 1
.
A point which already lies on S2 can then be rotated to any otherpoint on S2 by letting Rθ act on the point. Rotating the point x
(0, π4
)=
r1+r2√2
(1,−1, 0) gives the new point:
x(α) = Rαx(0,
π
4
)=
√2r13
3 cos(α) +√3 sin(α)
−3 cos(α) +√3 sin(α)
2√3 sin(α)
.
A rotated point, x(α) = (x, y, z), will correspond to two points on the torus,
62
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
x±, given by
x± =x
2
(1± r2
r1cos
(arcsin
(z
2r1
))−1),
y± =y
2
(1± r2
r1cos
(arcsin
(z
2r1
))−1),
z± =z
2.
Figures 2.5-2.7 illustrates this result for a regular, horn and spindle torus.
Figure 2.5: Plot of v3(x3) over a proper torus (r1 = 1, r2 = 3), 3D-plot with curvemarked (left), parametrised plot with curve marked (center), values ofv3(x(α)) along the curve (right).
Figure 2.6: Plot of v3(x3) over a horn torus (r1 = 1, r2 = 1), 3D-plot with curvemarked (left), parametrised plot with curve marked (center), values ofv3(x(α)) along the curve (right).
63
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Figure 2.7: Plot of v3(x3) over a spindle torus (r1 = 3, r2 = 1), 3D-plot with curvemarked (left), parametrised plot with curve marked (center), values ofv3(x(α)) along the curve (right).
The explicit expression for the curves are:
v3±(x(α)) =
√2 cos(α)
2(1 + 2 cos(α)2)√3 + 6 cos(α)2(
− 3√
3 + 6 cos(α)2r31 − 2√3 + 6 cos(α)2r31 cos(α)
2
+ 8√3 + 6 cos(α)2r31 cos(α)
4 ∓ 15r21r2 ∓ 6 cos(α)2r21r2
± 48 cos(α)4r21r2 − 7√3 + 6 cos(α)2r1r
22
+ 16 cos(α)2r22√3 + 6 cos(α)2)r1 ± 12 cos(α)2r32 ∓ 3r32
).
Note that as r2 = 0 the expression simplifies to v3±(x(α)) =1√2cos(3α)
which is the same expression that you would get if the same method wasused on the regular unit sphere.
2.1.4 Optimisation using Grobner bases
This section is based on Section 4 of Paper C
In this section we will find the extreme points of the Vandermonde de-terminant on a few different surfaces. This will be done using Lagrangemultipliers and Grobner bases but first we will make an observation aboutthe Vandermonde determinant that will be useful later.
Lemma 2.2. The Vandermonde determinant is a homogeneous polynomialof degree n(n−1)
2 .
64
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
Proof. Considering the expression for the Vandermonde determinant in The-
orem 1.2 the number of factor of vn(x) isn∑
i=1
i− 1 =n(n− 1)
2. Thus
vn(cx) = cn(n−1)
2 vn(x). (28)
Grobner bases together with algorithms to find them, and algorithmsfor solving a polynomial equation is an important tool that arises in manyapplications. One such application is the optimization of polynomials overaffine varieties through the method of Lagrange multipliers. We will heregive some main points and informal discussion on these methods as an in-troduction and to describe some notation.
Definition 2.1. ( [30]) Let f1, · · · , fm be polynomials in R[x1, · · · , xn]. Theaffine variety V (f1, · · · , fm) defined by f1, · · · , fm is the set of all points(x1, · · · , xn) ∈ Rn such that fi(x1, · · · , xn) = 0 for all 1 ≤ i ≤ m.
When n = 3 we will sometimes use the variables x, y, z instead ofx1, x2, x3. Affine varieties are this way the common zeros of a set of multi-variate polynomials. Such sets of polynomials will generate a greater set ofpolynomials [30] by
〈f1, · · · , fm〉 ≡
m∑i=1
hifi : h1, · · · , hm ∈ R[x1, · · · , xn]
,
and this larger set will define the same variety. But it will also define anideal (a set of polynomials that contains the zero-polynomial and is closedunder addition, and absorbs multiplication by any other polynomial) byI(f1, · · · , fm) = 〈f1, · · · , fm〉. A Grobner basis for this ideal is then a finiteset of polynomials g1, · · · , gk such that the ideal generated by the leadingterms of the polynomials g1, · · · , gk is the same ideal as that generated byall the leading terms of polynomials in I = 〈f1, · · · , fm〉.
In this paper we consider the optimization of the Vandermonde deter-minant vn(x) over surfaces defined by a polynomial equation on the form
sn(x1, · · · , xn ; p; a1, · · · , an) ≡n∑
i=1
ai|xi|p = 1, (29)
where we will select the constants ai and p to get ellipsoids in three di-mensions, cylinders in three dimensions, and spheres under the p-norm in
65
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Figure 2.7: Plot of v3(x3) over a spindle torus (r1 = 3, r2 = 1), 3D-plot with curvemarked (left), parametrised plot with curve marked (center), values ofv3(x(α)) along the curve (right).
The explicit expression for the curves are:
v3±(x(α)) =
√2 cos(α)
2(1 + 2 cos(α)2)√3 + 6 cos(α)2(
− 3√3 + 6 cos(α)2r31 − 2
√3 + 6 cos(α)2r31 cos(α)
2
+ 8√3 + 6 cos(α)2r31 cos(α)
4 ∓ 15r21r2 ∓ 6 cos(α)2r21r2
± 48 cos(α)4r21r2 − 7√3 + 6 cos(α)2r1r
22
+ 16 cos(α)2r22√3 + 6 cos(α)2)r1 ± 12 cos(α)2r32 ∓ 3r32
).
Note that as r2 = 0 the expression simplifies to v3±(x(α)) =1√2cos(3α)
which is the same expression that you would get if the same method wasused on the regular unit sphere.
2.1.4 Optimisation using Grobner bases
This section is based on Section 4 of Paper C
In this section we will find the extreme points of the Vandermonde de-terminant on a few different surfaces. This will be done using Lagrangemultipliers and Grobner bases but first we will make an observation aboutthe Vandermonde determinant that will be useful later.
Lemma 2.2. The Vandermonde determinant is a homogeneous polynomialof degree n(n−1)
2 .
64
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
Proof. Considering the expression for the Vandermonde determinant in The-
orem 1.2 the number of factor of vn(x) isn∑
i=1
i− 1 =n(n− 1)
2. Thus
vn(cx) = cn(n−1)
2 vn(x). (28)
Grobner bases together with algorithms to find them, and algorithmsfor solving a polynomial equation is an important tool that arises in manyapplications. One such application is the optimization of polynomials overaffine varieties through the method of Lagrange multipliers. We will heregive some main points and informal discussion on these methods as an in-troduction and to describe some notation.
Definition 2.1. ( [30]) Let f1, · · · , fm be polynomials in R[x1, · · · , xn]. Theaffine variety V (f1, · · · , fm) defined by f1, · · · , fm is the set of all points(x1, · · · , xn) ∈ Rn such that fi(x1, · · · , xn) = 0 for all 1 ≤ i ≤ m.
When n = 3 we will sometimes use the variables x, y, z instead ofx1, x2, x3. Affine varieties are this way the common zeros of a set of multi-variate polynomials. Such sets of polynomials will generate a greater set ofpolynomials [30] by
〈f1, · · · , fm〉 ≡
m∑i=1
hifi : h1, · · · , hm ∈ R[x1, · · · , xn]
,
and this larger set will define the same variety. But it will also define anideal (a set of polynomials that contains the zero-polynomial and is closedunder addition, and absorbs multiplication by any other polynomial) byI(f1, · · · , fm) = 〈f1, · · · , fm〉. A Grobner basis for this ideal is then a finiteset of polynomials g1, · · · , gk such that the ideal generated by the leadingterms of the polynomials g1, · · · , gk is the same ideal as that generated byall the leading terms of polynomials in I = 〈f1, · · · , fm〉.
In this paper we consider the optimization of the Vandermonde deter-minant vn(x) over surfaces defined by a polynomial equation on the form
sn(x1, · · · , xn ; p; a1, · · · , an) ≡n∑
i=1
ai|xi|p = 1, (29)
where we will select the constants ai and p to get ellipsoids in three di-mensions, cylinders in three dimensions, and spheres under the p-norm in
65
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
n dimensions. The cases of the ellipsoids and the cylinders are suitable forsolution by Grobner basis methods, but due to the existing symmetries forthe spheres other methods are more suitable, as provided in Section 2.2.4.
From (28) and the convexity of the interior of the sets defined by (29),under a suitable choice of the constant p and non-negative ai, it is easy
to see that the optimal value of vn onn∑
i=1
ai|xi|p ≤ 1 will be attained on
∑ni=1 ai|xi|p = 1. And so, by the method of Lagrange multipliers we have
that the minimal/maximal values of vn(x1, · · · , xn) on sn(x1, · · · , xn) ≤ 1will be attained at points such that ∂vn/∂xi − λ∂sn/∂xi = 0 for 1 ≤ i ≤ nand some constant λ and sn(x1, · · · , xn)− 1 = 0, [144].
For p = 2 the resulting set of equations will form a set of polynomials inλ, x1, · · · , xn. These polynomials will define an ideal over R[λ, x1, · · · , xn],and by finding a Grobner basis for this ideal we can use the especially niceproperties of Grobner bases to find analytical solutions to these problems,that is, to find roots for the polynomials in the computed basis.
2.1.5 Extreme points on the ellipsoid in three dimensions
This section is based on Section 5 of Paper C
In this section we will find the extreme points of the Vandermonde determi-nant on the three dimensional ellipsoid given by
ax2 + by2 + cz2 = 1 (30)
where a > 0, b > 0, c > 0.
Using the method of Lagrange multipliers together with (30) and somerewriting gives that all stationary points of the Vandermonde determinantlie in the variety
V = V(ax2 + by2 + cz2 − 1, ax+ by + cz,
ax(z − x)(y − x)− by(z − y)(y − x) + cz(z − y)(z − x)).
Computing a Grobner basis for V using the lexicographic order x > y > z
66
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
give the following three basis polynomials:
g1(z) =(a+ b)(a− b)2
−(4(a+ b)2(a+ c)(b+ c) + 3c2(a2 + ab+ b2) + 3c(a3 + b3)
)z2
+ 3c(a+ b+ c)(4(a+ b)(a+ c)(b+ c) + (a2 + b2)c+ (a+ b)c2
)z4
− c2(b+ c)(a+ c)(a+ b+ c)2z6, (31)
g2(y, z) =(2(a+ b)2(a+ c)(b+ c) + c(a2 + 2b2)(a+ b+ c) + 2bc2(a+ b)
)z
+ q1z5 − q2z
3 − b(a− b)(a+ b)(a+ b+ 3c)y, (32)
g3(x, z) =(2(a+ b)2(a+ c)(b+ c) + c(2a2 + b2)(a+ b+ c) + 2ac2(a+ b)
)z
− q1z5 + q2z
3 − a(a− b)(a+ b)(a+ b+ 3c)x, (33)
q1 = 9 c2(b+ c)(a+ c)(a+ b+ c)2,
q2 = 3c(a+ b+ c)(3a2b+ 4a2c+ 3ab2 + 6abc+ 4ac2 + 4b2c+ 4bc2).
This basis was calculated using software for symbolic computation [1].
Since g1 only depends on z and g2 and g3 are first degree polynomial iny and x respectively the stationary points can be found by finding the rootsof g1 and then calculate the corresponding x and y coordinates. A generalformula can be found in this case (since g1 only contains even powers of zit can be treated as a third degree polynomial) but it is quite cumbersomeand we will therefore not give it explicitly.
Lemma 2.3. The extreme points of v3 on an ellipsoid will have real coor-dinates.
Proof. The discriminant is a useful tool for determining how many real rootslow-level polynomials have. Following Irving [75] the discriminant, ∆(p), ofa third degree polynomial p(x) = c0 + c1x+ c2x
2 + c3x3 is
∆ = 18c1c2c3c4 − 4c32c4 + c22c23 − 4c1c
33 − 27c21c
24
and if p(x) is non-negative then all roots will be real (but not necessarilydistinct). Since the first basis polynomial g1 only contains terms with evenexponents and is of degree 6 the polynomial g1 defined by g1(z
2) = g1(z)will be a polynomial of degree 3 whose roots are the square roots of g1.Calculating the discriminant of g1 gives
∆(g1) = 9(a− b)2(a+ b+ 3c)2(a+ b+ c)4abc3(32(a3b2 + a3c2 + a2b3 + a2c3 + b3c2 + b2c3) + 61abc(a+ b+ c)2
).
67
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
n dimensions. The cases of the ellipsoids and the cylinders are suitable forsolution by Grobner basis methods, but due to the existing symmetries forthe spheres other methods are more suitable, as provided in Section 2.2.4.
From (28) and the convexity of the interior of the sets defined by (29),under a suitable choice of the constant p and non-negative ai, it is easy
to see that the optimal value of vn onn∑
i=1
ai|xi|p ≤ 1 will be attained on
∑ni=1 ai|xi|p = 1. And so, by the method of Lagrange multipliers we have
that the minimal/maximal values of vn(x1, · · · , xn) on sn(x1, · · · , xn) ≤ 1will be attained at points such that ∂vn/∂xi − λ∂sn/∂xi = 0 for 1 ≤ i ≤ nand some constant λ and sn(x1, · · · , xn)− 1 = 0, [144].
For p = 2 the resulting set of equations will form a set of polynomials inλ, x1, · · · , xn. These polynomials will define an ideal over R[λ, x1, · · · , xn],and by finding a Grobner basis for this ideal we can use the especially niceproperties of Grobner bases to find analytical solutions to these problems,that is, to find roots for the polynomials in the computed basis.
2.1.5 Extreme points on the ellipsoid in three dimensions
This section is based on Section 5 of Paper C
In this section we will find the extreme points of the Vandermonde determi-nant on the three dimensional ellipsoid given by
ax2 + by2 + cz2 = 1 (30)
where a > 0, b > 0, c > 0.
Using the method of Lagrange multipliers together with (30) and somerewriting gives that all stationary points of the Vandermonde determinantlie in the variety
V = V(ax2 + by2 + cz2 − 1, ax+ by + cz,
ax(z − x)(y − x)− by(z − y)(y − x) + cz(z − y)(z − x)).
Computing a Grobner basis for V using the lexicographic order x > y > z
66
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
give the following three basis polynomials:
g1(z) =(a+ b)(a− b)2
−(4(a+ b)2(a+ c)(b+ c) + 3c2(a2 + ab+ b2) + 3c(a3 + b3)
)z2
+ 3c(a+ b+ c)(4(a+ b)(a+ c)(b+ c) + (a2 + b2)c+ (a+ b)c2
)z4
− c2(b+ c)(a+ c)(a+ b+ c)2z6, (31)
g2(y, z) =(2(a+ b)2(a+ c)(b+ c) + c(a2 + 2b2)(a+ b+ c) + 2bc2(a+ b)
)z
+ q1z5 − q2z
3 − b(a− b)(a+ b)(a+ b+ 3c)y, (32)
g3(x, z) =(2(a+ b)2(a+ c)(b+ c) + c(2a2 + b2)(a+ b+ c) + 2ac2(a+ b)
)z
− q1z5 + q2z
3 − a(a− b)(a+ b)(a+ b+ 3c)x, (33)
q1 = 9 c2(b+ c)(a+ c)(a+ b+ c)2,
q2 = 3c(a+ b+ c)(3a2b+ 4a2c+ 3ab2 + 6abc+ 4ac2 + 4b2c+ 4bc2).
This basis was calculated using software for symbolic computation [1].
Since g1 only depends on z and g2 and g3 are first degree polynomial iny and x respectively the stationary points can be found by finding the rootsof g1 and then calculate the corresponding x and y coordinates. A generalformula can be found in this case (since g1 only contains even powers of zit can be treated as a third degree polynomial) but it is quite cumbersomeand we will therefore not give it explicitly.
Lemma 2.3. The extreme points of v3 on an ellipsoid will have real coor-dinates.
Proof. The discriminant is a useful tool for determining how many real rootslow-level polynomials have. Following Irving [75] the discriminant, ∆(p), ofa third degree polynomial p(x) = c0 + c1x+ c2x
2 + c3x3 is
∆ = 18c1c2c3c4 − 4c32c4 + c22c23 − 4c1c
33 − 27c21c
24
and if p(x) is non-negative then all roots will be real (but not necessarilydistinct). Since the first basis polynomial g1 only contains terms with evenexponents and is of degree 6 the polynomial g1 defined by g1(z
2) = g1(z)will be a polynomial of degree 3 whose roots are the square roots of g1.Calculating the discriminant of g1 gives
∆(g1) = 9(a− b)2(a+ b+ 3c)2(a+ b+ c)4abc3(32(a3b2 + a3c2 + a2b3 + a2c3 + b3c2 + b2c3) + 61abc(a+ b+ c)2
).
67
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Figure 2.8: Illustration of the ellipsoid defined byx2
9+
y2
4+z2 = 0 with the extreme
points of the Vandermonde determinant marked. Displayed in Cartesiancoordinates on the right and in ellipsoidal coordinates on the left.
Since a, b and c are all positive numbers it is clear that ∆(g1) is non-negative. Furthermore, since a, b and c are positive numbers all terms in g1with odd powers have negative coefficients and all terms with even powershave positive coefficients. Thus if w < 0 then g1(w) > 0 and thus all rootsmust be positive.
An illustration of an ellipsoid and the extreme points of the Vandermondedeterminant on its surface is shown in Figure 2.8.
2.1.6 Extreme points on the cylinder in three dimensions
This section is based on Section 6 of Paper C
In this section we will examine the local extreme points on an infinitely longcylinder aligned with the x-axis in 3 dimensions. In this case we do not needto use Grobner basis techniques since the problem can be reduced to a onedimensional polynomial equation.
The cylinder is defined by
by2 + cz2 = 1, where b > 0, c > 0. (34)
68
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
Using the method of Lagrange multipliers gives the equation system
∂v3∂x
= 0,
∂v3∂y
= 2λby,
∂v3∂z
= 2λcz.
Taking the sum of each expression gives
by + cz = 0 ⇔ y = −c
bz. (35)
Combining (34) and (35) gives
(cb+ 1
)cz2 = 1 ⇒ z = ±
√b
c
1√b+ c
⇒ y = ∓√
c
b
1√b+ c
.
Thus the plane defined by (35) intersects with the cylinder along thelines
1 =
(x,
√c
b
1√b+ c
,−√
b
c
1√b+ c
)∣∣∣∣x ∈ R
= (x, r,−s)|x ∈ R ,
2 =
(x,−
√c
b
1√b+ c
,
√b
c
1√b+ c
)∣∣∣∣x ∈ R
= (x,−r, s)|x ∈ R .
Finding the stationary points for v3 along 1:
v3 (x, r,−s) =
(x2 +
1√b+ c
(√b
c−√
c
b
)x+
1
b+ c
)(r + s) ,
∂v3∂x
(x, r,−s) =
(2x+
1√b+ c
(√b
c−√
c
b
))(r + s) .
From this it follows that
∂v3∂x
(x, r,−s) = 0 ⇔ x =1
2√b+ c
(√c
b−√
b
c
).
Thus
x1 =1√b+ c
(1
2
(√c
b−√
b
c
),
√c
b,−
√b
c
)(36)
is the only stationary point on 1. An analogous argument shows that x2 =−x1 is the only stationary point on 2.
An illustration of where these points are placed on the cylinder is shownin Figure 2.9.
69
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Figure 2.8: Illustration of the ellipsoid defined byx2
9+
y2
4+z2 = 0 with the extreme
points of the Vandermonde determinant marked. Displayed in Cartesiancoordinates on the right and in ellipsoidal coordinates on the left.
Since a, b and c are all positive numbers it is clear that ∆(g1) is non-negative. Furthermore, since a, b and c are positive numbers all terms in g1with odd powers have negative coefficients and all terms with even powershave positive coefficients. Thus if w < 0 then g1(w) > 0 and thus all rootsmust be positive.
An illustration of an ellipsoid and the extreme points of the Vandermondedeterminant on its surface is shown in Figure 2.8.
2.1.6 Extreme points on the cylinder in three dimensions
This section is based on Section 6 of Paper C
In this section we will examine the local extreme points on an infinitely longcylinder aligned with the x-axis in 3 dimensions. In this case we do not needto use Grobner basis techniques since the problem can be reduced to a onedimensional polynomial equation.
The cylinder is defined by
by2 + cz2 = 1, where b > 0, c > 0. (34)
68
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
Using the method of Lagrange multipliers gives the equation system
∂v3∂x
= 0,
∂v3∂y
= 2λby,
∂v3∂z
= 2λcz.
Taking the sum of each expression gives
by + cz = 0 ⇔ y = −c
bz. (35)
Combining (34) and (35) gives
(cb+ 1
)cz2 = 1 ⇒ z = ±
√b
c
1√b+ c
⇒ y = ∓√
c
b
1√b+ c
.
Thus the plane defined by (35) intersects with the cylinder along thelines
1 =
(x,
√c
b
1√b+ c
,−√
b
c
1√b+ c
)∣∣∣∣x ∈ R
= (x, r,−s)|x ∈ R ,
2 =
(x,−
√c
b
1√b+ c
,
√b
c
1√b+ c
)∣∣∣∣x ∈ R
= (x,−r, s)|x ∈ R .
Finding the stationary points for v3 along 1:
v3 (x, r,−s) =
(x2 +
1√b+ c
(√b
c−√
c
b
)x+
1
b+ c
)(r + s) ,
∂v3∂x
(x, r,−s) =
(2x+
1√b+ c
(√b
c−√
c
b
))(r + s) .
From this it follows that
∂v3∂x
(x, r,−s) = 0 ⇔ x =1
2√b+ c
(√c
b−√
b
c
).
Thus
x1 =1√b+ c
(1
2
(√c
b−√
b
c
),
√c
b,−
√b
c
)(36)
is the only stationary point on 1. An analogous argument shows that x2 =−x1 is the only stationary point on 2.
An illustration of where these points are placed on the cylinder is shownin Figure 2.9.
69
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Figure 2.9: Illustration of the cylinder defined by y2 +16
25z2 = 1 with the extreme
points of the Vandermonde determinant marked. Displayed in Cartesiancoordinates on the right and in cylindrical coordinates on the left.
2.1.7 Optimizing the Vandermonde determinant on a surfacedefined by a homogeneous polynomial
This section is based on Section 7 of Paper C
When using Lagrange multipliers it can be desirable to not have to considerthe λ-parameter (the scaling between the gradient and direction given bythe constraint). We demonstrate a simple way to remove this parameterwhen the surface is defined by an homogeneous polynomial.
Lemma 2.4. Let g : R → R be a homogeneous polynomial such thatg(cx) = ckg(x) with k = n(n−1)
2 . If g(x) = 1, x ∈ Cn defines a continu-ous bounded surface then any point on the surface that is a stationary pointfor the Vandermonde determinant, z ∈ Cn, can be written as z = cy where
∂vn∂xi
∣∣∣∣x=y
=∂g
∂xi
∣∣∣∣x=y
, i ∈ 1, 2, . . . , n (37)
and c = g(y)−1k .
Proof. By the method of Lagrange multipliers the point y ∈ x ∈ Rn|g(x) =1 is a stationary point for the Vandermonde determinant if
∂vn∂xk
∣∣∣∣x=y
= λ∂g
∂xk
∣∣∣∣x=y
, k ∈ 1, 2, . . . , n
for some λ ∈ R.
70
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
The stationary points on the surface given by g(cx) = ck will be givenby
cn(n−1)
2∂vn∂xk
∣∣∣∣x=y
= ckλ∂g
∂xk
∣∣∣∣x=y
, k ∈ 1, 2, . . . , n
and if c is chosen such that λ = cn(1−n)
2 ck then the stationary points aredefined by
∂vn∂xk
=∂g
∂xk, k ∈ 1, 2, . . . , n.
Suppose that y ∈ x ∈ Rn|g(x) = ck is a stationary point for vn then
the point given by z = cy where c = g(y)−1k will be a stationary point
for the Vandermonde determinant and will lie on the surface defined byg(x) = 1.
Lemma 2.5. If z is a stationary point for the Vandermonde determinanton the surface g(x) = 1 where g(x) is a homogeneous polynomial then −z iseither a stationary point or does not lie on the surface.
Proof. Since g(−x) = (−1)kg(x) is either 1 or −1 then |vn(x)| = |vn(−x)|for any point, including z and the points in a neighbourhood around it whichmeans that if g(−x) = g(x) then the stationary points are preserved andotherwise the point will lie on the surface defined by g(x) = −1 instead ofg(x) = 1.
A well-known example of homogeneous polynomials are quadratic forms.If we let
g(x) = xaSx
then g(x) is a quadratic form which in turn is a homogeneous polynomialwith k = 2. If S is a positive definite matrix then g(x) = 1 defines anellipsoid. Here we will demonstrate the use of Lemma 2.4 to find the extremepoints on a rotated ellipsoid.
Consider the ellipsoid defined by
1
9x2 +
5
8y2 +
3
4yz +
5
8z2 = 1 (38)
then by Lemma 2 we can instead consider the points in the variety
V = V(− 2xy + 2xz + y2 − z2 − 2
9x,
− x2 + 2xy − 2yz + z2 − 5
4y − 3
4z,
− 2xz − y2 + 2yz + x2 − 3
4y − 5
4z).
71
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Figure 2.9: Illustration of the cylinder defined by y2 +16
25z2 = 1 with the extreme
points of the Vandermonde determinant marked. Displayed in Cartesiancoordinates on the right and in cylindrical coordinates on the left.
2.1.7 Optimizing the Vandermonde determinant on a surfacedefined by a homogeneous polynomial
This section is based on Section 7 of Paper C
When using Lagrange multipliers it can be desirable to not have to considerthe λ-parameter (the scaling between the gradient and direction given bythe constraint). We demonstrate a simple way to remove this parameterwhen the surface is defined by an homogeneous polynomial.
Lemma 2.4. Let g : R → R be a homogeneous polynomial such thatg(cx) = ckg(x) with k = n(n−1)
2 . If g(x) = 1, x ∈ Cn defines a continu-ous bounded surface then any point on the surface that is a stationary pointfor the Vandermonde determinant, z ∈ Cn, can be written as z = cy where
∂vn∂xi
∣∣∣∣x=y
=∂g
∂xi
∣∣∣∣x=y
, i ∈ 1, 2, . . . , n (37)
and c = g(y)−1k .
Proof. By the method of Lagrange multipliers the point y ∈ x ∈ Rn|g(x) =1 is a stationary point for the Vandermonde determinant if
∂vn∂xk
∣∣∣∣x=y
= λ∂g
∂xk
∣∣∣∣x=y
, k ∈ 1, 2, . . . , n
for some λ ∈ R.
70
2.1. EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS ON
VARIOUS SURFACES IN THREE DIMENSIONS
The stationary points on the surface given by g(cx) = ck will be givenby
cn(n−1)
2∂vn∂xk
∣∣∣∣x=y
= ckλ∂g
∂xk
∣∣∣∣x=y
, k ∈ 1, 2, . . . , n
and if c is chosen such that λ = cn(1−n)
2 ck then the stationary points aredefined by
∂vn∂xk
=∂g
∂xk, k ∈ 1, 2, . . . , n.
Suppose that y ∈ x ∈ Rn|g(x) = ck is a stationary point for vn then
the point given by z = cy where c = g(y)−1k will be a stationary point
for the Vandermonde determinant and will lie on the surface defined byg(x) = 1.
Lemma 2.5. If z is a stationary point for the Vandermonde determinanton the surface g(x) = 1 where g(x) is a homogeneous polynomial then −z iseither a stationary point or does not lie on the surface.
Proof. Since g(−x) = (−1)kg(x) is either 1 or −1 then |vn(x)| = |vn(−x)|for any point, including z and the points in a neighbourhood around it whichmeans that if g(−x) = g(x) then the stationary points are preserved andotherwise the point will lie on the surface defined by g(x) = −1 instead ofg(x) = 1.
A well-known example of homogeneous polynomials are quadratic forms.If we let
g(x) = xaSx
then g(x) is a quadratic form which in turn is a homogeneous polynomialwith k = 2. If S is a positive definite matrix then g(x) = 1 defines anellipsoid. Here we will demonstrate the use of Lemma 2.4 to find the extremepoints on a rotated ellipsoid.
Consider the ellipsoid defined by
1
9x2 +
5
8y2 +
3
4yz +
5
8z2 = 1 (38)
then by Lemma 2 we can instead consider the points in the variety
V = V(− 2xy + 2xz + y2 − z2 − 2
9x,
− x2 + 2xy − 2yz + z2 − 5
4y − 3
4z,
− 2xz − y2 + 2yz + x2 − 3
4y − 5
4z).
71
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Finding the Grobner basis of V gives
g1(z) = z(6z + 1)(260642z2 − 27436z + 697),
g2(y, z) = − 1138484256z3 − 127275604z2 + 16689841z + 6277879y,
g3(x, z) = 10246358304z3 + 1145480436z2 − 93707658z + 6277879x.
This system is not difficult to solve and the resulting points are:
p0 = (0, 0, 0),
p1 =
(0,
1
6,−1
6
),
p2 =
(45√2
361,− 1
19− 5
√2
722,1
19− 5
√2
722
),
p3 =
(45√2
361,− 1
19+
5√2
722,1
19+
5√2
722
).
The point p0 is an artifact of the rewrite and does not lie on any ellipsoidand can therefore be discarded. By Lemma 4 there are also three morestationary points p4 = −p1, p5 = −p2 and p6 = −p3. Rescaling each ofthese points according to Lemma 2 gives qi =
√g(pi) which are all points
on the ellipsoid defined by g(x) = 1. The result is illustrated in Figure 2.10.Note that this example gives a simple case with a Grobner basis that is
small and easy to find. Using this technique for other polynomials and inhigher dimensions can require significant computational resources.
2.2 Optimization of the Vandermondedeterminant on some n-dimensional surfaces
In this section we will consider the extreme points of the Vandermondedeterminant on the n-dimensional unit sphere in Rn. We want both to findan analytical solution and to identify some properties of the determinantthat can help us to visualize it in some area around the extreme points indimensions n > 3.
72
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Figure 2.10: Illustration of the ellipsoid defined by (38) with the extreme points of theVandermonde determinant marked. Displayed in Cartesian coordinateson the right and in ellipsoidal coordinates on the left.
2.2.1 The extreme points on the sphere given by roots of apolynomial
This section is based on Section 2.1 of Paper A
The extreme points of the Vandermonde determinant on the unit sphere inRn are known and given by Theorem 2.3 where we present a special case ofTheorem 6.7.3 in [158]. We will also provide a proof that is more explicitthan the one in [158] and that exposes more of the rich symmetric prop-erties of the Vandermonde determinant. For the sake of convenience someproperties related to the extreme points of the Vandermonde determinantdefined by real vectors xn will be presented before Theorem 2.3.
Theorem 2.1. For any 1 ≤ k ≤ n
∂vn∂xk
=
n∑i=1i =k
vn(xn)
xk − xi(39)
This theorem will be proven after the introduction of the following usefullemma:
Lemma 2.6. For any 1 ≤ k ≤ n− 1
∂vn∂xk
= − vn(xn)
xn − xk+
[n−1∏i=1
(xn − xi)
]∂vn−1
∂xk(40)
73
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Finding the Grobner basis of V gives
g1(z) = z(6z + 1)(260642z2 − 27436z + 697),
g2(y, z) = − 1138484256z3 − 127275604z2 + 16689841z + 6277879y,
g3(x, z) = 10246358304z3 + 1145480436z2 − 93707658z + 6277879x.
This system is not difficult to solve and the resulting points are:
p0 = (0, 0, 0),
p1 =
(0,
1
6,−1
6
),
p2 =
(45√2
361,− 1
19− 5
√2
722,1
19− 5
√2
722
),
p3 =
(45√2
361,− 1
19+
5√2
722,1
19+
5√2
722
).
The point p0 is an artifact of the rewrite and does not lie on any ellipsoidand can therefore be discarded. By Lemma 4 there are also three morestationary points p4 = −p1, p5 = −p2 and p6 = −p3. Rescaling each ofthese points according to Lemma 2 gives qi =
√g(pi) which are all points
on the ellipsoid defined by g(x) = 1. The result is illustrated in Figure 2.10.Note that this example gives a simple case with a Grobner basis that is
small and easy to find. Using this technique for other polynomials and inhigher dimensions can require significant computational resources.
2.2 Optimization of the Vandermondedeterminant on some n-dimensional surfaces
In this section we will consider the extreme points of the Vandermondedeterminant on the n-dimensional unit sphere in Rn. We want both to findan analytical solution and to identify some properties of the determinantthat can help us to visualize it in some area around the extreme points indimensions n > 3.
72
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Figure 2.10: Illustration of the ellipsoid defined by (38) with the extreme points of theVandermonde determinant marked. Displayed in Cartesian coordinateson the right and in ellipsoidal coordinates on the left.
2.2.1 The extreme points on the sphere given by roots of apolynomial
This section is based on Section 2.1 of Paper A
The extreme points of the Vandermonde determinant on the unit sphere inRn are known and given by Theorem 2.3 where we present a special case ofTheorem 6.7.3 in [158]. We will also provide a proof that is more explicitthan the one in [158] and that exposes more of the rich symmetric prop-erties of the Vandermonde determinant. For the sake of convenience someproperties related to the extreme points of the Vandermonde determinantdefined by real vectors xn will be presented before Theorem 2.3.
Theorem 2.1. For any 1 ≤ k ≤ n
∂vn∂xk
=
n∑i=1i =k
vn(xn)
xk − xi(39)
This theorem will be proven after the introduction of the following usefullemma:
Lemma 2.6. For any 1 ≤ k ≤ n− 1
∂vn∂xk
= − vn(xn)
xn − xk+
[n−1∏i=1
(xn − xi)
]∂vn−1
∂xk(40)
73
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and
∂vn∂xn
=n−1∑i=1
vn(xn)
xn − xi. (41)
Proof. Note that the determinant can be described recursively
vn(xn) =
[n−1∏i=1
(xn − xi)
] ∏1≤i<j≤n−1
(xj − xi)
=
[n−1∏i=1
(xn − xi)
]vn−1(xn−1). (42)
Formula (40) follows immediately from applying the differentiation formulafor products on (42). Formula (41) follows from (42), the differentiation rulefor products and that vn−1(xn−1) is independent of xn.
∂vn∂xn
=vn−1(xn−1)
xn − x1
n−1∏i=1
(xn − xi)
+ (xn − x1)∂
∂xn
(vn−1(xn−1)
xn − x1
n−1∏i=1
(xn − xi)
)
=vn(xn)
xn − x1+
vn(xn)
xn − x2
+ (xn − x1)(xn − x2)∂
∂xn
(vn(xn)
(xn − x1)(xn − x2)
)
=n−1∑i=1
vn(xn)
xn − xi+
[n−1∏i=1
(xn − xi)
]∂vn−1
∂xn=
n−1∑i=1
vn(xn)
xn − xi.
Proof of Theorem 2.1. Using Lemma 2.6 we can see that for k = n equation(39) follows immediately from (41). The case 1 ≤ k < n will be proved usinginduction. Using (40) gives
∂vn∂xk
= − vn(xn)
xn − xk+
[n−1∏i=1
(xn − xi)
]∂vn−1
∂xk.
74
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Supposing that formula (39) is true for n− 1 results in
∂vn∂xk
=− vn(xn)
xn − xk+
[n−1∏i=1
(xn − xi)
]n−1∑i=1i =k
vn−1(xn−1)
xk − xi
=vn(xn)
xk − xn+
n−1∑i=1i =k
vn(xn)
xk − xi=
n∑i=1i =k
vn(xn)
xk − xi.
Showing that (39) is true for n = 2 completes the proof
∂v2∂x1
=∂
∂x1(x2 − x1) = −1 =
x2 − x1x1 − x2
=
2∑i=1i =1
v2(x2)
x1 − xi
∂v2∂x2
=∂
∂x2(x2 − x1) = 1 =
x2 − x1x2 − x1
=
2∑i=1i =2
v2(x2)
x2 − xi.
Theorem 2.2. The extreme points of vn(xn) on the unit sphere can all befound in the hyperplane defined by
n∑i=1
xi = 0. (43)
This theorem will be proved after the introduction of the following usefullemma:
Lemma 2.7. For any n ≥ 2 the sum of the partial derivatives of vn(xn)will be zero.
n∑k=1
∂vn∂xk
= 0. (44)
Proof. This lemma is easily proven using Lemma 2.6 and induction:
n∑k=1
∂vn∂xk
=n−1∑k=1
(− vn(xn)
xn − xk+
[n−1∏i=1
(xn − xi)
]∂vn−1
∂xk
)+
n−1∑i=1
vn(xn)
xn − xi
=
[n−1∏i=1
(xn − xi)
]n−1∑k=1
∂vn−1
∂xk.
75
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and
∂vn∂xn
=n−1∑i=1
vn(xn)
xn − xi. (41)
Proof. Note that the determinant can be described recursively
vn(xn) =
[n−1∏i=1
(xn − xi)
] ∏1≤i<j≤n−1
(xj − xi)
=
[n−1∏i=1
(xn − xi)
]vn−1(xn−1). (42)
Formula (40) follows immediately from applying the differentiation formulafor products on (42). Formula (41) follows from (42), the differentiation rulefor products and that vn−1(xn−1) is independent of xn.
∂vn∂xn
=vn−1(xn−1)
xn − x1
n−1∏i=1
(xn − xi)
+ (xn − x1)∂
∂xn
(vn−1(xn−1)
xn − x1
n−1∏i=1
(xn − xi)
)
=vn(xn)
xn − x1+
vn(xn)
xn − x2
+ (xn − x1)(xn − x2)∂
∂xn
(vn(xn)
(xn − x1)(xn − x2)
)
=n−1∑i=1
vn(xn)
xn − xi+
[n−1∏i=1
(xn − xi)
]∂vn−1
∂xn=
n−1∑i=1
vn(xn)
xn − xi.
Proof of Theorem 2.1. Using Lemma 2.6 we can see that for k = n equation(39) follows immediately from (41). The case 1 ≤ k < n will be proved usinginduction. Using (40) gives
∂vn∂xk
= − vn(xn)
xn − xk+
[n−1∏i=1
(xn − xi)
]∂vn−1
∂xk.
74
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Supposing that formula (39) is true for n− 1 results in
∂vn∂xk
=− vn(xn)
xn − xk+
[n−1∏i=1
(xn − xi)
]n−1∑i=1i =k
vn−1(xn−1)
xk − xi
=vn(xn)
xk − xn+
n−1∑i=1i =k
vn(xn)
xk − xi=
n∑i=1i =k
vn(xn)
xk − xi.
Showing that (39) is true for n = 2 completes the proof
∂v2∂x1
=∂
∂x1(x2 − x1) = −1 =
x2 − x1x1 − x2
=
2∑i=1i =1
v2(x2)
x1 − xi
∂v2∂x2
=∂
∂x2(x2 − x1) = 1 =
x2 − x1x2 − x1
=
2∑i=1i =2
v2(x2)
x2 − xi.
Theorem 2.2. The extreme points of vn(xn) on the unit sphere can all befound in the hyperplane defined by
n∑i=1
xi = 0. (43)
This theorem will be proved after the introduction of the following usefullemma:
Lemma 2.7. For any n ≥ 2 the sum of the partial derivatives of vn(xn)will be zero.
n∑k=1
∂vn∂xk
= 0. (44)
Proof. This lemma is easily proven using Lemma 2.6 and induction:
n∑k=1
∂vn∂xk
=n−1∑k=1
(− vn(xn)
xn − xk+
[n−1∏i=1
(xn − xi)
]∂vn−1
∂xk
)+
n−1∑i=1
vn(xn)
xn − xi
=
[n−1∏i=1
(xn − xi)
]n−1∑k=1
∂vn−1
∂xk.
75
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Thus if equation (44) is true for n− 1 it is also true for n. Showing that theequation holds for n = 2 is very simple
∂v2∂x1
+∂v2∂x2
= −1 + 1 = 0.
Proof of Theorem 2.2. Using the method of Lagrange multipliers it followsthat any xn on the unit sphere that is an extreme point of the Vandermondedeterminant will also be a stationary point for the Lagrange function
Λn(xn, λ) = v(xn)− λ
(n∑
i=1
x2i − 1
)
for some λ. Explicitly this requirement becomes
∂Λn
∂xk= 0 for all 1 ≤ k ≤ n, (45)
∂Λn
∂λ= 0. (46)
Equation (46) corresponds to the restriction to the unit sphere and is there-fore immediately satisfied. Since all the partial derivatives of the Lagrangefunction should be equal to zero it is obvious that the sum of the partialderivatives will also be equal to zero. Combining this with Lemma 2.7 gives
n∑k=1
∂Λn
∂xk=
n∑k=1
(∂vn∂xk
− 2λxk
)= −2λ
n∑k=1
xk = 0. (47)
There are two ways to satisfy condition (47) either λ = 0 orn∑
k=1
xk = 0.
When λ = 0 equation (45) reduces to
∂vn∂xk
= 0 for all 1 ≤ k ≤ n,
and by equation (19) this can only be true if vn(xn) = 0, which is of no
interest to us, and so all extreme points must lie in the hyperplane
n∑k=1
xk =
0.
76
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Theorem 2.3. A point on the unit sphere in Rn, xn = (x1, x2, . . . xn),is an extreme point of the Vandermonde determinant if and only if all xi,i ∈ 1, 2, . . . n, are distinct roots of the rescaled Hermite polynomial
Pn(x) = (2n(n− 1))−n2 Hn
(√n(n− 1)
2x
). (48)
Remark 2.1. Note that if xn = (x1, x2, . . . xn) is an extreme point of theVandermonde determinant then any other point whose coordinates are apermutation of the coordinates of xn is also an extreme point. This followsfrom the determinant function being, by definition, alternating with respectto the columns of the matrix and the xis defines the columns of the Vander-monde matrix. Thus any permutation of the xis will give the same value for|vn(xn)|. Since there are n! permutations there will be at least n! extremepoints. The roots of the polynomial (48) defines the set of xis fully and thusthere are exactly n! extreme points, n!/2 positive and n!/2 negative.
Remark 2.2. All terms in Pn(x) are of even order if n is even and of oddorder when n is odd. This means that the roots of Pn(x) will be symmetricalin the sense that if xi is a root then −xi is also a root.
Proof of Theorem 2.3. By the method of Lagrange multipliers condition (45)must be satisfied for any extreme point. If xn is a fixed extreme point sothat
vn(xn) = vmax,
then (45) can be written explicitly, using (39), as
∂Λn
∂xk=
n∑i=1i =k
vmax
xk − xi− 2λxk = 0 for all 1 ≤ k ≤ n,
or alternatively by introducing a new multiplier ρ as
n∑i=1i =k
1
xk − xi=
2λ
vmaxxk =
ρ
nxk for all 1 ≤ k ≤ n. (49)
77
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Thus if equation (44) is true for n− 1 it is also true for n. Showing that theequation holds for n = 2 is very simple
∂v2∂x1
+∂v2∂x2
= −1 + 1 = 0.
Proof of Theorem 2.2. Using the method of Lagrange multipliers it followsthat any xn on the unit sphere that is an extreme point of the Vandermondedeterminant will also be a stationary point for the Lagrange function
Λn(xn, λ) = v(xn)− λ
(n∑
i=1
x2i − 1
)
for some λ. Explicitly this requirement becomes
∂Λn
∂xk= 0 for all 1 ≤ k ≤ n, (45)
∂Λn
∂λ= 0. (46)
Equation (46) corresponds to the restriction to the unit sphere and is there-fore immediately satisfied. Since all the partial derivatives of the Lagrangefunction should be equal to zero it is obvious that the sum of the partialderivatives will also be equal to zero. Combining this with Lemma 2.7 gives
n∑k=1
∂Λn
∂xk=
n∑k=1
(∂vn∂xk
− 2λxk
)= −2λ
n∑k=1
xk = 0. (47)
There are two ways to satisfy condition (47) either λ = 0 orn∑
k=1
xk = 0.
When λ = 0 equation (45) reduces to
∂vn∂xk
= 0 for all 1 ≤ k ≤ n,
and by equation (19) this can only be true if vn(xn) = 0, which is of no
interest to us, and so all extreme points must lie in the hyperplane
n∑k=1
xk =
0.
76
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Theorem 2.3. A point on the unit sphere in Rn, xn = (x1, x2, . . . xn),is an extreme point of the Vandermonde determinant if and only if all xi,i ∈ 1, 2, . . . n, are distinct roots of the rescaled Hermite polynomial
Pn(x) = (2n(n− 1))−n2 Hn
(√n(n− 1)
2x
). (48)
Remark 2.1. Note that if xn = (x1, x2, . . . xn) is an extreme point of theVandermonde determinant then any other point whose coordinates are apermutation of the coordinates of xn is also an extreme point. This followsfrom the determinant function being, by definition, alternating with respectto the columns of the matrix and the xis defines the columns of the Vander-monde matrix. Thus any permutation of the xis will give the same value for|vn(xn)|. Since there are n! permutations there will be at least n! extremepoints. The roots of the polynomial (48) defines the set of xis fully and thusthere are exactly n! extreme points, n!/2 positive and n!/2 negative.
Remark 2.2. All terms in Pn(x) are of even order if n is even and of oddorder when n is odd. This means that the roots of Pn(x) will be symmetricalin the sense that if xi is a root then −xi is also a root.
Proof of Theorem 2.3. By the method of Lagrange multipliers condition (45)must be satisfied for any extreme point. If xn is a fixed extreme point sothat
vn(xn) = vmax,
then (45) can be written explicitly, using (39), as
∂Λn
∂xk=
n∑i=1i =k
vmax
xk − xi− 2λxk = 0 for all 1 ≤ k ≤ n,
or alternatively by introducing a new multiplier ρ as
n∑i=1i =k
1
xk − xi=
2λ
vmaxxk =
ρ
nxk for all 1 ≤ k ≤ n. (49)
77
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
By forming the polynomial f(x) = (x− x1)(x− x2) · · · (x− xn) and notingthat
f ′(xk) =n∑
j=1
n∏i=1i =j
(x− xi)
∣∣∣∣x=xk
=n∏
i=1i =k
(xk − xi),
f ′′(xk) =n∑
l=1
n∑j=1j =l
n∏i=1i =ji =l
(x− xi)
∣∣∣∣x=xk
=n∑
j=1j =k
n∏i=1i =ji =k
(xk − xi) +n∑
l=1l =k
n∏i=1i =li =k
(xk − xi)
= 2n∑
j=1j =k
n∏i=1i =ji =k
(xk − xi),
we can rewrite (49) as
1
2
f ′′(xk)
f ′(xk)=
ρ
nxk,
or
f ′′(xk)−2ρ
nxkf
′(xk) = 0.
And since the last equation must vanish for all k we must have
f ′′(x)− 2ρ
nxf ′(x) = cf(x), (50)
for some constant c. To find c the xn-terms of the right and left part ofequation (50) are compared to each other,
c · cnxn = −2ρ
nxncnx
n−1 = −2ρ · cnxn ⇒ c = −2ρ.
Thus the following differential equation for f(x) must be satisfied
f ′′(x)− 2ρ
nxf ′(x) + 2ρf(x) = 0. (51)
Choosing x = az gives
f ′′(az)− 2ρ
(n− 1)a2zf ′(az) + 2ρf(az)
=1
a2d2f
dz2(az)− 2ρ
naz
1
a
df
dz(az) + 2ρf(az) = 0.
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2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
By setting g(z) = f(az) and choosing a =√
nρ a differential equation that
matches the definition for the Hermite polynomials is found:
g′′(z)− 2zg′(z) + 2ng(z) = 0. (52)
By definition the solution to (52) is g(z) = bHn(z) where b is a constant.An exact expression for the constant a can be found using Lemma 2.8 (forthe sake of convenience the lemma is stated and proved after this theorem).We get
n∑i=1
x2i =n∑
i=1
a2z2i = 1 ⇒ a2n(n− 1)
2= 1,
and so
a =
√2
n(n− 1).
Thus condition (45) is satisfied when xi are the roots of
Pn(x) = bHn (z) = bHn
(√n(n− 1)
2x
).
Choosing b = (2n(n− 1))−n2 gives Pn(x) with leading coefficient 1. This can
be confirmed by calculating the leading coefficient of P (x) using the explicitexpression for the Hermite polynomial (54). This completes the proof.
Lemma 2.8. Let xi, i = 1, 2, . . . , n be roots of the Hermite polynomialHn(x). Then
n∑i=1
x2i =n(n− 1)
2.
Proof. By letting ek(x1, . . . xn) denote the elementary symmetric polynomi-als Hn(x) can be written as
Hn(x) = An(x− x1) · · · (x− xn)
= An(xn − e1(x1, . . . , xn)x
n−1 + e2(x1, . . . , xn)xn−2 + q(x))
where q(x) is a polynomial of degree n− 3. Noting that
n∑i=1
x2i = (x1 + . . .+ xn)2 − 2
∑1≤i<j≤n
xixj
= e1(x1, . . . , xn)2 − 2e2(x1, . . . , xn), (53)
79
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
By forming the polynomial f(x) = (x− x1)(x− x2) · · · (x− xn) and notingthat
f ′(xk) =n∑
j=1
n∏i=1i =j
(x− xi)
∣∣∣∣x=xk
=n∏
i=1i =k
(xk − xi),
f ′′(xk) =n∑
l=1
n∑j=1j =l
n∏i=1i =ji =l
(x− xi)
∣∣∣∣x=xk
=n∑
j=1j =k
n∏i=1i =ji =k
(xk − xi) +n∑
l=1l =k
n∏i=1i =li =k
(xk − xi)
= 2n∑
j=1j =k
n∏i=1i =ji =k
(xk − xi),
we can rewrite (49) as
1
2
f ′′(xk)
f ′(xk)=
ρ
nxk,
or
f ′′(xk)−2ρ
nxkf
′(xk) = 0.
And since the last equation must vanish for all k we must have
f ′′(x)− 2ρ
nxf ′(x) = cf(x), (50)
for some constant c. To find c the xn-terms of the right and left part ofequation (50) are compared to each other,
c · cnxn = −2ρ
nxncnx
n−1 = −2ρ · cnxn ⇒ c = −2ρ.
Thus the following differential equation for f(x) must be satisfied
f ′′(x)− 2ρ
nxf ′(x) + 2ρf(x) = 0. (51)
Choosing x = az gives
f ′′(az)− 2ρ
(n− 1)a2zf ′(az) + 2ρf(az)
=1
a2d2f
dz2(az)− 2ρ
naz
1
a
df
dz(az) + 2ρf(az) = 0.
78
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
By setting g(z) = f(az) and choosing a =√
nρ a differential equation that
matches the definition for the Hermite polynomials is found:
g′′(z)− 2zg′(z) + 2ng(z) = 0. (52)
By definition the solution to (52) is g(z) = bHn(z) where b is a constant.An exact expression for the constant a can be found using Lemma 2.8 (forthe sake of convenience the lemma is stated and proved after this theorem).We get
n∑i=1
x2i =n∑
i=1
a2z2i = 1 ⇒ a2n(n− 1)
2= 1,
and so
a =
√2
n(n− 1).
Thus condition (45) is satisfied when xi are the roots of
Pn(x) = bHn (z) = bHn
(√n(n− 1)
2x
).
Choosing b = (2n(n− 1))−n2 gives Pn(x) with leading coefficient 1. This can
be confirmed by calculating the leading coefficient of P (x) using the explicitexpression for the Hermite polynomial (54). This completes the proof.
Lemma 2.8. Let xi, i = 1, 2, . . . , n be roots of the Hermite polynomialHn(x). Then
n∑i=1
x2i =n(n− 1)
2.
Proof. By letting ek(x1, . . . xn) denote the elementary symmetric polynomi-als Hn(x) can be written as
Hn(x) = An(x− x1) · · · (x− xn)
= An(xn − e1(x1, . . . , xn)x
n−1 + e2(x1, . . . , xn)xn−2 + q(x))
where q(x) is a polynomial of degree n− 3. Noting that
n∑i=1
x2i = (x1 + . . .+ xn)2 − 2
∑1≤i<j≤n
xixj
= e1(x1, . . . , xn)2 − 2e2(x1, . . . , xn), (53)
79
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
it is clear the the sum of the square of the roots can be described using thecoefficients for xn, xn−1 and xn−2. The explicit expression for Hn(x) is [158]
Hn(x) = n!
n2 ∑
i=0
(−1)i
i!
(2x)n−2i
(n− 2i)!
= 2nxn − 2n−2n(n− 1)xn−2 + n!
n2 ∑
i=3
(−1)i
i!
(2x)n−2i
(n− 2i)!. (54)
Comparing the coefficients in the two expressions for Hn(x) gives
An = 2n,
Ane1(x1, . . . , xn) = 0,
Ane2(x1, . . . , xn) = −n(n− 1)2n−2.
Thus by (53)n∑
i=1
x2i =n(n− 1)
2.
Theorem 2.4. The coefficients, ak, for the term xk in Pn(x) given by (48)are given by the following relations
an = 1, an−1 = 0, an−2 =1
2,
ak = − (k + 1)(k + 2)
n(n− 1)(n− k)ak+2, 1 ≤ k ≤ n− 3. (55)
Proof. equation (51) tells us that
Pn(x) =1
2ρP ′′n (x)−
1
nxP ′
n(x). (56)
That an = 1 follows from the definition of Pn and an−1 = 0 follows from theHermite polynomials only having terms of odd powers when n is odd andeven powers when n is even. That an−2 = 1
2 can be easily shown using thedefinition of Pn and the explicit formula for the Hermite polynomials (54).The value of the ρ can be found by comparing the xn−2 terms in (56)
an−2 =1
2ρn(n− 1)an +
1
n(n− 2)an−2.
80
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
From this follows1
2ρ=
−1
n2(n− 1).
Comparing the xn−l terms in (56) gives the following relation
an−l =1
2ρ(n− l + 2)(n− l)an−l+2 + (n− l)aa−l
1
n
which is equivalent to
an−l = an−l+2−(n− l + 2)(n− l + 1)
ln2(n− 1).
Letting k = n− l gives (55).
2.2.2 Further visual exploration on the sphere
This section is based on Section 2.4 of Paper A
Visualization of the determinant v3(x3) on the unit sphere is straightforward,as well as visualizations for g3(x3,a) for different a. In three dimensions allpoints on the sphere can be mapped to the plane. In higher dimensionswe need to reduce the set of visualized points somehow. In this sectionwe provide visualizations for v4, . . . , v7 by using symmetry properties of theVandermonde determinant.
Four dimensions
By Theorem 2.2 we know that the extreme points of v4(x4) on the sphereall lie in the hyperplane x1 + x2 + x3 + x4 = 0. The intersection of thishyperplane with the unit sphere in R4 can be described as a unit sphere inR3, under a suitable basis, and can then be easily visualized.
This can be realized using the transformation
x =
−1 −1 0−1 1 01 0 −11 0 1
1/√4 0 0
0 1/√2 0
0 0 1/√2
t. (57)
81
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
it is clear the the sum of the square of the roots can be described using thecoefficients for xn, xn−1 and xn−2. The explicit expression for Hn(x) is [158]
Hn(x) = n!
n2 ∑
i=0
(−1)i
i!
(2x)n−2i
(n− 2i)!
= 2nxn − 2n−2n(n− 1)xn−2 + n!
n2 ∑
i=3
(−1)i
i!
(2x)n−2i
(n− 2i)!. (54)
Comparing the coefficients in the two expressions for Hn(x) gives
An = 2n,
Ane1(x1, . . . , xn) = 0,
Ane2(x1, . . . , xn) = −n(n− 1)2n−2.
Thus by (53)n∑
i=1
x2i =n(n− 1)
2.
Theorem 2.4. The coefficients, ak, for the term xk in Pn(x) given by (48)are given by the following relations
an = 1, an−1 = 0, an−2 =1
2,
ak = − (k + 1)(k + 2)
n(n− 1)(n− k)ak+2, 1 ≤ k ≤ n− 3. (55)
Proof. equation (51) tells us that
Pn(x) =1
2ρP ′′n (x)−
1
nxP ′
n(x). (56)
That an = 1 follows from the definition of Pn and an−1 = 0 follows from theHermite polynomials only having terms of odd powers when n is odd andeven powers when n is even. That an−2 = 1
2 can be easily shown using thedefinition of Pn and the explicit formula for the Hermite polynomials (54).The value of the ρ can be found by comparing the xn−2 terms in (56)
an−2 =1
2ρn(n− 1)an +
1
n(n− 2)an−2.
80
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
From this follows1
2ρ=
−1
n2(n− 1).
Comparing the xn−l terms in (56) gives the following relation
an−l =1
2ρ(n− l + 2)(n− l)an−l+2 + (n− l)aa−l
1
n
which is equivalent to
an−l = an−l+2−(n− l + 2)(n− l + 1)
ln2(n− 1).
Letting k = n− l gives (55).
2.2.2 Further visual exploration on the sphere
This section is based on Section 2.4 of Paper A
Visualization of the determinant v3(x3) on the unit sphere is straightforward,as well as visualizations for g3(x3,a) for different a. In three dimensions allpoints on the sphere can be mapped to the plane. In higher dimensionswe need to reduce the set of visualized points somehow. In this sectionwe provide visualizations for v4, . . . , v7 by using symmetry properties of theVandermonde determinant.
Four dimensions
By Theorem 2.2 we know that the extreme points of v4(x4) on the sphereall lie in the hyperplane x1 + x2 + x3 + x4 = 0. The intersection of thishyperplane with the unit sphere in R4 can be described as a unit sphere inR3, under a suitable basis, and can then be easily visualized.
This can be realized using the transformation
x =
−1 −1 0−1 1 01 0 −11 0 1
1/√4 0 0
0 1/√2 0
0 0 1/√2
t. (57)
81
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
(a) Plot with t-basis given by (57). (b) Plot with θ and φ given by (21).
Figure 2.11: Plot of v4(x4) over points on the unit sphere.
The results of plotting the v4(x4) after performing this transformationcan be seen in Figure 2.11. All 24 = 4! extreme points are clearly visible.
From Figure 2.11 we see that whenever we have a local maxima we havea local maxima at the opposite side of the sphere as well, and the same forminima. This is due to the occurrence of the exponents in the rows of Vn.From equation (19) we have
vn((−1)xn) = (−1)n(n−1)
2 vn(xn),
and so opposite points are both maxima or both minima if n = 4k orn = 4k + 1 for some k ∈ Z+ and opposite points are of different typesif n = 4k − 2 or n = 4k − 1 for some k ∈ Z+.
x2 =
(− 1√
2,1√2
).
P2(x) = x2 − 1.
We will see later on that the extreme points are the six points acquiredfrom permuting the coordinates in 2.1.1
x3 =1√2(−1, 0, 1) .
82
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
For reasons that will become clear in Section 2.2.1 it is also useful to thinkabout these coordinates as the roots of the polynomial
P3(x) = x3 − 1
2x.
By Theorem 2.3 the extreme points on the unit sphere for v4(x4) isdescribed by the roots of this polynomial
P4(x) = x4 − 1
2x2 +
1
48.
The roots of P4(x) are:
x41 = −1
2
√1 +
√2
3, x42 = −1
2
√1−
√2
3,
x43 =1
2
√1−
√2
3, x44 =
1
2
√1 +
√2
3.
Five dimensions
By Theorem 2.3 or 2.4 we see that the polynomials providing the coordinatesof the extreme points have all even or all odd powers. From this it is easyto see that all coordinates of the extreme points must come in pairs xi,−xi.Furthermore, by Theorem 2.2 we know that the extreme points of v5(x5) onthe sphere all lie in the hyperplane x1 + x2 + x3 + x4 + x5 = 0.
We use this to visualize v5(x5) by selecting a subspace of R5 that containsall points that have coordinates which are symmetrically placed on the realline, (x1, x2, 0,−x2,−x1).
The coordinates in Figure 2.12 (a) are related to x5 by
x5 =
−1 0 10 −1 10 0 10 1 11 0 1
1/√2 0 0
0 1/√2 0
0 0 1/√5
t. (58)
83
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
(a) Plot with t-basis given by (57). (b) Plot with θ and φ given by (21).
Figure 2.11: Plot of v4(x4) over points on the unit sphere.
The results of plotting the v4(x4) after performing this transformationcan be seen in Figure 2.11. All 24 = 4! extreme points are clearly visible.
From Figure 2.11 we see that whenever we have a local maxima we havea local maxima at the opposite side of the sphere as well, and the same forminima. This is due to the occurrence of the exponents in the rows of Vn.From equation (19) we have
vn((−1)xn) = (−1)n(n−1)
2 vn(xn),
and so opposite points are both maxima or both minima if n = 4k orn = 4k + 1 for some k ∈ Z+ and opposite points are of different typesif n = 4k − 2 or n = 4k − 1 for some k ∈ Z+.
x2 =
(− 1√
2,1√2
).
P2(x) = x2 − 1.
We will see later on that the extreme points are the six points acquiredfrom permuting the coordinates in 2.1.1
x3 =1√2(−1, 0, 1) .
82
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
For reasons that will become clear in Section 2.2.1 it is also useful to thinkabout these coordinates as the roots of the polynomial
P3(x) = x3 − 1
2x.
By Theorem 2.3 the extreme points on the unit sphere for v4(x4) isdescribed by the roots of this polynomial
P4(x) = x4 − 1
2x2 +
1
48.
The roots of P4(x) are:
x41 = −1
2
√1 +
√2
3, x42 = −1
2
√1−
√2
3,
x43 =1
2
√1−
√2
3, x44 =
1
2
√1 +
√2
3.
Five dimensions
By Theorem 2.3 or 2.4 we see that the polynomials providing the coordinatesof the extreme points have all even or all odd powers. From this it is easyto see that all coordinates of the extreme points must come in pairs xi,−xi.Furthermore, by Theorem 2.2 we know that the extreme points of v5(x5) onthe sphere all lie in the hyperplane x1 + x2 + x3 + x4 + x5 = 0.
We use this to visualize v5(x5) by selecting a subspace of R5 that containsall points that have coordinates which are symmetrically placed on the realline, (x1, x2, 0,−x2,−x1).
The coordinates in Figure 2.12 (a) are related to x5 by
x5 =
−1 0 10 −1 10 0 10 1 11 0 1
1/√2 0 0
0 1/√2 0
0 0 1/√5
t. (58)
83
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
(a) Plot with t-basis given by (58). (b) Plot with θ and φ given by (21).
Figure 2.12: Plot of v5(x5) over points on the unit sphere.
The result, see Figure 2.12, is a visualization of a subspace containing8 of the 120 extreme points. Note that to satisfy the condition that thecoordinates should be symmetrically distributed pairs can be fulfilled in twoother subspaces with points that can be described in the following ways:(x1, x2, 0,−x1,−x2) and (x2,−x2, 0, x1,−x1). This means that a transfor-mation similar to (58) can be used to describe 3 · 8 = 24 different extremepoints.
The transformation (58) corresponds to choosing x3 = 0. Choosinganother coordinate to be zero will give a different subspace of R5 whichbehaves identically to the visualized one. This multiplies the number ofextreme points by five to the expected 5 · 4! = 120.By Theorem 2.3 the extreme points on the unit sphere for v5(x5) is describedby the roots of this polynomial
P5(x) = x5 − 1
2x3 +
3
80x.
The roots of P5(x) are:
x51 = −x55, x52 = −x54, x53 = 0,
x54 =1
2
√1−
√2
5, x55 =
1
2
√1 +
√2
5.
84
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Six dimensions
As for v5(x5) we use symmetry to visualize v6(x6). We select a subspaceof R6 that contains all symmetrical points (x1, x2, x3,−x3,−x2,−x1) on thesphere.
The coordinates in Figure 2.13 (a) are related to x6 by
x6 =
−1 0 00 −1 00 0 −10 0 10 1 01 0 0
1/√2 0 0
0 1/√2 0
0 0 1/√2
t. (59)
(a) Plot with t-basis given by (59). (b) Plot with θ and φ given by (21).
Figure 2.13: Plot of v6(x6) over points on the unit sphere.
In Figure 2.13 there are 48 visible extreme points. The remaining ex-treme points can be found using arguments analogous the five-dimensionalcase.
By Theorem 2.3 the extreme points on the unit sphere for v6(x6) isdescribed by the roots of this polynomial
P6(x) = x6 − 1
2x4 +
1
20x2 − 1
1800.
85
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
(a) Plot with t-basis given by (58). (b) Plot with θ and φ given by (21).
Figure 2.12: Plot of v5(x5) over points on the unit sphere.
The result, see Figure 2.12, is a visualization of a subspace containing8 of the 120 extreme points. Note that to satisfy the condition that thecoordinates should be symmetrically distributed pairs can be fulfilled in twoother subspaces with points that can be described in the following ways:(x1, x2, 0,−x1,−x2) and (x2,−x2, 0, x1,−x1). This means that a transfor-mation similar to (58) can be used to describe 3 · 8 = 24 different extremepoints.
The transformation (58) corresponds to choosing x3 = 0. Choosinganother coordinate to be zero will give a different subspace of R5 whichbehaves identically to the visualized one. This multiplies the number ofextreme points by five to the expected 5 · 4! = 120.By Theorem 2.3 the extreme points on the unit sphere for v5(x5) is describedby the roots of this polynomial
P5(x) = x5 − 1
2x3 +
3
80x.
The roots of P5(x) are:
x51 = −x55, x52 = −x54, x53 = 0,
x54 =1
2
√1−
√2
5, x55 =
1
2
√1 +
√2
5.
84
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Six dimensions
As for v5(x5) we use symmetry to visualize v6(x6). We select a subspaceof R6 that contains all symmetrical points (x1, x2, x3,−x3,−x2,−x1) on thesphere.
The coordinates in Figure 2.13 (a) are related to x6 by
x6 =
−1 0 00 −1 00 0 −10 0 10 1 01 0 0
1/√2 0 0
0 1/√2 0
0 0 1/√2
t. (59)
(a) Plot with t-basis given by (59). (b) Plot with θ and φ given by (21).
Figure 2.13: Plot of v6(x6) over points on the unit sphere.
In Figure 2.13 there are 48 visible extreme points. The remaining ex-treme points can be found using arguments analogous the five-dimensionalcase.
By Theorem 2.3 the extreme points on the unit sphere for v6(x6) isdescribed by the roots of this polynomial
P6(x) = x6 − 1
2x4 +
1
20x2 − 1
1800.
85
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The roots of P6(x) are:
x61 =− x66, x62 = −x65, x63 = −x64,
x64 =(−1)
34
2√15
(10i− 3
√10
(z6w
136 + z6w
136
)) 12
=1
2√15
√10− 2
√10
(√3l6 − k6
), (60)
x65 =(−1)
14
2√15
(−10i− 3
√10
(z6w
136 + z6w
136
)) 12
=1
2√15
√10− 2
√10
(√3l6 + k6
), (61)
x66 =
(1
30
(3√10
(w
136 + w
136
)+ 5
)) 12
=
√1
30
(2√10 · k6 + 5
), (62)
z6 =√3 + i, w6 = 2 + i
√6
k6 =cos
(1
3arctan
(√3
2
)), l6 = sin
(1
3arctan
(√3
2
)).
86
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Seven dimensions
As for v6(x6) we use symmetry to visualize v7(x7). We select a subspace ofR7 that contains all symmetrical points (x1, x2, x3, 0,−x3,−x2,−x1) on thesphere.
The coordinates in Figure 2.14 (a) are related to x7 by
x7 =
−1 0 00 −1 00 0 −10 0 00 0 10 1 01 0 0
1/√2 0 0
0 1/√2 0
0 0 1/√2
t. (63)
(a) Plot with t-basis given by (63). (b) Plot with θ and φ given by (21).
Figure 2.14: Plot of v7(x7) over points on the unit sphere.
In Figure 2.14 48 extreme points are visible just like it was for the six-dimensional case. This is expected since the transformation correspondsto choosing x4 = 0 which restricts us to a six-dimensional subspace of R7
which can then be visualized in the same way as the six-dimensional case.The remaining extreme points can be found using arguments analogous thefive-dimensional case.
By Theorem 2.3 the extreme points on the unit sphere for v4 is describedby the roots of this polynomial
P7(x) = x7 − 1
2x5 +
5
84x3 − 5
3528x.
87
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The roots of P6(x) are:
x61 =− x66, x62 = −x65, x63 = −x64,
x64 =(−1)
34
2√15
(10i− 3
√10
(z6w
136 + z6w
136
)) 12
=1
2√15
√10− 2
√10
(√3l6 − k6
), (60)
x65 =(−1)
14
2√15
(−10i− 3
√10
(z6w
136 + z6w
136
)) 12
=1
2√15
√10− 2
√10
(√3l6 + k6
), (61)
x66 =
(1
30
(3√10
(w
136 + w
136
)+ 5
)) 12
=
√1
30
(2√10 · k6 + 5
), (62)
z6 =√3 + i, w6 = 2 + i
√6
k6 =cos
(1
3arctan
(√3
2
)), l6 = sin
(1
3arctan
(√3
2
)).
86
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Seven dimensions
As for v6(x6) we use symmetry to visualize v7(x7). We select a subspace ofR7 that contains all symmetrical points (x1, x2, x3, 0,−x3,−x2,−x1) on thesphere.
The coordinates in Figure 2.14 (a) are related to x7 by
x7 =
−1 0 00 −1 00 0 −10 0 00 0 10 1 01 0 0
1/√2 0 0
0 1/√2 0
0 0 1/√2
t. (63)
(a) Plot with t-basis given by (63). (b) Plot with θ and φ given by (21).
Figure 2.14: Plot of v7(x7) over points on the unit sphere.
In Figure 2.14 48 extreme points are visible just like it was for the six-dimensional case. This is expected since the transformation correspondsto choosing x4 = 0 which restricts us to a six-dimensional subspace of R7
which can then be visualized in the same way as the six-dimensional case.The remaining extreme points can be found using arguments analogous thefive-dimensional case.
By Theorem 2.3 the extreme points on the unit sphere for v4 is describedby the roots of this polynomial
P7(x) = x7 − 1
2x5 +
5
84x3 − 5
3528x.
87
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The roots of P7(x) are:
x71 =− x77, x72 = −x76, x73 = −x75, x74 = 0,
x75 =(−1)
34
2√21
(14i− 3
√14
(z6w
136 + z6w
136
)) 12
=1
2√21
√14− 2
√14
(√3l6 − k6
), (64)
x76 =(−1)
14
2√21
(−14i− 3
√14
(z6w
136 + z6w
136
)) 12
=1
2√21
√14− 2
√14
(√3l7 + k7
), (65)
x77 =
√1
42
(3√14
(w
136 + w
136
)+ 5
) 12
=
√1
42
(2√14k7 + 5
), (66)
z6 =√3 + i, w6 = 2 + i
√10
k7 =cos
(1
3arctan
(√5
2
)),
l7 =sin
(1
3arctan
(√5
2
)).
2.2.3 The extreme points on spheres defined by the p-normsgiven by roots of a polynomial
This section is based on page 630–635 of Paper B
Consider the unit (n−1)-sphere under the p-norm (p ∈ Z, p ≥ 1), we define
Sn−1p = xn ∈ Rn : |x1|p + · · ·+ |xn|p = 1 .
When p = 2 we have the Euclidean norm and thus the usual (n− 1)-sphere,we also let p = ∞ which gives the cube defined by the boundary of [−1, 1]n.The 2-sphere for some different norms are depicted in Figure 2.15.
The four separate plots in Figure 2.15 are all rotated slightly by thesame transformation for visual clarity and the mappings of color to valueare slightly different between the figures. The positive maxima can be seento lie in the dark red regions and the minima can be found in the dark blue
88
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Figure 2.15: Value of v3(x3) over: S22 (left), S2
4 (middle left), S28 (middle right) and
S2∞ (right).
regions. As we soon will see the extreme points on S22 are
(−1/
√2, 0, 1/
√2)
and the vectors constructed by permutating the coordinates of this vector,making a total of 6 = 3! extreme points. Similarly, for the cube we have thevectors formed by the six permutations of the coordinates in (−1, 0, 1). Thetwo stated vectors are both maxima and correspond to the top left maximain the figures, odd permutations of these vectors will give minima and evenpermutations will again give maxima. This follows directly from the anti-symmetry of the Vandermonde determinant, given a permutation σ ∈ Sn
we have vn(xσ1 , · · · , xσn) = sgn(σ)vn(x1, · · · , xn).We are thus faced with the problem of maximizing |vn(xn)| over Sn−1
p .By the symmetry of |vn| we do not loose any generality by assuming thatthe coordinates are ordered and pairwise distinct, x1 < · · · < xn.
Theorem 2.5. The coordinates x1 < · · · < xn, n ≥ 2, that maximize theabsolute value of the Vandermonde determinant over the unit sphere in Rn,under the p-norm, that is Sn−1
p , are unique. Furthermore, the coordinatesare symmetric so that xi = −xn−i+1 for 1 ≤ i ≤ n and the total number ofmaxima, counting permutations, are n!.
This is an extension of a result presented by Szego [158, p.140].
Proof. Suppose that we have two different sequences of coordinates xn andx′n:
x1 < · · · < xn,
x′1 < · · · < x′n,
both maximizing |vn| on Sn−1p , so |vn(x1, · · · , xn)| = |vn(x′1, · · · , x′n)| = vmax
n
is the maximum value. Now define a new configuration zn defined by,
zi =xi + x′i2σ
, 1 ≤ i ≤ n,
89
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The roots of P7(x) are:
x71 =− x77, x72 = −x76, x73 = −x75, x74 = 0,
x75 =(−1)
34
2√21
(14i− 3
√14
(z6w
136 + z6w
136
)) 12
=1
2√21
√14− 2
√14
(√3l6 − k6
), (64)
x76 =(−1)
14
2√21
(−14i− 3
√14
(z6w
136 + z6w
136
)) 12
=1
2√21
√14− 2
√14
(√3l7 + k7
), (65)
x77 =
√1
42
(3√14
(w
136 + w
136
)+ 5
) 12
=
√1
42
(2√14k7 + 5
), (66)
z6 =√3 + i, w6 = 2 + i
√10
k7 =cos
(1
3arctan
(√5
2
)),
l7 =sin
(1
3arctan
(√5
2
)).
2.2.3 The extreme points on spheres defined by the p-normsgiven by roots of a polynomial
This section is based on page 630–635 of Paper B
Consider the unit (n−1)-sphere under the p-norm (p ∈ Z, p ≥ 1), we define
Sn−1p = xn ∈ Rn : |x1|p + · · ·+ |xn|p = 1 .
When p = 2 we have the Euclidean norm and thus the usual (n− 1)-sphere,we also let p = ∞ which gives the cube defined by the boundary of [−1, 1]n.The 2-sphere for some different norms are depicted in Figure 2.15.
The four separate plots in Figure 2.15 are all rotated slightly by thesame transformation for visual clarity and the mappings of color to valueare slightly different between the figures. The positive maxima can be seento lie in the dark red regions and the minima can be found in the dark blue
88
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Figure 2.15: Value of v3(x3) over: S22 (left), S2
4 (middle left), S28 (middle right) and
S2∞ (right).
regions. As we soon will see the extreme points on S22 are
(−1/
√2, 0, 1/
√2)
and the vectors constructed by permutating the coordinates of this vector,making a total of 6 = 3! extreme points. Similarly, for the cube we have thevectors formed by the six permutations of the coordinates in (−1, 0, 1). Thetwo stated vectors are both maxima and correspond to the top left maximain the figures, odd permutations of these vectors will give minima and evenpermutations will again give maxima. This follows directly from the anti-symmetry of the Vandermonde determinant, given a permutation σ ∈ Sn
we have vn(xσ1 , · · · , xσn) = sgn(σ)vn(x1, · · · , xn).We are thus faced with the problem of maximizing |vn(xn)| over Sn−1
p .By the symmetry of |vn| we do not loose any generality by assuming thatthe coordinates are ordered and pairwise distinct, x1 < · · · < xn.
Theorem 2.5. The coordinates x1 < · · · < xn, n ≥ 2, that maximize theabsolute value of the Vandermonde determinant over the unit sphere in Rn,under the p-norm, that is Sn−1
p , are unique. Furthermore, the coordinatesare symmetric so that xi = −xn−i+1 for 1 ≤ i ≤ n and the total number ofmaxima, counting permutations, are n!.
This is an extension of a result presented by Szego [158, p.140].
Proof. Suppose that we have two different sequences of coordinates xn andx′n:
x1 < · · · < xn,
x′1 < · · · < x′n,
both maximizing |vn| on Sn−1p , so |vn(x1, · · · , xn)| = |vn(x′1, · · · , x′n)| = vmax
n
is the maximum value. Now define a new configuration zn defined by,
zi =xi + x′i2σ
, 1 ≤ i ≤ n,
89
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
where σ > 0 is a normalization constant to assure that zn lies on the sphere.It is easy to see that we have
z1 < · · · < zn.
Note that σ ≤ 1 by necessity since Sn−1p is the boundary of a convex set. This
follows directly from the absolute homogeneity and the triangle inequalityassociated with the p-norm:
∥∥∥∥xn + x′
n
2
∥∥∥∥p
≤∥∥∥xn
2
∥∥∥p+
∥∥∥∥x′n
2
∥∥∥∥p
= 1.
We have established that zn lies on Sn−1p . For each 1 ≤ i < j ≤ n we now
have
|zj − zi| =
∣∣∣xj + x′j − xi − x′i
∣∣∣2σ
=|xj − xi|+
∣∣∣x′j − x′i
∣∣∣2σ
≥|xj − xi|12∣∣x′j − x′i
∣∣ 12 ,
where the last step follows from σ < 1 and the general relation
(a+ b
2
)2
=
(a− b
2
)2
+ ab ≥ ab.
It follows that
|vn(zn)| ≥ |vn(xn)|12∣∣vn(x′
n)∣∣ 12 = vmax
n ,
and to not have a contradiction, we must have that the equality holds, thatis xn = x′
n, and so the maximum is unique.
Now, consider
vn(−xn) = (−1)n(n−1)
2 vn(xn),
which follows easily from the degree of the expansion of vn, where everyterm is of degree n(n−1)
2 . If follows that if xn is a maximum of |vn| onthe sphere then −xn is also a maximum. Now, since the maximum withordered coordinates x1 < · · · < xn is unique we must have that −xn =x1, · · · ,−x1 = xn, that is xi = −xn−i+1 for 1 ≤ i ≤ n and so the maximaare symmetric.
From xi = −xn−i+1 and the pairwise distinctness of the coordinates (thedeterminant is non-zero at the maximum) we have that the n! permutations(xσ1 , · · · , xσn) are the distinct maxima.
90
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
The condition xi = −xn−i+1 for the ordered maximum x1 < · · · < xnimplies that the maxima all lie in the hyperplane x1 + · · ·+ xn = 0. Thesefacts can be used to visualize vn on Sn−1
p for n = 4, 5, 6, 7, while we for n ≥ 8have more than three degrees of freedom.
Motivated by the fact that there is only one set of coordinates and thatthe maxima are constructed from different orderings of these, it makes senseto instead consider the polynomial constructed from these coordinates. Ourtask now is to find the polynomials that define the optimizing coordinatesfor different n ≥ 2 and p ≥ 1,
Pnp (x) =
n∏i=1
(x− xi),
where xi are the distinct maximizing coordinates that depend on n and p.We have
Pnp′(xk) =
n∑j=1
n∏i=1i =j
(x− xi)
∣∣∣∣x=xk
=
n∏i=1i =k
(xk − xi),
Pnp′′(xk) =
n∑l=1
n∑j=1j =l
n∏i=1i =ji =l
(x− xi)
∣∣∣∣x=xk
=n∑
j=1j =k
n∏i=1i =ji =k
(xk − xi) +n∑
l=1l =k
n∏i=1i =li =k
(xk − xi)
= 2n∑
j=1j =k
n∏i=1i =ji =k
(xk − xi),
and it is easy to show that
Pnp′′(xk)
Pnp′(xk)
= 2n∑
i=1i =k
1
xk − xi. (67)
Define
snp (xn) =
(n∑
i=1
|xi|p)
− 1,
so that snp vanishes exactly on the unit sphere under the p-norm, that is
Sn−1p =
xn ∈ Rn : snp (xn) = 0
. We have the partial derivatives
∂snp (xn)
∂xk= p |xk|p−1 sgn(xk).
91
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
where σ > 0 is a normalization constant to assure that zn lies on the sphere.It is easy to see that we have
z1 < · · · < zn.
Note that σ ≤ 1 by necessity since Sn−1p is the boundary of a convex set. This
follows directly from the absolute homogeneity and the triangle inequalityassociated with the p-norm:
∥∥∥∥xn + x′
n
2
∥∥∥∥p
≤∥∥∥xn
2
∥∥∥p+
∥∥∥∥x′n
2
∥∥∥∥p
= 1.
We have established that zn lies on Sn−1p . For each 1 ≤ i < j ≤ n we now
have
|zj − zi| =
∣∣∣xj + x′j − xi − x′i
∣∣∣2σ
=|xj − xi|+
∣∣∣x′j − x′i
∣∣∣2σ
≥|xj − xi|12∣∣x′j − x′i
∣∣ 12 ,
where the last step follows from σ < 1 and the general relation
(a+ b
2
)2
=
(a− b
2
)2
+ ab ≥ ab.
It follows that
|vn(zn)| ≥ |vn(xn)|12∣∣vn(x′
n)∣∣ 12 = vmax
n ,
and to not have a contradiction, we must have that the equality holds, thatis xn = x′
n, and so the maximum is unique.
Now, consider
vn(−xn) = (−1)n(n−1)
2 vn(xn),
which follows easily from the degree of the expansion of vn, where everyterm is of degree n(n−1)
2 . If follows that if xn is a maximum of |vn| onthe sphere then −xn is also a maximum. Now, since the maximum withordered coordinates x1 < · · · < xn is unique we must have that −xn =x1, · · · ,−x1 = xn, that is xi = −xn−i+1 for 1 ≤ i ≤ n and so the maximaare symmetric.
From xi = −xn−i+1 and the pairwise distinctness of the coordinates (thedeterminant is non-zero at the maximum) we have that the n! permutations(xσ1 , · · · , xσn) are the distinct maxima.
90
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
The condition xi = −xn−i+1 for the ordered maximum x1 < · · · < xnimplies that the maxima all lie in the hyperplane x1 + · · ·+ xn = 0. Thesefacts can be used to visualize vn on Sn−1
p for n = 4, 5, 6, 7, while we for n ≥ 8have more than three degrees of freedom.
Motivated by the fact that there is only one set of coordinates and thatthe maxima are constructed from different orderings of these, it makes senseto instead consider the polynomial constructed from these coordinates. Ourtask now is to find the polynomials that define the optimizing coordinatesfor different n ≥ 2 and p ≥ 1,
Pnp (x) =
n∏i=1
(x− xi),
where xi are the distinct maximizing coordinates that depend on n and p.We have
Pnp′(xk) =
n∑j=1
n∏i=1i =j
(x− xi)
∣∣∣∣x=xk
=
n∏i=1i =k
(xk − xi),
Pnp′′(xk) =
n∑l=1
n∑j=1j =l
n∏i=1i =ji =l
(x− xi)
∣∣∣∣x=xk
=n∑
j=1j =k
n∏i=1i =ji =k
(xk − xi) +n∑
l=1l =k
n∏i=1i =li =k
(xk − xi)
= 2n∑
j=1j =k
n∏i=1i =ji =k
(xk − xi),
and it is easy to show that
Pnp′′(xk)
Pnp′(xk)
= 2n∑
i=1i =k
1
xk − xi. (67)
Define
snp (xn) =
(n∑
i=1
|xi|p)
− 1,
so that snp vanishes exactly on the unit sphere under the p-norm, that is
Sn−1p =
xn ∈ Rn : snp (xn) = 0
. We have the partial derivatives
∂snp (xn)
∂xk= p |xk|p−1 sgn(xk).
91
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
To maximize |vn(xn)| over the surface snp (xn) = 0 we transform theobjective function by applying a (strictly increasing) logarithm:
wn(xn) = log (|vn(xn)|) =∑
1≤i≤j
log (|xj − xi|) ,
with partial derivatives
∂wn(xn)
∂xk=
n∑i=1i =k
1
xk − xi=
1
2
Pnp′′(xk)
Pnp′(xk)
.
By the method of Lagrange multipliers we now have that the maxima of|vn(xn)| on Sn−1
p must be stationary points to the Lagrangian
Λ(xn, λ) = wn(xn)− λsnp (xn), (68)
which explicitly means
∂wn(xn)
∂xk= λ
∂snp (xn)
∂xk,
for some multiplier λ ∈ R. We then get
1
2
Pnp′′(xk)
Pnp′(xk)
= λp |xk|p−1 sgn(xk).
Letting ρ = −2λp we then get
Pnp′′(xk) + ρ |xk|p−1 sgn(xk)P
np′(xk) = 0. (69)
This leads us to our first set of polynomials defining the solution to ourmaximization problem.
Theorem 2.6. The polynomial Pn2 , of degree n > 2 and with a leading co-
efficient of 1, whose roots form the coordinates in the points xn ∈ Rn thatmaximize |vn(xn)| over the Euclidean hypersphere Sn−1 satisfy the differen-tial equation
Pn2′′(x) + n(1− n)xPn
2′(x) + n2(n− 1)Pn
2 (x) = 0. (70)
Furthermore, the coefficients for the three terms of highest degree are
cn = 1, cn−1 = 0, cn−2 = −1
2, (71)
and the subsequent coefficients are defined recursively by
ck = − (k + 1)(k + 2)
n(n− 1)(n− k)ck+2. (72)
92
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Proof. By equation (69) we have
Pn2′′(x) + ρxPn
2′(x)
∣∣∣∣x=xk
= 0, 1 ≤ k ≤ n.
The left part of this equation represents n evaluations of a polynomial ofdegree n that vanishes on x1, · · · , xn and must thus be a constant multipleof Pn
2 , that we defined as the polynomial that vanishes on x1, · · · , xn, andso
Pn2′′(x) + ρxPn
2′(x) + σPn
2 (x) = 0.
Two find the coefficients σ and ρ we need to adapt this polynomial to
the sphere. We haven∑
i=1
xi = 0. The condition cn = 1 is by choice. The
condition cn−1 = 0 follows from the expansion of the coefficients in Pn2 .
P pn(x) =
n∏i=1
(x− xi) =
n∑k=0
(−1)n−ken−k(x1, · · · , xn)xk,
where ek is the elementary symmetric polynomial defined by
ek(x1, · · · , xn) =∑
i1<···<ik
xi1xi2 · · ·xik .
We have cn−1 = −e1(x1, · · · , xn) = −(x1 + · · · + xn) = 0. The conditioncn−2 = −1
2 places us on the unit sphere
e1(x1, · · · , xn)2 − 2e2(x1, · · · , xn) = x21 + · · ·+ x2n = 1,
−2e2(x1, · · · , xn) = 1,
cn−2 = e2(x1, · · · , xn) = −1
2.
This establishes equation (71) in the theorem.Now, the coefficients cn, · · · , c0 for any polynomial solution p(x) of degree
n to a differential equation on the form
pn′′(x) + ρxpn′(x) + σpn(x) = 0,
must satisfy
ρncn + σcn = 0
ρ(n− 1)cn−1 + σcn−1 = 0
n(n− 1)cn + ρ(n− 2)cn−2 + σcn−2 = 0.
93
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
To maximize |vn(xn)| over the surface snp (xn) = 0 we transform theobjective function by applying a (strictly increasing) logarithm:
wn(xn) = log (|vn(xn)|) =∑
1≤i≤j
log (|xj − xi|) ,
with partial derivatives
∂wn(xn)
∂xk=
n∑i=1i =k
1
xk − xi=
1
2
Pnp′′(xk)
Pnp′(xk)
.
By the method of Lagrange multipliers we now have that the maxima of|vn(xn)| on Sn−1
p must be stationary points to the Lagrangian
Λ(xn, λ) = wn(xn)− λsnp (xn), (68)
which explicitly means
∂wn(xn)
∂xk= λ
∂snp (xn)
∂xk,
for some multiplier λ ∈ R. We then get
1
2
Pnp′′(xk)
Pnp′(xk)
= λp |xk|p−1 sgn(xk).
Letting ρ = −2λp we then get
Pnp′′(xk) + ρ |xk|p−1 sgn(xk)P
np′(xk) = 0. (69)
This leads us to our first set of polynomials defining the solution to ourmaximization problem.
Theorem 2.6. The polynomial Pn2 , of degree n > 2 and with a leading co-
efficient of 1, whose roots form the coordinates in the points xn ∈ Rn thatmaximize |vn(xn)| over the Euclidean hypersphere Sn−1 satisfy the differen-tial equation
Pn2′′(x) + n(1− n)xPn
2′(x) + n2(n− 1)Pn
2 (x) = 0. (70)
Furthermore, the coefficients for the three terms of highest degree are
cn = 1, cn−1 = 0, cn−2 = −1
2, (71)
and the subsequent coefficients are defined recursively by
ck = − (k + 1)(k + 2)
n(n− 1)(n− k)ck+2. (72)
92
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
Proof. By equation (69) we have
Pn2′′(x) + ρxPn
2′(x)
∣∣∣∣x=xk
= 0, 1 ≤ k ≤ n.
The left part of this equation represents n evaluations of a polynomial ofdegree n that vanishes on x1, · · · , xn and must thus be a constant multipleof Pn
2 , that we defined as the polynomial that vanishes on x1, · · · , xn, andso
Pn2′′(x) + ρxPn
2′(x) + σPn
2 (x) = 0.
Two find the coefficients σ and ρ we need to adapt this polynomial to
the sphere. We haven∑
i=1
xi = 0. The condition cn = 1 is by choice. The
condition cn−1 = 0 follows from the expansion of the coefficients in Pn2 .
P pn(x) =
n∏i=1
(x− xi) =
n∑k=0
(−1)n−ken−k(x1, · · · , xn)xk,
where ek is the elementary symmetric polynomial defined by
ek(x1, · · · , xn) =∑
i1<···<ik
xi1xi2 · · ·xik .
We have cn−1 = −e1(x1, · · · , xn) = −(x1 + · · · + xn) = 0. The conditioncn−2 = −1
2 places us on the unit sphere
e1(x1, · · · , xn)2 − 2e2(x1, · · · , xn) = x21 + · · ·+ x2n = 1,
−2e2(x1, · · · , xn) = 1,
cn−2 = e2(x1, · · · , xn) = −1
2.
This establishes equation (71) in the theorem.Now, the coefficients cn, · · · , c0 for any polynomial solution p(x) of degree
n to a differential equation on the form
pn′′(x) + ρxpn′(x) + σpn(x) = 0,
must satisfy
ρncn + σcn = 0
ρ(n− 1)cn−1 + σcn−1 = 0
n(n− 1)cn + ρ(n− 2)cn−2 + σcn−2 = 0.
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Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The second of these equations is trivial since cn−1 = 0. The first and thirdequation simplifies to
ρn+ σ = 0,
n(n− 1)− 1
2ρ(n− 2)− 1
2σ = 0,
and so
ρ =−2n(n− 1)
n− (n− 2)= −n(n− 1)
σ = n2(n− 1),
which establishes equation (70) in the theorem.For the recursive formula for the coefficients cn−3, · · · , c0, equation (72),
we again look at the slightly more general case and retain ρ, σ. The coeffi-cients for the polynomial p satisfying
pn′′(x) + ρxpn′(x) + σpn(x) = 0,
must satisfyck+2(k + 1)(k + 2) + ρkck + σck = 0,
that is
ck =−(k + 1)(k + 2)
ρk + σck+2.
For Pn2 we then have
ck =−(k + 1)(k + 2)
−n(n− 1)k + n2(n− 1)ck+2, (73)
and equation (72) follows.
The case p = ∞ follows in a similar manner.
Theorem 2.7. The polynomial Pn∞, of degree n > 2 and with a leading
coefficient of 1, whose roots form the coordinates in the points xn ∈ Rn thatmaximize |vn(xn)| over the cube Sn−1
∞ satisfy
Pn∞(x) = (x− 1)(x+ 1)pn∞(x). (74)
where pn∞ is defined by the differential equation:
(1− x2)pn∞′′(x)− 4xpn∞
′(x) +m(m+ 3)pn∞(x) = 0,
94
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
where m = n− 2 is the degree of the polynomial pn∞. Furthermore, the firsttwo coefficients for pn∞ are cm = 1, cm−1 = 0 and the subsequent coefficientssatisfy
ck =(k + 1)(k + 2)
k(k + 3)−m(m+ 3)ck+2. (75)
Proof. It is easy to show that the coordinates −1 and +1 must be presentin the maxima points, if they were not then we could rescale the point sothat the value of |vn(xn)| is increased, which is not allowed. We may thusassume the ordered sequence of coordinates
−1 = x1 < · · · < xn = +1.
The absolute value of the Vandermonde determinant then becomes
|vn(xn)| = 2
n−1∏i=2
(|1 + xi| |1− xi|)∏
1<i<j<n
|xj − xi| .
We now take the logarithm of this, differentiate and equate the partialderivatives to zero to find the stationary points (actually maxima), andarrive at
1
xk + 1+
1
xk − 1+
n−1∑i=2i =k
1
xk − xi= 0, 1 < k < n,
which similarly to equation (67) can be written
1
xk + 1+
1
xk − 1+
1
2
pn∞′′(xk)
pn∞′(xk)
= 0, 1 < k < n, (76)
for some polynomial pn∞ constructed from the roots x2, · · · , xn−1.
The left part of equation (76) now identifies a differential expression onpn∞ which we by the same method as for p = 2 identify by a multiple of pn∞,that is
(1− x2)pn∞′′(x)− 4xpn∞
′(x) + σpn∞(x) = 0. (77)
The constant σ is found by considering the coefficient for xm:
−m(m− 1)− 4m+ σ = 0 ⇔ σ = m(m+ 3).
Finally
(1− x2)pn∞′′(x)− 4xpn∞
′(x) +m(m+ 3)pn∞(x) = 0.
95
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The second of these equations is trivial since cn−1 = 0. The first and thirdequation simplifies to
ρn+ σ = 0,
n(n− 1)− 1
2ρ(n− 2)− 1
2σ = 0,
and so
ρ =−2n(n− 1)
n− (n− 2)= −n(n− 1)
σ = n2(n− 1),
which establishes equation (70) in the theorem.For the recursive formula for the coefficients cn−3, · · · , c0, equation (72),
we again look at the slightly more general case and retain ρ, σ. The coeffi-cients for the polynomial p satisfying
pn′′(x) + ρxpn′(x) + σpn(x) = 0,
must satisfyck+2(k + 1)(k + 2) + ρkck + σck = 0,
that is
ck =−(k + 1)(k + 2)
ρk + σck+2.
For Pn2 we then have
ck =−(k + 1)(k + 2)
−n(n− 1)k + n2(n− 1)ck+2, (73)
and equation (72) follows.
The case p = ∞ follows in a similar manner.
Theorem 2.7. The polynomial Pn∞, of degree n > 2 and with a leading
coefficient of 1, whose roots form the coordinates in the points xn ∈ Rn thatmaximize |vn(xn)| over the cube Sn−1
∞ satisfy
Pn∞(x) = (x− 1)(x+ 1)pn∞(x). (74)
where pn∞ is defined by the differential equation:
(1− x2)pn∞′′(x)− 4xpn∞
′(x) +m(m+ 3)pn∞(x) = 0,
94
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
where m = n− 2 is the degree of the polynomial pn∞. Furthermore, the firsttwo coefficients for pn∞ are cm = 1, cm−1 = 0 and the subsequent coefficientssatisfy
ck =(k + 1)(k + 2)
k(k + 3)−m(m+ 3)ck+2. (75)
Proof. It is easy to show that the coordinates −1 and +1 must be presentin the maxima points, if they were not then we could rescale the point sothat the value of |vn(xn)| is increased, which is not allowed. We may thusassume the ordered sequence of coordinates
−1 = x1 < · · · < xn = +1.
The absolute value of the Vandermonde determinant then becomes
|vn(xn)| = 2
n−1∏i=2
(|1 + xi| |1− xi|)∏
1<i<j<n
|xj − xi| .
We now take the logarithm of this, differentiate and equate the partialderivatives to zero to find the stationary points (actually maxima), andarrive at
1
xk + 1+
1
xk − 1+
n−1∑i=2i =k
1
xk − xi= 0, 1 < k < n,
which similarly to equation (67) can be written
1
xk + 1+
1
xk − 1+
1
2
pn∞′′(xk)
pn∞′(xk)
= 0, 1 < k < n, (76)
for some polynomial pn∞ constructed from the roots x2, · · · , xn−1.
The left part of equation (76) now identifies a differential expression onpn∞ which we by the same method as for p = 2 identify by a multiple of pn∞,that is
(1− x2)pn∞′′(x)− 4xpn∞
′(x) + σpn∞(x) = 0. (77)
The constant σ is found by considering the coefficient for xm:
−m(m− 1)− 4m+ σ = 0 ⇔ σ = m(m+ 3).
Finally
(1− x2)pn∞′′(x)− 4xpn∞
′(x) +m(m+ 3)pn∞(x) = 0.
95
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The polynomial that provide all coordinates for the maxima is then
Pn∞(x) = (x− 1)(x+ 1)pn∞. (78)
Equation (75) follows by the same methods as for p = 2, that is, by identi-fying all coefficients in the differential equation to be identically zero.
We have provided the means to describe the coordinates of the extremepoints of the Vandermonde determinant over the unit spheres under theEuclidean norm and under the infinity norm. The resulting polynomials
can be identified by rescaled Hermite polynomials, Hn
(x√n(n− 1)/2
),
and Gegenbauer polynomials, C(3/2)n (x), respectively. In the next section we
will discuss the case p = 4.
2.2.4 The Vandermonde determinant on spheres defined bythe 4-norm
This section is based on Section 8 of Paper C
The optimization of the Vandermonde determinant on the sphere (p = 2)and on the cube (p = ∞) lends themselves to methods in orthogonal poly-nomials. In fact, as shown by Stieltjes and recaptured by Szego [158], andpresented in more detail in and extended in Section 2.2.1 there is a fairlystraightforward solution derived from electrostatic considerations, which im-plicitly deals with the Vandermonde determinant.
Consider the optimization of vn(x) over the sphere
sn(x) =n∑
i=1
xpi = 1,
for suitable choices for p (even). Instead of optimizing vn we are free tooptimize ln |vn| over the sphere to the same effect (vn(x) = 0 is not a solution,all xi are pair-wise distinct). This leaves us with the set of equations byLagrange multipliers.
∂ ln |vn|∂xk
= λ∂sn∂xk
, sn = 1, (79)
where the left-most equation holds for 1 ≤ k ≤ n. It is easy to show thatthe partial derivatives can be written
∂ ln |vn|∂xk
=
n∑i=1i =k
1
xk − xi. (80)
96
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
From this it is easy to show that (80) can be rewritten by introducing the
univariate polynomial Pn(x) =n∏
i=1
(x− xi) as
∂ ln |vn|∂xk
=1
2
P ′′(xk)
P ′(xk).
Now the leftmost equation of (79) can be written
1
2
P ′′(xk)
P ′(xk)= λ
∂sn∂xk
,
or more succinct
P ′′(xk)− 2λ∂sn∂xk
P ′(xk) = 0, (81)
In the case p = 2 we are lucky since (81) becomes, by introducing the new“multiplier” ρ (
P ′′(x) + ρnxP′(x)
)|x=xk
= 0, (82)
and since the left part of this equation is a polynomial of degree n and hasroots x1, · · · , xn we must have
P ′′(x) + ρnxP′(x) + σnP (x) = 0, (83)
for some ρn, σn that may depend on n. Now if choose P (x) to be monic,note that ρn = 0, and require us to be on the sphere we get P (x) = xn −12x
n−2 + · · · , and by identifying coefficients we get ρn and σn:
P ′′(x) + n(1− n)xP ′(x) + n2(n− 1)P (x) = 0,
which is a nice and well known form of differential equation and defines asequence of orthogonal polynomials that are rescaled Hermite polynomials[158], so we can find a recurrence relation for Pn+1 in terms of Pn and Pn−1,and we can, for a fixed n, construct the coefficients of Pn recursively, withoutexplicitly finding P1, · · · , Pn−1.
Now, this is for p = 2. For p = 4 we continue from (81) instead with
(P ′′(x) + ρnx
3P ′(x))|x=xk
= 0. (84)
Now the polynomial in x in the left part of this equation has shared rootswith P (x) and so by the same method as for p = 2 we get:
P ′′(x) + ρnx3P ′(x) + (σnx
2 + τnx+ υn)P (x) = 0. (85)
97
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The polynomial that provide all coordinates for the maxima is then
Pn∞(x) = (x− 1)(x+ 1)pn∞. (78)
Equation (75) follows by the same methods as for p = 2, that is, by identi-fying all coefficients in the differential equation to be identically zero.
We have provided the means to describe the coordinates of the extremepoints of the Vandermonde determinant over the unit spheres under theEuclidean norm and under the infinity norm. The resulting polynomials
can be identified by rescaled Hermite polynomials, Hn
(x√n(n− 1)/2
),
and Gegenbauer polynomials, C(3/2)n (x), respectively. In the next section we
will discuss the case p = 4.
2.2.4 The Vandermonde determinant on spheres defined bythe 4-norm
This section is based on Section 8 of Paper C
The optimization of the Vandermonde determinant on the sphere (p = 2)and on the cube (p = ∞) lends themselves to methods in orthogonal poly-nomials. In fact, as shown by Stieltjes and recaptured by Szego [158], andpresented in more detail in and extended in Section 2.2.1 there is a fairlystraightforward solution derived from electrostatic considerations, which im-plicitly deals with the Vandermonde determinant.
Consider the optimization of vn(x) over the sphere
sn(x) =n∑
i=1
xpi = 1,
for suitable choices for p (even). Instead of optimizing vn we are free tooptimize ln |vn| over the sphere to the same effect (vn(x) = 0 is not a solution,all xi are pair-wise distinct). This leaves us with the set of equations byLagrange multipliers.
∂ ln |vn|∂xk
= λ∂sn∂xk
, sn = 1, (79)
where the left-most equation holds for 1 ≤ k ≤ n. It is easy to show thatthe partial derivatives can be written
∂ ln |vn|∂xk
=
n∑i=1i =k
1
xk − xi. (80)
96
2.2. OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SOME N -DIMENSIONAL SURFACES
From this it is easy to show that (80) can be rewritten by introducing the
univariate polynomial Pn(x) =n∏
i=1
(x− xi) as
∂ ln |vn|∂xk
=1
2
P ′′(xk)
P ′(xk).
Now the leftmost equation of (79) can be written
1
2
P ′′(xk)
P ′(xk)= λ
∂sn∂xk
,
or more succinct
P ′′(xk)− 2λ∂sn∂xk
P ′(xk) = 0, (81)
In the case p = 2 we are lucky since (81) becomes, by introducing the new“multiplier” ρ (
P ′′(x) + ρnxP′(x)
)|x=xk
= 0, (82)
and since the left part of this equation is a polynomial of degree n and hasroots x1, · · · , xn we must have
P ′′(x) + ρnxP′(x) + σnP (x) = 0, (83)
for some ρn, σn that may depend on n. Now if choose P (x) to be monic,note that ρn = 0, and require us to be on the sphere we get P (x) = xn −12x
n−2 + · · · , and by identifying coefficients we get ρn and σn:
P ′′(x) + n(1− n)xP ′(x) + n2(n− 1)P (x) = 0,
which is a nice and well known form of differential equation and defines asequence of orthogonal polynomials that are rescaled Hermite polynomials[158], so we can find a recurrence relation for Pn+1 in terms of Pn and Pn−1,and we can, for a fixed n, construct the coefficients of Pn recursively, withoutexplicitly finding P1, · · · , Pn−1.
Now, this is for p = 2. For p = 4 we continue from (81) instead with
(P ′′(x) + ρnx
3P ′(x))|x=xk
= 0. (84)
Now the polynomial in x in the left part of this equation has shared rootswith P (x) and so by the same method as for p = 2 we get:
P ′′(x) + ρnx3P ′(x) + (σnx
2 + τnx+ υn)P (x) = 0. (85)
97
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
It is easy to show for the sphere under any p-norm that the extremepoints of ln |vn(x)| where x1 < · · · < xn are unique, see [158] or argue anal-ogously to the p = 2 case in Section 2.2.1, this coupled with the symmetryrelation ln |vn(x)| = ln |vn(−x)|, provides us with the property that the ex-treme points are symmetric in the sense that for all 1 ≤ i ≤ n we have thatthere exists a 1 ≤ j ≤ n such that xi = −xj , for odd n we then have thatxi = 0 for some i. We thus get polynomials P (x) on the form:
P (x) = xn + cn−2xn−2 + cn−4x
n−4 + · · · ,
with every other coefficient zero, for even n we have only even powers, forodd n we have odd powers. By identifying powers in (85) we get thatτnxP (x) will not share any powers with any other part of the equation andso τn = 0. We can also by identifying coefficients get nρn+σn = 0. We nowhave
P ′′(x) + ρnx3P ′(x) + (−nρnx
2 + υn)P (x) = 0. (86)
For n = 2 we get the specific system
2 + ρx3(2x) + (−2ρx2 + υ)(x2 + c0) = 0,
but we actually don’t need to calculate much here since it is easy do adaptthe roots of x2 + c0 to the sphere with p = 4, we get:
P 42 (x) = x2 − 1√
2.
The case n = 3 is also easy and by symmetry we get a zero coordinate:
P 43 (x) = x3 − 1√
2x.
The case n = 4 becomes a bit more interesting:
(12x2 + 2c2) + ρx3(4x3 + 2c2x) + (−4ρx2 + υ)(x4 + c2x2 + c0) = 0,
(υ − 2ρc2)x4 + (12 + υc2 − 4ρc0)x
2 + (2c2 + υc0) = 0,
This provides three equations. Now letting t = x2 so that
P (t) = t2 + c2t+ c0 = (t− t1)(t− t2) = t2 − (t1 + t2)t+ t1t2,
gives us the last equation∑
x4i = 2∑
t2i = 2(c22 − 2c0) = 1. Solving thisgives us
P 44 (x) = x4 − 2√
6x2 +
1
12.
98
Chapter 3
Approximation ofelectrostatic dischargecurrents using theanalytically extendedfunction
This chapter is based on Papers D, E and F:
Paper D. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov.On some properties of the multi-peaked analytically extended functionfor approximation of lightning discharge currents. Sergei Silvestrovand Milica Rancic, editors, Engineering Mathematics I: Electromag-netics, Fluid Mechanics, Material Physics and Financial Engineer-ing, volume 178 of Springer Proceedings in Mathematics & Statistics.Springer International Publishing, 2016.
Paper E. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov.Estimation of parameters for the multi-peaked AEF current functions.Methodology and Computing in Applied Probability, pages 1–15, 2016.
Paper F. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov.Electrostatic discharge currents representation using the multi-peakedanalytically extended function by interpolation on a D-optimal design.Preprint: arXiv:1701.03728 [physics.comp-ph], 2017.
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
It is easy to show for the sphere under any p-norm that the extremepoints of ln |vn(x)| where x1 < · · · < xn are unique, see [158] or argue anal-ogously to the p = 2 case in Section 2.2.1, this coupled with the symmetryrelation ln |vn(x)| = ln |vn(−x)|, provides us with the property that the ex-treme points are symmetric in the sense that for all 1 ≤ i ≤ n we have thatthere exists a 1 ≤ j ≤ n such that xi = −xj , for odd n we then have thatxi = 0 for some i. We thus get polynomials P (x) on the form:
P (x) = xn + cn−2xn−2 + cn−4x
n−4 + · · · ,
with every other coefficient zero, for even n we have only even powers, forodd n we have odd powers. By identifying powers in (85) we get thatτnxP (x) will not share any powers with any other part of the equation andso τn = 0. We can also by identifying coefficients get nρn+σn = 0. We nowhave
P ′′(x) + ρnx3P ′(x) + (−nρnx
2 + υn)P (x) = 0. (86)
For n = 2 we get the specific system
2 + ρx3(2x) + (−2ρx2 + υ)(x2 + c0) = 0,
but we actually don’t need to calculate much here since it is easy do adaptthe roots of x2 + c0 to the sphere with p = 4, we get:
P 42 (x) = x2 − 1√
2.
The case n = 3 is also easy and by symmetry we get a zero coordinate:
P 43 (x) = x3 − 1√
2x.
The case n = 4 becomes a bit more interesting:
(12x2 + 2c2) + ρx3(4x3 + 2c2x) + (−4ρx2 + υ)(x4 + c2x2 + c0) = 0,
(υ − 2ρc2)x4 + (12 + υc2 − 4ρc0)x
2 + (2c2 + υc0) = 0,
This provides three equations. Now letting t = x2 so that
P (t) = t2 + c2t+ c0 = (t− t1)(t− t2) = t2 − (t1 + t2)t+ t1t2,
gives us the last equation∑
x4i = 2∑
t2i = 2(c22 − 2c0) = 1. Solving thisgives us
P 44 (x) = x4 − 2√
6x2 +
1
12.
98
Chapter 3
Approximation ofelectrostatic dischargecurrents using theanalytically extendedfunction
This chapter is based on Papers D, E and F:
Paper D. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov.On some properties of the multi-peaked analytically extended functionfor approximation of lightning discharge currents. Sergei Silvestrovand Milica Rancic, editors, Engineering Mathematics I: Electromag-netics, Fluid Mechanics, Material Physics and Financial Engineer-ing, volume 178 of Springer Proceedings in Mathematics & Statistics.Springer International Publishing, 2016.
Paper E. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov.Estimation of parameters for the multi-peaked AEF current functions.Methodology and Computing in Applied Probability, pages 1–15, 2016.
Paper F. Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov.Electrostatic discharge currents representation using the multi-peakedanalytically extended function by interpolation on a D-optimal design.Preprint: arXiv:1701.03728 [physics.comp-ph], 2017.
3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)
3.1 The analytically extended function (AEF)
In this section we consider least square approximation using a particularfunction we call the power exponential function, defined in Definition 3.1, asa basis.
Definition 3.1. Here we will refer to the function defined by (87) as thepower exponential function,
x(β; t) =(te1−t
)β, 0 ≤ t. (87)
For non-negative values of t and β the power exponential function hasa steeply rising initial part followed by a more slowly decaying part, seeFigure 3.1. This makes it qualitatively similar to several functions thatare popular for approximation of important phenomena in different fieldssuch as approximation of lightning discharge currents and pharmacokinetics.Examples include the biexponential function [22], [151], the Heidler function[65] and the Pulse function [178].
Figure 3.1: An illustration of how the steepness of the power exponential functionvaries with β.
The power exponential function has been used in applications, for ex-ample to model attack rate of predatory fish, see [135,136].
Here we examine linear combinations of piecewise power exponentialfunctions that in later sections will be used to approximate electrostaticdischarge current functions.
101
3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)
3.1 The analytically extended function (AEF)
In this section we consider least square approximation using a particularfunction we call the power exponential function, defined in Definition 3.1, asa basis.
Definition 3.1. Here we will refer to the function defined by (87) as thepower exponential function,
x(β; t) =(te1−t
)β, 0 ≤ t. (87)
For non-negative values of t and β the power exponential function hasa steeply rising initial part followed by a more slowly decaying part, seeFigure 3.1. This makes it qualitatively similar to several functions thatare popular for approximation of important phenomena in different fieldssuch as approximation of lightning discharge currents and pharmacokinetics.Examples include the biexponential function [22], [151], the Heidler function[65] and the Pulse function [178].
Figure 3.1: An illustration of how the steepness of the power exponential functionvaries with β.
The power exponential function has been used in applications, for ex-ample to model attack rate of predatory fish, see [135,136].
Here we examine linear combinations of piecewise power exponentialfunctions that in later sections will be used to approximate electrostaticdischarge current functions.
101
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
3.1.1 The p-peak analytically extended function
This section is based on Section 2 of Paper D
The p -peaked AEF is constructed using the power exponential functiongiven in Definition 3.1. In order to get a function with multiple peaks andwhere the steepness of the rise between each peak as well as the slope ofthe decaying part is not dependent on each other, we define the analyti-cally extended function (AEF) as a function that consist of piecewise linearcombinations of the power exponential function that has been scaled andtranslated so that the resulting function is continuous. With a given differ-ence in height between subsequent peaks Im1 , Im2 , . . . , Imp , correspondingtimes tm1 , tm2 , . . . , tmp , integers nq > 0, real values βq,k, ηq,k, 1 ≤ q ≤ p+1,1 ≤ k ≤ nq such that the sum over k of ηq,k is equal to one, the p -peakedAEF i(t) is given by (88).
Definition 3.2. Given Imq ∈ R and tmq ∈ R, q = 1, 2, . . . , p such thattm0 = 0 < tm1 < tm2 < . . . < tmp along with ηq,k, βq,k ∈ R and 0 < nq ∈ Z
for q = 1, 2, . . . , p+ 1, k = 1, 2, . . . , nq such that
nq∑k=1
ηq,k = 1.
The analytically extended function (AEF), i(t), with p peaks is defined as
i(t)=
(q−1∑k=1
Imk
)+Imq
nq∑k=1
ηq,kxq(t)β2q,k+1, tmq−1 ≤ t ≤ tmq , 1≤q≤p,
(p∑
k=1
Imk
) np+1∑k=1
ηp+1,kxp+1(t)β2p+1,k , tmp ≤ t,
(88)
where
xq(t) =
t− tmq−1
∆tmq
exp
(tmq − t
∆tmq
), 1 ≤ q ≤ p,
t
tmq
exp
(1− t
tmq
), q = p+ 1,
and ∆tmq = tmq − tmq−1 .
Sometimes the notation i(t;β,η) with
β =[β1,1 β1,2 . . . βq,k . . . βp+1,np+1
],
η =[η1,1 η1,2 . . . ηq,k . . . ηp+1,np+1
],
will be used to clarify what the particular parameters for a certain AEF are.
102
3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)
Remark 3.1. The p -peak AEF can be written more compactly if we intro-duce the vectors
ηq = [ηq,1 ηq,2 . . . ηq,nq ], (89)
xq(t) =
[xq(t)
β2q,1+1 xq(t)
β2q,2+1 . . . xq(t)
β2q,nq
+1]
, 1 ≤ q ≤ p,[xq(t)
β2q,1 xq(t)
β2q,2 . . . xq(t)
β2q,nq
], q = p+ 1.
(90)
The more compact form is
i(t) =
(q−1∑k=1
Imk
)+ Imq · η
q xq(t), tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p,
(q∑
k=1
Imk
)· η
q xq(t), tmq ≤ t, q = p+ 1.
(91)
If the AEF is used to model an electrical current, than the derivativeof the AEF determines the induced electrical voltage in conductive loops inthe lightning field. For this reason it is desirable to guarantee that the firstderivative of the AEF is continuous.
Since the AEF is a linear function of elementary functions its derivativecan be found using standard methods.
Theorem 3.1. The derivative of the p -peak AEF is
di(t)
dt=
Imq
tmq − t
t− tmq−1
xq(t)
∆tmq
ηq Bq xq(t), tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p,
Imq
xq(t)
t
tmq − t
tmq
ηq Bq xq(t), tmq ≤ t, q = p+ 1,
(92)
where
Bq =
β2q,1 + 1 0 . . . 0
0 β2q,2 + 1 . . . 0
......
. . ....
0 0 . . . β2q,nq
+ 1
,
Bp+1 =
β2p+1,1 0 . . . 0
0 β2p+1,2 . . . 0
......
. . ....
0 0 . . . β2p+1,np+1
,
for 1 ≤ q ≤ p.
103
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
3.1.1 The p-peak analytically extended function
This section is based on Section 2 of Paper D
The p -peaked AEF is constructed using the power exponential functiongiven in Definition 3.1. In order to get a function with multiple peaks andwhere the steepness of the rise between each peak as well as the slope ofthe decaying part is not dependent on each other, we define the analyti-cally extended function (AEF) as a function that consist of piecewise linearcombinations of the power exponential function that has been scaled andtranslated so that the resulting function is continuous. With a given differ-ence in height between subsequent peaks Im1 , Im2 , . . . , Imp , correspondingtimes tm1 , tm2 , . . . , tmp , integers nq > 0, real values βq,k, ηq,k, 1 ≤ q ≤ p+1,1 ≤ k ≤ nq such that the sum over k of ηq,k is equal to one, the p -peakedAEF i(t) is given by (88).
Definition 3.2. Given Imq ∈ R and tmq ∈ R, q = 1, 2, . . . , p such thattm0 = 0 < tm1 < tm2 < . . . < tmp along with ηq,k, βq,k ∈ R and 0 < nq ∈ Z
for q = 1, 2, . . . , p+ 1, k = 1, 2, . . . , nq such that
nq∑k=1
ηq,k = 1.
The analytically extended function (AEF), i(t), with p peaks is defined as
i(t)=
(q−1∑k=1
Imk
)+Imq
nq∑k=1
ηq,kxq(t)β2q,k+1, tmq−1 ≤ t ≤ tmq , 1≤q≤p,
(p∑
k=1
Imk
) np+1∑k=1
ηp+1,kxp+1(t)β2p+1,k , tmp ≤ t,
(88)
where
xq(t) =
t− tmq−1
∆tmq
exp
(tmq − t
∆tmq
), 1 ≤ q ≤ p,
t
tmq
exp
(1− t
tmq
), q = p+ 1,
and ∆tmq = tmq − tmq−1 .
Sometimes the notation i(t;β,η) with
β =[β1,1 β1,2 . . . βq,k . . . βp+1,np+1
],
η =[η1,1 η1,2 . . . ηq,k . . . ηp+1,np+1
],
will be used to clarify what the particular parameters for a certain AEF are.
102
3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)
Remark 3.1. The p -peak AEF can be written more compactly if we intro-duce the vectors
ηq = [ηq,1 ηq,2 . . . ηq,nq ], (89)
xq(t) =
[xq(t)
β2q,1+1 xq(t)
β2q,2+1 . . . xq(t)
β2q,nq
+1]
, 1 ≤ q ≤ p,[xq(t)
β2q,1 xq(t)
β2q,2 . . . xq(t)
β2q,nq
], q = p+ 1.
(90)
The more compact form is
i(t) =
(q−1∑k=1
Imk
)+ Imq · η
q xq(t), tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p,
(q∑
k=1
Imk
)· η
q xq(t), tmq ≤ t, q = p+ 1.
(91)
If the AEF is used to model an electrical current, than the derivativeof the AEF determines the induced electrical voltage in conductive loops inthe lightning field. For this reason it is desirable to guarantee that the firstderivative of the AEF is continuous.
Since the AEF is a linear function of elementary functions its derivativecan be found using standard methods.
Theorem 3.1. The derivative of the p -peak AEF is
di(t)
dt=
Imq
tmq − t
t− tmq−1
xq(t)
∆tmq
ηq Bq xq(t), tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p,
Imq
xq(t)
t
tmq − t
tmq
ηq Bq xq(t), tmq ≤ t, q = p+ 1,
(92)
where
Bq =
β2q,1 + 1 0 . . . 0
0 β2q,2 + 1 . . . 0
......
. . ....
0 0 . . . β2q,nq
+ 1
,
Bp+1 =
β2p+1,1 0 . . . 0
0 β2p+1,2 . . . 0
......
. . ....
0 0 . . . β2p+1,np+1
,
for 1 ≤ q ≤ p.
103
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Proof. From the definition of the AEF (see (88)) and the derivative of thepower exponential function (87) given by
d
dtx(β; t) = β(1− t)tβ−1eβ(1−t),
expression (92) can easily be derived since differentiation is a linear operationand the result can be rewritten in the compact form analogously to (91).
Illustration of the AEF function and its derivative for various values ofβq,k-parameters is shown in Fig. 3.2.
Figure 3.2: Illustration of the AEF (solid line) and its derivative (dashedline) with different βq,k-parameters but the same Imq
and tmq. (a) 0 < βq,k < 1, (b) 4 < βq,k < 5,
(c) 12 < βq,k < 13, (d) a mixture of large and small βq,k-parameters
Lemma 3.1. The AEF is continuous and at each tmq the derivative is equalto zero.
Proof. Within each interval tmq−1 ≤ t ≤ tmq the AEF is a linear combinationof continuous functions and at each tmq the function will approach the same
104
3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)
value from both directions unless all ηq,k ≤ 0, but if all ηq,k ≤ 0 thennq∑k=1
ηq,k = 1.
Noting that for any diagonal matrix B the expression
ηq Bxq(t) =
nq∑k=1
ηq,kBkkxq(t)β2q,k+1, 1 ≤ q ≤ p,
is well-defined and that the equivalent statement holds for q = p it is easyto see from (92) that the factor (tmq − t) in the derivative ensures that thederivative is zero every time t = tmq .
When interpolating a waveform with p peaks it is natural to require thatthere will not appear new peaks between the chosen peaks. This correspondsto requiring monotonicity in each interval. One way to achieve this is givenin lemma 3.2.
Lemma 3.2. If ηq,k ≥ 0, k = 1, . . . , nq the AEF, i(t), is strictly monotonicon the interval tmq−1 < t < tmq .
Proof. The AEF will be strictly monotonic in an interval if the derivative hasthe same sign everywhere in the interval. That this is the case follows from(92) since every term in η
q Bq xq(t) is non-negative if ηq,k ≥ 0, k = 1, . . . , nq,so the sign of the derivative it determined by Imq .
If we allow some of the ηq,k-parameters to be negative, the derivativecan change sign and the function might get an extra peak between twoother peaks, see Fig. 3.3.
The integral of the electric current represents the charge flow. Unlikethe Heidler function the integral of the AEF is relatively straightforward tofind. How to do this is detailed in lemma 3.3, lemma 3.4, theorem 3.2, andtheorem 3.3.
Lemma 3.3. For any tmq−1 ≤ t0 ≤ t1 ≤ tmq , 1 ≤ q ≤ p,
∫ t1
t0
xq(t)β dt =
eβ
ββ+1∆γ
(β + 1,
t1 − tmq
β∆tmq
,t0 − tmq
β∆tmq
)(93)
with ∆tmq = tmq − tmq−1 and
∆γ(β, t0, t1) = γ (β + 1, βt1)− γ (β + 1, βt0) ,
105
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Proof. From the definition of the AEF (see (88)) and the derivative of thepower exponential function (87) given by
d
dtx(β; t) = β(1− t)tβ−1eβ(1−t),
expression (92) can easily be derived since differentiation is a linear operationand the result can be rewritten in the compact form analogously to (91).
Illustration of the AEF function and its derivative for various values ofβq,k-parameters is shown in Fig. 3.2.
Figure 3.2: Illustration of the AEF (solid line) and its derivative (dashedline) with different βq,k-parameters but the same Imq
and tmq. (a) 0 < βq,k < 1, (b) 4 < βq,k < 5,
(c) 12 < βq,k < 13, (d) a mixture of large and small βq,k-parameters
Lemma 3.1. The AEF is continuous and at each tmq the derivative is equalto zero.
Proof. Within each interval tmq−1 ≤ t ≤ tmq the AEF is a linear combinationof continuous functions and at each tmq the function will approach the same
104
3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)
value from both directions unless all ηq,k ≤ 0, but if all ηq,k ≤ 0 thennq∑k=1
ηq,k = 1.
Noting that for any diagonal matrix B the expression
ηq Bxq(t) =
nq∑k=1
ηq,kBkkxq(t)β2q,k+1, 1 ≤ q ≤ p,
is well-defined and that the equivalent statement holds for q = p it is easyto see from (92) that the factor (tmq − t) in the derivative ensures that thederivative is zero every time t = tmq .
When interpolating a waveform with p peaks it is natural to require thatthere will not appear new peaks between the chosen peaks. This correspondsto requiring monotonicity in each interval. One way to achieve this is givenin lemma 3.2.
Lemma 3.2. If ηq,k ≥ 0, k = 1, . . . , nq the AEF, i(t), is strictly monotonicon the interval tmq−1 < t < tmq .
Proof. The AEF will be strictly monotonic in an interval if the derivative hasthe same sign everywhere in the interval. That this is the case follows from(92) since every term in η
q Bq xq(t) is non-negative if ηq,k ≥ 0, k = 1, . . . , nq,so the sign of the derivative it determined by Imq .
If we allow some of the ηq,k-parameters to be negative, the derivativecan change sign and the function might get an extra peak between twoother peaks, see Fig. 3.3.
The integral of the electric current represents the charge flow. Unlikethe Heidler function the integral of the AEF is relatively straightforward tofind. How to do this is detailed in lemma 3.3, lemma 3.4, theorem 3.2, andtheorem 3.3.
Lemma 3.3. For any tmq−1 ≤ t0 ≤ t1 ≤ tmq , 1 ≤ q ≤ p,
∫ t1
t0
xq(t)β dt =
eβ
ββ+1∆γ
(β + 1,
t1 − tmq
β∆tmq
,t0 − tmq
β∆tmq
)(93)
with ∆tmq = tmq − tmq−1 and
∆γ(β, t0, t1) = γ (β + 1, βt1)− γ (β + 1, βt0) ,
105
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Figure 3.3: An example of a two-peaked AEF where some of the ηq,k-parameters arenegative, so that it has points where the first derivative changes signbetween two peaks. The solid line is the AEF and the dashed lines is thederivative of the AEF.
where
γ(β, t) =
∫ t
0τβ−1e−τ dτ
is the lower incomplete Gamma function [2].
If t0 = tmq−1 and t1 = tmq then
∫ tmq
tmq−1
xq(t)β dt =
eβ
ββ+1γ (β + 1, β) . (94)
Proof.
∫ t1
t0
xq(t)β dt =
∫ t1
t0
(t− tmq
∆tmq
exp
(1−
t− tmq
∆tmq
))β
dt
=eβ−1
ββ
∫ t1
t0
(βt− tmq
∆tmq
)β
exp
(1− β
t− tmq
∆tmq
)dt.
106
3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)
Changing variables according to τ = βt−tmq
∆tmqgives
∫ t1
t0
xq(t)β dt =
eβ
ββ+1
∫ τ1
τ0
τβe−τ dt =
=eβ
ββ+1(γ(β + 1, τ1)− γ(β + 1, τ0))
=eβ
ββ+1∆γ(β + 1, τ1, τ0)
=eβ
ββ+1∆γ
(β + 1, β
t1 − tmq
∆tmq
, βt0 − tmq
∆tmq
).
When t0 = tmq−1 and t1 = tmq then
∫ t1
t0
xq(t)β dt =
eβ
ββ+1∆γ (β + 1, β)
and with γ(β + 1, 0) = 0 we get (94).
Lemma 3.4. For any tmq−1 ≤ t0 ≤ t1 ≤ tmq , 1 ≤ q ≤ p,
∫ t1
t0
i(t) dt = (t1 − t0)
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,k gq(t1, t0), (95)
where
gq(t1, t0) =eβ
2q,k
(β2q,k + 1
)β2q,k+1
∆γ
(β2q,k + 2,
t1 − tmq−1
∆tmq
,t0 − tmq−1
∆tmq
)
with ∆γ(β, t0, t1) defined as in (93).
Proof.
∫ t1
t0
i(t) dt =
∫ t1
t0
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,kxq(t)β2q,k+1 dt
= (t1 − t0)
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,k
∫ t1
t0
xq(t)β2q,k+1 dt
= (t1 − t0)
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,k gq(t0, t1).
107
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Figure 3.3: An example of a two-peaked AEF where some of the ηq,k-parameters arenegative, so that it has points where the first derivative changes signbetween two peaks. The solid line is the AEF and the dashed lines is thederivative of the AEF.
where
γ(β, t) =
∫ t
0τβ−1e−τ dτ
is the lower incomplete Gamma function [2].
If t0 = tmq−1 and t1 = tmq then
∫ tmq
tmq−1
xq(t)β dt =
eβ
ββ+1γ (β + 1, β) . (94)
Proof.
∫ t1
t0
xq(t)β dt =
∫ t1
t0
(t− tmq
∆tmq
exp
(1−
t− tmq
∆tmq
))β
dt
=eβ−1
ββ
∫ t1
t0
(βt− tmq
∆tmq
)β
exp
(1− β
t− tmq
∆tmq
)dt.
106
3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)
Changing variables according to τ = βt−tmq
∆tmqgives
∫ t1
t0
xq(t)β dt =
eβ
ββ+1
∫ τ1
τ0
τβe−τ dt =
=eβ
ββ+1(γ(β + 1, τ1)− γ(β + 1, τ0))
=eβ
ββ+1∆γ(β + 1, τ1, τ0)
=eβ
ββ+1∆γ
(β + 1, β
t1 − tmq
∆tmq
, βt0 − tmq
∆tmq
).
When t0 = tmq−1 and t1 = tmq then
∫ t1
t0
xq(t)β dt =
eβ
ββ+1∆γ (β + 1, β)
and with γ(β + 1, 0) = 0 we get (94).
Lemma 3.4. For any tmq−1 ≤ t0 ≤ t1 ≤ tmq , 1 ≤ q ≤ p,
∫ t1
t0
i(t) dt = (t1 − t0)
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,k gq(t1, t0), (95)
where
gq(t1, t0) =eβ
2q,k
(β2q,k + 1
)β2q,k+1
∆γ
(β2q,k + 2,
t1 − tmq−1
∆tmq
,t0 − tmq−1
∆tmq
)
with ∆γ(β, t0, t1) defined as in (93).
Proof.
∫ t1
t0
i(t) dt =
∫ t1
t0
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,kxq(t)β2q,k+1 dt
= (t1 − t0)
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,k
∫ t1
t0
xq(t)β2q,k+1 dt
= (t1 − t0)
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,k gq(t0, t1).
107
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Theorem 3.2. If tma−1 ≤ ta ≤ tma , tmb−1≤ tb ≤ tmb
and 0 ≤ ta ≤ tb ≤ tmp
then
∫ tb
ta
i(t) dt = (tma − ta)
(a−1∑k=1
Imk
)+ Ima
na∑k=1
ηa,k ga(ta, tma)
+b−1∑
q=a+1
(∆tmq
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,k g(β2q,k + 1
))
+ (tb − tmb)
(b−1∑k=1
Imk
)+ Imb
nb∑k=1
ηb,k gb(tmb, tb), (96)
where gq(t0, t1) is defined as in lemma 3.4 and
g(β) =eβ
ββ+1γ (β + 1, β) .
Proof. This theorem follows from integration being linear and lemma 3.4.
Theorem 3.3. For tmp ≤ t0 < t1 < ∞ the integral of the AEF is
∫ t1
t0
i(t) dt =
(p∑
k=1
Imk
) np+1∑k=1
ηp+1,k gp+1(t1, t0), (97)
where gq(t0, t1) is defined as in Lemma 3.4.When t0 = tmp and t1 → ∞ the integral becomes
∫ ∞
tmp
i(t) dt =
(p∑
k=1
Imk
) np+1∑k=1
ηp+1,k g(β2p+1,k
), (98)
where
g(β) =eβ
ββ+1(Γ(β + 1)− γ (β + 1, β))
with
Γ(β) =
∫ ∞
0tβ−1e−t dt
is the Gamma function [2].
Proof. This theorem follows from integration being linear and Lemma 3.4.
108
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
In the next section we will estimate the parameters of the AEF that givesthe best fit with respect to some data and for this the partial derivativeswith respect to the βmq parameters will be useful. Since the AEF is a linearfunction of elementary functions these partial derivatives can easily be foundusing standard methods.
Theorem 3.4. The partial derivatives of the p-peak AEF with respect tothe β parameters are
∂i
∂βq,k=
0, 0 ≤ t ≤ tmq−1 ,
2 Imqηq,k βq,k hq(t)xq(t)β2q,k+1, tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p,
0, tmq ≤ t,
(99)
∂i
∂βp+1,k=
0, 0 ≤ t ≤ tmp ,
2 Imp+1ηp+1,k βp+1,k hp+1(t)xp+1(t)β2p+1,k , tmp ≤ t,
(100)where
hq(t) =
ln
(t− tmq−1
∆tmq
)−
t− tmq−1
∆tmq
+ 1, 1 ≤ q ≤ p,
ln
(t
tmq
)− t
tmq
+ 1, q = p+ 1.
Proof. Since the βq,k parameters are independent, differentiation with re-spect to βq,k will annihilate all terms but one in each linear combination.The expressions (99) and (100) then follow from the standard rules for dif-ferentiation of composite functions and products of functions.
3.2 Approximation of lightning discharge currentfunctions
This section is based on Section 3 of Paper E
Many different types of systems, objects and equipment are susceptibleto damage from lightning discharges. Lightning effects are usually anal-ysed using lightning discharge models. Most of the engineering and electro-magnetic models imply channel-base current functions. Various single andmulti-peaked functions are proposed in the literature for modelling lightningchannel-base currents, examples include [65, 66, 78, 79, 83]. For engineering
109
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Theorem 3.2. If tma−1 ≤ ta ≤ tma , tmb−1≤ tb ≤ tmb
and 0 ≤ ta ≤ tb ≤ tmp
then
∫ tb
ta
i(t) dt = (tma − ta)
(a−1∑k=1
Imk
)+ Ima
na∑k=1
ηa,k ga(ta, tma)
+b−1∑
q=a+1
(∆tmq
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,k g(β2q,k + 1
))
+ (tb − tmb)
(b−1∑k=1
Imk
)+ Imb
nb∑k=1
ηb,k gb(tmb, tb), (96)
where gq(t0, t1) is defined as in lemma 3.4 and
g(β) =eβ
ββ+1γ (β + 1, β) .
Proof. This theorem follows from integration being linear and lemma 3.4.
Theorem 3.3. For tmp ≤ t0 < t1 < ∞ the integral of the AEF is
∫ t1
t0
i(t) dt =
(p∑
k=1
Imk
) np+1∑k=1
ηp+1,k gp+1(t1, t0), (97)
where gq(t0, t1) is defined as in Lemma 3.4.When t0 = tmp and t1 → ∞ the integral becomes
∫ ∞
tmp
i(t) dt =
(p∑
k=1
Imk
) np+1∑k=1
ηp+1,k g(β2p+1,k
), (98)
where
g(β) =eβ
ββ+1(Γ(β + 1)− γ (β + 1, β))
with
Γ(β) =
∫ ∞
0tβ−1e−t dt
is the Gamma function [2].
Proof. This theorem follows from integration being linear and Lemma 3.4.
108
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
In the next section we will estimate the parameters of the AEF that givesthe best fit with respect to some data and for this the partial derivativeswith respect to the βmq parameters will be useful. Since the AEF is a linearfunction of elementary functions these partial derivatives can easily be foundusing standard methods.
Theorem 3.4. The partial derivatives of the p-peak AEF with respect tothe β parameters are
∂i
∂βq,k=
0, 0 ≤ t ≤ tmq−1 ,
2 Imqηq,k βq,k hq(t)xq(t)β2q,k+1, tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p,
0, tmq ≤ t,
(99)
∂i
∂βp+1,k=
0, 0 ≤ t ≤ tmp ,
2 Imp+1ηp+1,k βp+1,k hp+1(t)xp+1(t)β2p+1,k , tmp ≤ t,
(100)where
hq(t) =
ln
(t− tmq−1
∆tmq
)−
t− tmq−1
∆tmq
+ 1, 1 ≤ q ≤ p,
ln
(t
tmq
)− t
tmq
+ 1, q = p+ 1.
Proof. Since the βq,k parameters are independent, differentiation with re-spect to βq,k will annihilate all terms but one in each linear combination.The expressions (99) and (100) then follow from the standard rules for dif-ferentiation of composite functions and products of functions.
3.2 Approximation of lightning discharge currentfunctions
This section is based on Section 3 of Paper E
Many different types of systems, objects and equipment are susceptibleto damage from lightning discharges. Lightning effects are usually anal-ysed using lightning discharge models. Most of the engineering and electro-magnetic models imply channel-base current functions. Various single andmulti-peaked functions are proposed in the literature for modelling lightningchannel-base currents, examples include [65, 66, 78, 79, 83]. For engineering
109
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and electromagnetic models, a general function that would be able to re-produce desired waveshapes is needed, such that analytical solutions for itsderivatives, integrals, and integral transformations, exist. A multi-peakedchannel-base current function has been proposed by Javor [78] as a gen-eralization of the so-called TRF (two-rise front) function from [79], whichpossesses such properties.
In this paper we analyse a modification of such multi-peaked function,a so-called p -peak analytically extended function (AEF). The possibilityof application of the AEF to approximation of various multi-peaked wave-shapes is investigated. Estimation of its parameters has been performedusing the Marquardt least-squares method (MLSM), an efficient method forthe estimation of non-linear function parameters, see Section 1.3.3. It hasbeen applied in many fields, including lightning research, e.g. for optimiz-ing parameters of the Heidler function in Lovric et al. [103], or the Pulsefunction in Lundengard et al. [105]- [106].
Some numerical results are presented, including those for the StandardIEC 62305 [74] current of the first-positive strokes, and an example of a fast-decaying lightning current waveform. Fitting a p-peaked AEF to recordedcurrent data (from [152]) is also illustrated.
3.2.1 Fitting the AEF
Suppose that we have kq points (tq,k, iq,k) ordered with respect to tq,k,tmq−1 < tq,1 < tq,2 < . . . < tq,kq < tmq , and we wish to choose parame-ters ηq,k and βq,k such that the sum of the squares of the residuals,
Sq =
kq∑k=1
(i(tq,k)− iq,k)2 , (101)
is minimized. One way to estimate these parameters is to use the Marquardtleast-square method described in Section 1.3.3.
In order to fit the AEF it is sufficient that kq ≥ nq. Suppose we have someestimate of the β-parameters which is collected in the vector b. It is thenfairly simple to calculate an estimate for the η-parameters, see Section 3.2.4,which we collect in h. We define a residual vector by (e)k = i(tq,k;b,h)−iq,kwhere i(t;b,h) is the AEF with the estimated parameters.
110
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
The Jacobian matrix, J, can in this case be described as
J =
∂i∂βq,1
∣∣∣t=tq,1
∂i∂βq,2
∣∣∣t=tq,1
. . . ∂i∂βq,nq
∣∣∣t=tq,1
∂i∂βq,1
∣∣∣t=tq,2
∂i∂βq,2
∣∣∣t=tq,2
. . . ∂i∂βq,nq
∣∣∣t=tq,2
......
. . ....
∂i∂βq,1
∣∣∣t=tq,kq
∂i∂βq,2
∣∣∣t=tq,kq
. . . ∂i∂βq,nq
∣∣∣t=tq,kq
(102)
where the partial derivatives are given by (99) and (100).
3.2.2 Estimating parameters for underdetermined systems
This section is based on Section 3.2 of Paper D
For the Marquardt least-squares method to work at least one data point perunknown parameter is needed, m ≥ k. It can still be possible to estimateall unknown parameters if there is insufficient data, m < k if we know somefurther relations between the parameters.
Suppose that k − m = p and let γj = βm+j , j = 1, 2, · · · , p. If thereare at least p known relations between the unknown parameters such thatγj = γj(β1, · · · , βm) for j = 1, 2, · · · , p then the Marquardt least-squaresmethod can be used to give estimates on β1, · · · , βm and the still unknownparameters can be estimated from these. Denoting the estimated parametersb = (b1, · · · , bm) and c = (c1, · · · , cp) the following algorithm can be used:
• Input: v > 1 and initial values b(0), λ(0).
• r = 0
Find c(r) using b(r) together with extra relations.
• Find b(r+1) and δ(r) using MLSM.
• Check chosen termination condition for MLSM, if it is not satisfied goto .
• Output: b, c.
The algorithm is illustrated in figure 3.4.
111
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and electromagnetic models, a general function that would be able to re-produce desired waveshapes is needed, such that analytical solutions for itsderivatives, integrals, and integral transformations, exist. A multi-peakedchannel-base current function has been proposed by Javor [78] as a gen-eralization of the so-called TRF (two-rise front) function from [79], whichpossesses such properties.
In this paper we analyse a modification of such multi-peaked function,a so-called p -peak analytically extended function (AEF). The possibilityof application of the AEF to approximation of various multi-peaked wave-shapes is investigated. Estimation of its parameters has been performedusing the Marquardt least-squares method (MLSM), an efficient method forthe estimation of non-linear function parameters, see Section 1.3.3. It hasbeen applied in many fields, including lightning research, e.g. for optimiz-ing parameters of the Heidler function in Lovric et al. [103], or the Pulsefunction in Lundengard et al. [105]- [106].
Some numerical results are presented, including those for the StandardIEC 62305 [74] current of the first-positive strokes, and an example of a fast-decaying lightning current waveform. Fitting a p-peaked AEF to recordedcurrent data (from [152]) is also illustrated.
3.2.1 Fitting the AEF
Suppose that we have kq points (tq,k, iq,k) ordered with respect to tq,k,tmq−1 < tq,1 < tq,2 < . . . < tq,kq < tmq , and we wish to choose parame-ters ηq,k and βq,k such that the sum of the squares of the residuals,
Sq =
kq∑k=1
(i(tq,k)− iq,k)2 , (101)
is minimized. One way to estimate these parameters is to use the Marquardtleast-square method described in Section 1.3.3.
In order to fit the AEF it is sufficient that kq ≥ nq. Suppose we have someestimate of the β-parameters which is collected in the vector b. It is thenfairly simple to calculate an estimate for the η-parameters, see Section 3.2.4,which we collect in h. We define a residual vector by (e)k = i(tq,k;b,h)−iq,kwhere i(t;b,h) is the AEF with the estimated parameters.
110
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
The Jacobian matrix, J, can in this case be described as
J =
∂i∂βq,1
∣∣∣t=tq,1
∂i∂βq,2
∣∣∣t=tq,1
. . . ∂i∂βq,nq
∣∣∣t=tq,1
∂i∂βq,1
∣∣∣t=tq,2
∂i∂βq,2
∣∣∣t=tq,2
. . . ∂i∂βq,nq
∣∣∣t=tq,2
......
. . ....
∂i∂βq,1
∣∣∣t=tq,kq
∂i∂βq,2
∣∣∣t=tq,kq
. . . ∂i∂βq,nq
∣∣∣t=tq,kq
(102)
where the partial derivatives are given by (99) and (100).
3.2.2 Estimating parameters for underdetermined systems
This section is based on Section 3.2 of Paper D
For the Marquardt least-squares method to work at least one data point perunknown parameter is needed, m ≥ k. It can still be possible to estimateall unknown parameters if there is insufficient data, m < k if we know somefurther relations between the parameters.
Suppose that k − m = p and let γj = βm+j , j = 1, 2, · · · , p. If thereare at least p known relations between the unknown parameters such thatγj = γj(β1, · · · , βm) for j = 1, 2, · · · , p then the Marquardt least-squaresmethod can be used to give estimates on β1, · · · , βm and the still unknownparameters can be estimated from these. Denoting the estimated parametersb = (b1, · · · , bm) and c = (c1, · · · , cp) the following algorithm can be used:
• Input: v > 1 and initial values b(0), λ(0).
• r = 0
Find c(r) using b(r) together with extra relations.
• Find b(r+1) and δ(r) using MLSM.
• Check chosen termination condition for MLSM, if it is not satisfied goto .
• Output: b, c.
The algorithm is illustrated in figure 3.4.
111
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Input: choose v andinitial values for b(0) and λ(0) r = 0
Find b(r+1) and δ(r)
using MLSMFind h(r) using b(r)
together with extra relations
Termination conditionsatisfied
r = r + 1
Output: b, h
YES
NO
Figure 3.4: Schematic description of the parameter estimation algorithm
3.2.3 Fitting with data points as well as charge flow andspecific energy conditions
By considering the charge flow at the striking point, Q0, unitary resistanceR and the specific energy, W0, we get two further conditions:
Q0 =
∫ ∞
0i(t) dt, (103)
W0 =
∫ ∞
0i(t)2 dt. (104)
First we will define
Q(b,h) =
∫ ∞
0i(t;b,h) dt
W (b,h) =
∫ ∞
0i(t;b,h)2 dt.
These two quantities can be calculated as follows.
Theorem 3.5.
Q(b,h) =
p∑q=1
(∆tmq
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,k g(β2q,k + 1)
)
+
(p∑
k=1
Imk
) np+1∑k=1
ηp+1,k g(β2p+1,k), (105)
112
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
W (b,h) =
p∑q=1
(
q−1∑k=1
Imk
)2
+
(q−1∑k=1
Imk
)Imq
nq∑k=1
ηq,k g(β2q,k + 1)
+ I2mq
nq∑k=1
η2q,k g(2β2
q,k + 2)
+ 2 I2mq
nq−1∑r=1
nq∑s=r+1
ηq,r ηq,s g(β2q,r + β2
q,s + 2)
+
(p∑
k=1
Imk
)2( np∑k=1
η2p,k g(2β2
p,k
)
+2
np+1−1∑r=1
np+1∑s=r+1
ηp+1,r ηp+1,s g(β2p+1,r + β2
p+1,s
)
(106)
where g(β) and g(β) are defined in Theorem 3.2 and 3.3.
Proof. Formula (105) is found by combining (96) and (97).Formula (106) is found by noting that
(n∑
k=1
ak
)2
=n∑
k=1
a2k +n−1∑r=1
n∑s=r+1
ar as
and then reasoning analogously to the proofs for (96) and (97).
We can calculate the charge flow and specific energy given by the AEFwith formulas (105) and (106), respectively, and get two additional residualterms EQ0 = Q(b,h)−Q0 and EW0 = W (b,h)−W0. Since these are globalconditions this means that the parameters η and β no longer can be fittedseparately in each interval. This means that we need to consider all datapoints simultaneously. The resulting J-matrix is
J =
J1 . . . 0...
. . ....
0 . . . Jp+1∂EQ0∂β1,1
. . .∂EQ0∂β1,n1
. . .∂EQ0∂βp+1,1
. . .∂EQ0
∂βp+1,np+1∂EW0∂β1,1
. . .∂EW0∂β1,n1
. . .∂EW0∂βp+1,1
. . .∂EW0
∂βp+1,np+1
(107)
113
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Input: choose v andinitial values for b(0) and λ(0) r = 0
Find b(r+1) and δ(r)
using MLSMFind h(r) using b(r)
together with extra relations
Termination conditionsatisfied
r = r + 1
Output: b, h
YES
NO
Figure 3.4: Schematic description of the parameter estimation algorithm
3.2.3 Fitting with data points as well as charge flow andspecific energy conditions
By considering the charge flow at the striking point, Q0, unitary resistanceR and the specific energy, W0, we get two further conditions:
Q0 =
∫ ∞
0i(t) dt, (103)
W0 =
∫ ∞
0i(t)2 dt. (104)
First we will define
Q(b,h) =
∫ ∞
0i(t;b,h) dt
W (b,h) =
∫ ∞
0i(t;b,h)2 dt.
These two quantities can be calculated as follows.
Theorem 3.5.
Q(b,h) =
p∑q=1
(∆tmq
(q−1∑k=1
Imk
)+ Imq
nq∑k=1
ηq,k g(β2q,k + 1)
)
+
(p∑
k=1
Imk
) np+1∑k=1
ηp+1,k g(β2p+1,k), (105)
112
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
W (b,h) =
p∑q=1
(
q−1∑k=1
Imk
)2
+
(q−1∑k=1
Imk
)Imq
nq∑k=1
ηq,k g(β2q,k + 1)
+ I2mq
nq∑k=1
η2q,k g(2β2
q,k + 2)
+ 2 I2mq
nq−1∑r=1
nq∑s=r+1
ηq,r ηq,s g(β2q,r + β2
q,s + 2)
+
(p∑
k=1
Imk
)2( np∑k=1
η2p,k g(2β2
p,k
)
+2
np+1−1∑r=1
np+1∑s=r+1
ηp+1,r ηp+1,s g(β2p+1,r + β2
p+1,s
)
(106)
where g(β) and g(β) are defined in Theorem 3.2 and 3.3.
Proof. Formula (105) is found by combining (96) and (97).Formula (106) is found by noting that
(n∑
k=1
ak
)2
=n∑
k=1
a2k +n−1∑r=1
n∑s=r+1
ar as
and then reasoning analogously to the proofs for (96) and (97).
We can calculate the charge flow and specific energy given by the AEFwith formulas (105) and (106), respectively, and get two additional residualterms EQ0 = Q(b,h)−Q0 and EW0 = W (b,h)−W0. Since these are globalconditions this means that the parameters η and β no longer can be fittedseparately in each interval. This means that we need to consider all datapoints simultaneously. The resulting J-matrix is
J =
J1 . . . 0...
. . ....
0 . . . Jp+1∂EQ0∂β1,1
. . .∂EQ0∂β1,n1
. . .∂EQ0∂βp+1,1
. . .∂EQ0
∂βp+1,np+1∂EW0∂β1,1
. . .∂EW0∂β1,n1
. . .∂EW0∂βp+1,1
. . .∂EW0
∂βp+1,np+1
(107)
113
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
where
Jq =
∂i∂βq,1
∣∣∣t=tq,1
∂i∂βq,2
∣∣∣t=tq,1
. . . ∂i∂βq,nq
∣∣∣t=tq,1
∂i∂βq,1
∣∣∣t=tq,2
∂i∂βq,2
∣∣∣t=tq,2
. . . ∂i∂βq,nq
∣∣∣t=tq,2
......
. . ....
∂i∂βq,1
∣∣∣t=tq,kq
∂i∂βq,2
∣∣∣t=tq,kq
. . . ∂i∂βq,nq
∣∣∣t=tq,kq
and the partial derivatives in the last two rows are given by
∂Q
∂βq,s=
2 Imqηq,s βq,sdg
dβ
∣∣∣∣β=β2
q,s+1
, 1 ≤ q ≤ p,
2 Impηp+1,s βp+1,sdg
dβ
∣∣∣∣β=β2
p+1,s
, q = p+ 1.
For 1 ≤ q ≤ p
∂W
∂βq,s= 2
(q−1∑k=1
Imk
)Imqηq,s βq,s
dg
dβ
∣∣∣∣β=β2
q,s+1
+4 I2mqηq,sβq,s
ηq,s
dg
dβ
∣∣∣∣β=2β2
q,s+2
+
nq∑k=1k =s
ηq,kdg
dβ
∣∣∣∣β=β2
q,s+β2q,k+2
and
∂W
∂βp+1,s= 4
(p∑
k=1
Imk
)ηp+1,sβp+1,s
ηp+1,s
dg
dβ
∣∣∣∣β=2β2
p+1,s
+
nq∑k=1k =s
ηp+1,kdg
dβ
∣∣∣∣β=β2
p+1,s+β2p+1,k
.
The derivatives of g(β) and g(β) are
dg
dβ=1
e
(1 +
eβ
ββ
(Γ(β + 1)
(Ψ(β)− ln(β)
)−G(β)
)), (108)
dg
dβ=1
e
(eβ
ββG(β)− 1
), (109)
114
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
where Γ(β) is the Gamma function, Ψ(β) is the digamma function, see forexample [2], and G(β) is a special case of the Meijer G-function and can bedefined as
G(β) = G3,02,3
(β
∣∣∣∣1, 1
0, 0, β + 1
)
using the notation from [139]. When evaluating this function it might bemore practical to rewrite G using other special functions
G(β) = G3,02,3
(β
∣∣∣∣1, 1
0, 0, β + 1
)=
ββ+1
(β + 1)22F2(β + 1, β + 1; β + 2, β + 2; −β)
−(Ψ(β) + π cot(πβ) + ln(β)
)π csc (πβ)
Γ (−β)
where
2F2(β + 1, β + 1; β + 2, β + 2; −β) =∞∑k=0
(−1)kβk (β + 1)2
(β + k + 1)2
is a special case of the hypergeometric function. These partial derivativeswere found using software foe symbolic computation [1].
Note that all η-parameters must be recalculated for each step and howthis is done is detailed in Section 3.2.4.
3.2.4 Calculating the η-parameters from the β-parameters
Suppose that we have nq − 1 points (tq,k, iq,k) such that
tmq−1 < tq,1 < tq,2 < . . . < tq,nq−1 < tmq .
For an AEF that interpolates these points it must be true that
q−1∑k=1
Imk+ Imq
nq∑s=1
ηq,sxq(tq,k)βq,s = iq,k, k = 1, 2, . . . , nq − 1. (110)
Since ηq,1 + ηq,2 + . . .+ ηq,nq = 1 equation (110) can be rewritten as
Imq
nq−1∑s=1
ηq,s
(xq(tq,k)
βq,s − xq(tq,k)βq,nq
)= iq,k − xq(tq,k)
βq,nq −q−1∑s=1
Ims
(111)
115
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
where
Jq =
∂i∂βq,1
∣∣∣t=tq,1
∂i∂βq,2
∣∣∣t=tq,1
. . . ∂i∂βq,nq
∣∣∣t=tq,1
∂i∂βq,1
∣∣∣t=tq,2
∂i∂βq,2
∣∣∣t=tq,2
. . . ∂i∂βq,nq
∣∣∣t=tq,2
......
. . ....
∂i∂βq,1
∣∣∣t=tq,kq
∂i∂βq,2
∣∣∣t=tq,kq
. . . ∂i∂βq,nq
∣∣∣t=tq,kq
and the partial derivatives in the last two rows are given by
∂Q
∂βq,s=
2 Imqηq,s βq,sdg
dβ
∣∣∣∣β=β2
q,s+1
, 1 ≤ q ≤ p,
2 Impηp+1,s βp+1,sdg
dβ
∣∣∣∣β=β2
p+1,s
, q = p+ 1.
For 1 ≤ q ≤ p
∂W
∂βq,s= 2
(q−1∑k=1
Imk
)Imqηq,s βq,s
dg
dβ
∣∣∣∣β=β2
q,s+1
+4 I2mqηq,sβq,s
ηq,s
dg
dβ
∣∣∣∣β=2β2
q,s+2
+
nq∑k=1k =s
ηq,kdg
dβ
∣∣∣∣β=β2
q,s+β2q,k+2
and
∂W
∂βp+1,s= 4
(p∑
k=1
Imk
)ηp+1,sβp+1,s
ηp+1,s
dg
dβ
∣∣∣∣β=2β2
p+1,s
+
nq∑k=1k =s
ηp+1,kdg
dβ
∣∣∣∣β=β2
p+1,s+β2p+1,k
.
The derivatives of g(β) and g(β) are
dg
dβ=1
e
(1 +
eβ
ββ
(Γ(β + 1)
(Ψ(β)− ln(β)
)−G(β)
)), (108)
dg
dβ=1
e
(eβ
ββG(β)− 1
), (109)
114
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
where Γ(β) is the Gamma function, Ψ(β) is the digamma function, see forexample [2], and G(β) is a special case of the Meijer G-function and can bedefined as
G(β) = G3,02,3
(β
∣∣∣∣1, 1
0, 0, β + 1
)
using the notation from [139]. When evaluating this function it might bemore practical to rewrite G using other special functions
G(β) = G3,02,3
(β
∣∣∣∣1, 1
0, 0, β + 1
)=
ββ+1
(β + 1)22F2(β + 1, β + 1; β + 2, β + 2; −β)
−(Ψ(β) + π cot(πβ) + ln(β)
)π csc (πβ)
Γ (−β)
where
2F2(β + 1, β + 1; β + 2, β + 2; −β) =∞∑k=0
(−1)kβk (β + 1)2
(β + k + 1)2
is a special case of the hypergeometric function. These partial derivativeswere found using software foe symbolic computation [1].
Note that all η-parameters must be recalculated for each step and howthis is done is detailed in Section 3.2.4.
3.2.4 Calculating the η-parameters from the β-parameters
Suppose that we have nq − 1 points (tq,k, iq,k) such that
tmq−1 < tq,1 < tq,2 < . . . < tq,nq−1 < tmq .
For an AEF that interpolates these points it must be true that
q−1∑k=1
Imk+ Imq
nq∑s=1
ηq,sxq(tq,k)βq,s = iq,k, k = 1, 2, . . . , nq − 1. (110)
Since ηq,1 + ηq,2 + . . .+ ηq,nq = 1 equation (110) can be rewritten as
Imq
nq−1∑s=1
ηq,s
(xq(tq,k)
βq,s − xq(tq,k)βq,nq
)= iq,k − xq(tq,k)
βq,nq −q−1∑s=1
Ims
(111)
115
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
for k = 1, 2, . . . , nq − 1. This can be written as a matrix equation
ImqXqηq = iq, (112)
ηq =[ηq,1 ηq,2 . . . ηq,nq−1
],(iq
)k= iq,k − xq(tq,k)
βq,nq −q−1∑s=1
Ims ,
(Xq
)k,s
= xq(k, s) = xq(tq,k)βq,s − xq(tq,k)
βq,nq ,
and xq(t) given by (89).When all βq,k, k = 1, 2, . . . , nq are known then ηq,k, k = 1, 2, . . . , nq − 1 can
be found by solving equation (112) and ηq,nq = 1−nq−1∑k=1
ηq,k.
If we have kq > nq−1 data points then the parameters can be estimatedwith the least-squares solution to (112), more specifically the solution to
I2mqX
q Xqηq = Xq iq.
3.2.5 Explicit formulas for a single-peak AEF
Consider the case where p = 1, n1 = n2 = 2 and τ = ttm1
. Then the explicit
formula for the AEF is
i(τ)
Im1
=
η1,1 τ
β21,1+1e(β
21,1+1)(1−τ)+ η1,2 τ
β21,2+1e(β
21,2+1)(1−τ), 0≤τ≤1,
η2,1 τβ22,1 eβ
22,1(1−τ)+ η2,2 τ
β22,2 eβ
22,2(1−τ) , 1≤τ.
(113)
Assume that four datapoints, (ik, τk), k = 1, 2, 3, 4, as well as the chargeflow Q0 and specific energy W0, are known.
If we want to fit the AEF to this data using MLSM equation (107) gives
J =
f1(τ1) f2(τ1) 0 0f1(τ2) f2(τ2) 0 0
0 0 g1(τ3) g2(τ3)0 0 g1(τ4) g2(τ4)
∂
∂β1,1Q(β,η)
∂
∂β1,2Q(β,η)
∂
∂β2,1Q(β,η)
∂
∂β2,2Q(β,η)
∂
∂β1,1W (β,η)
∂
∂β1,2W (β,η)
∂
∂β2,1W (β,η)
∂
∂β2,2W (β,η)
,
116
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
fk(τ) = 2 η1,k β1,kτβ21,k+1e(β
21,k+1)(1−τ)( ln(τ) + 1− τ
),
η1,1 =i1Im1
− τβ21,2
1 e(β21,2+1)(1−τ1), η1,2 = 1− η1,1,
gk(τ) = 2 η2,k β2,kτβ22,keβ
22,k(1−τ)( ln(τ) + 1− τ
),
η2,1 =i3Im1
− τβ22,2
3 eβ21,2(1−τ3), η2,2 = 1− η2,1,
β =[(β21,1 + 1
) (β21,2 + 1
)β22,1 β2
2,2
],
η =[η1,1 η1,2 η2,1 η2,2
],
Q(β,η)
Im1
=
2∑s=1
η1,seβ
21,s
(β21,s + 1
)β21,s+1
γ(β21,s + 2, β2
2,s + 1)
+
2∑s=1
η2,seβ
22,s
β2β2
2,s+1
2,s
(Γ(β22,s + 1
)− γ
(β22,s + 1, β2
2,s
)),
∂Q
∂βq,s=
2 Im1η1,s β1,sdg
dβ
∣∣∣∣β=β2
1,s+1
, q = 1,
2 Imqηp,s β2,sdg
dβ
∣∣∣∣β=β2
2,s
, q = 2,
with derivatives of g(β) and g(β) given by (108) and (109),
β =[(β21,1 + β2
1,2 + 2) (
β21,1 + β2
1,2 + 2)
(β22,1 + β2
2,2) (β22,1 + β2
2,2)],
η =[η21,1 η21,2 η22,1 η22,2
],
η =[(η1,1η1,2) (η1,1η1,2)(η2,1η2,2) (η2,1η2,2)
],
∂
∂βq,sW (β,η) = 2βq,s
∂
∂βq,sQ (2β, η) + β
q,((s−1 mod 2)+1
) ∂
∂βq,sQ(β, η
).
3.2.6 Fitting to lightning discharge currents
This section is based on Section 4 of Paper E
Some results of fitting the AEF to a few waveforms will be given. Somesingle-peak waveforms given by Heidler functions in IEC 62305-1 standard[74] will be approximated using the AEF, and furthermore, fitting the multi-peaked waveform to experimental data will be presented.
117
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
for k = 1, 2, . . . , nq − 1. This can be written as a matrix equation
ImqXqηq = iq, (112)
ηq =[ηq,1 ηq,2 . . . ηq,nq−1
],(iq
)k= iq,k − xq(tq,k)
βq,nq −q−1∑s=1
Ims ,
(Xq
)k,s
= xq(k, s) = xq(tq,k)βq,s − xq(tq,k)
βq,nq ,
and xq(t) given by (89).When all βq,k, k = 1, 2, . . . , nq are known then ηq,k, k = 1, 2, . . . , nq − 1 can
be found by solving equation (112) and ηq,nq = 1−nq−1∑k=1
ηq,k.
If we have kq > nq−1 data points then the parameters can be estimatedwith the least-squares solution to (112), more specifically the solution to
I2mqX
q Xqηq = Xq iq.
3.2.5 Explicit formulas for a single-peak AEF
Consider the case where p = 1, n1 = n2 = 2 and τ = ttm1
. Then the explicit
formula for the AEF is
i(τ)
Im1
=
η1,1 τ
β21,1+1e(β
21,1+1)(1−τ)+ η1,2 τ
β21,2+1e(β
21,2+1)(1−τ), 0≤τ≤1,
η2,1 τβ22,1 eβ
22,1(1−τ)+ η2,2 τ
β22,2 eβ
22,2(1−τ) , 1≤τ.
(113)
Assume that four datapoints, (ik, τk), k = 1, 2, 3, 4, as well as the chargeflow Q0 and specific energy W0, are known.
If we want to fit the AEF to this data using MLSM equation (107) gives
J =
f1(τ1) f2(τ1) 0 0f1(τ2) f2(τ2) 0 0
0 0 g1(τ3) g2(τ3)0 0 g1(τ4) g2(τ4)
∂
∂β1,1Q(β,η)
∂
∂β1,2Q(β,η)
∂
∂β2,1Q(β,η)
∂
∂β2,2Q(β,η)
∂
∂β1,1W (β,η)
∂
∂β1,2W (β,η)
∂
∂β2,1W (β,η)
∂
∂β2,2W (β,η)
,
116
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
fk(τ) = 2 η1,k β1,kτβ21,k+1e(β
21,k+1)(1−τ)( ln(τ) + 1− τ
),
η1,1 =i1Im1
− τβ21,2
1 e(β21,2+1)(1−τ1), η1,2 = 1− η1,1,
gk(τ) = 2 η2,k β2,kτβ22,keβ
22,k(1−τ)( ln(τ) + 1− τ
),
η2,1 =i3Im1
− τβ22,2
3 eβ21,2(1−τ3), η2,2 = 1− η2,1,
β =[(β21,1 + 1
) (β21,2 + 1
)β22,1 β2
2,2
],
η =[η1,1 η1,2 η2,1 η2,2
],
Q(β,η)
Im1
=
2∑s=1
η1,seβ
21,s
(β21,s + 1
)β21,s+1
γ(β21,s + 2, β2
2,s + 1)
+
2∑s=1
η2,seβ
22,s
β2β2
2,s+1
2,s
(Γ(β22,s + 1
)− γ
(β22,s + 1, β2
2,s
)),
∂Q
∂βq,s=
2 Im1η1,s β1,sdg
dβ
∣∣∣∣β=β2
1,s+1
, q = 1,
2 Imqηp,s β2,sdg
dβ
∣∣∣∣β=β2
2,s
, q = 2,
with derivatives of g(β) and g(β) given by (108) and (109),
β =[(β21,1 + β2
1,2 + 2) (
β21,1 + β2
1,2 + 2)
(β22,1 + β2
2,2) (β22,1 + β2
2,2)],
η =[η21,1 η21,2 η22,1 η22,2
],
η =[(η1,1η1,2) (η1,1η1,2)(η2,1η2,2) (η2,1η2,2)
],
∂
∂βq,sW (β,η) = 2βq,s
∂
∂βq,sQ (2β, η) + β
q,((s−1 mod 2)+1
) ∂
∂βq,sQ(β, η
).
3.2.6 Fitting to lightning discharge currents
This section is based on Section 4 of Paper E
Some results of fitting the AEF to a few waveforms will be given. Somesingle-peak waveforms given by Heidler functions in IEC 62305-1 standard[74] will be approximated using the AEF, and furthermore, fitting the multi-peaked waveform to experimental data will be presented.
117
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Single-peak waveforms
In this section some numerical results of fitting the AEF function to single-peak waveshapes are presented and compared with the corresponding fittingof the Heidler function. The AEF given by (113) is used to model few light-ning current waveshapes whose parameters (rise/decay time ratio, T1/T2,peak current value, Im1, time to peak current, tm1, charge flow at the strik-ing point, Q0, specific energy, W0, and time to 0.1Im1, t1) are given in table1. Data points were chosen as follows:
(i1, τ1) = (0.1 Im1 , t1),
(i2, τ2) = (0.9 Im1 , t2 = t1 + 0.8T1),
(i3, τ3) = (0.5 Im1 , th = t1 − 0.1T1 + T2),
(i4, τ4) = (i(1.5 th), 1.5 th).
Figure 3.5: First-positive stroke represented by the AEF function. Here it is fittedwith respect to both the data points as well as Q0 and W0.
Figure 3.6: First-negative stroke represented by the AEF function. Here it is fittedwith the extra constraint 0 ≤ η ≤ 1 for all η-parameters.
118
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
Figure 3.7: Fast-decaying waveshape represented by the AEF function. Here it isfitted with the extra constraint 0 ≤ η ≤ 1 for all η-parameters.
The AEF representation of the waveshape denoted as the first positivestroke current in IEC 62305 standard [74], is shown in figure 3.5. Rising anddecaying parts of the first negative stroke current from IEC 62305 standard[74] are shown in figure 3.6 - left and right, respectively. β and η parametersof both waveshapes optimized by the MLSM are given in table 3.1.
We have also observed a so-called fast-decaying waveshape whose param-eters are given in table 3.1. It’s representation using the AEF function isshown in figure 3.7, and corresponding β and η parameter values in table 3.1.
Apart from the AEF function (solid line), the Heidler function represen-tation of the same waveshapes (dashed line), and used data points (red solidcircles) are also shown in the figures.
Multi-peaked AEF waveforms for measured data
In this section the AEF will be constructed by fitting to measured datarather than approximation of the Heidler function. We will use data basedon the measurements of flash number 23 in [152]. Two AEFs have beenconstructed, one by choosing peaks corresponding to local maxima, see figure3.8, and one by choosing peaks corresponding to local maxima and localminima, see figure 3.9. For both AEFs there are two terms in each intervalwhich means that for each peak there are two parameters that are chosenmanually (time and current for each peak) and for each interval there aretwo parameters that are fitted using the MLSM.
The AEF in figure 3.8 demonstrates that the AEF can handle caseswhere the function is not constant or monotonically increasing/decreasingbetween peaks. This is only possible if the AEF has more than one term inthe interval.
119
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Single-peak waveforms
In this section some numerical results of fitting the AEF function to single-peak waveshapes are presented and compared with the corresponding fittingof the Heidler function. The AEF given by (113) is used to model few light-ning current waveshapes whose parameters (rise/decay time ratio, T1/T2,peak current value, Im1, time to peak current, tm1, charge flow at the strik-ing point, Q0, specific energy, W0, and time to 0.1Im1, t1) are given in table1. Data points were chosen as follows:
(i1, τ1) = (0.1 Im1 , t1),
(i2, τ2) = (0.9 Im1 , t2 = t1 + 0.8T1),
(i3, τ3) = (0.5 Im1 , th = t1 − 0.1T1 + T2),
(i4, τ4) = (i(1.5 th), 1.5 th).
Figure 3.5: First-positive stroke represented by the AEF function. Here it is fittedwith respect to both the data points as well as Q0 and W0.
Figure 3.6: First-negative stroke represented by the AEF function. Here it is fittedwith the extra constraint 0 ≤ η ≤ 1 for all η-parameters.
118
3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS
Figure 3.7: Fast-decaying waveshape represented by the AEF function. Here it isfitted with the extra constraint 0 ≤ η ≤ 1 for all η-parameters.
The AEF representation of the waveshape denoted as the first positivestroke current in IEC 62305 standard [74], is shown in figure 3.5. Rising anddecaying parts of the first negative stroke current from IEC 62305 standard[74] are shown in figure 3.6 - left and right, respectively. β and η parametersof both waveshapes optimized by the MLSM are given in table 3.1.
We have also observed a so-called fast-decaying waveshape whose param-eters are given in table 3.1. It’s representation using the AEF function isshown in figure 3.7, and corresponding β and η parameter values in table 3.1.
Apart from the AEF function (solid line), the Heidler function represen-tation of the same waveshapes (dashed line), and used data points (red solidcircles) are also shown in the figures.
Multi-peaked AEF waveforms for measured data
In this section the AEF will be constructed by fitting to measured datarather than approximation of the Heidler function. We will use data basedon the measurements of flash number 23 in [152]. Two AEFs have beenconstructed, one by choosing peaks corresponding to local maxima, see figure3.8, and one by choosing peaks corresponding to local maxima and localminima, see figure 3.9. For both AEFs there are two terms in each intervalwhich means that for each peak there are two parameters that are chosenmanually (time and current for each peak) and for each interval there aretwo parameters that are fitted using the MLSM.
The AEF in figure 3.8 demonstrates that the AEF can handle caseswhere the function is not constant or monotonically increasing/decreasingbetween peaks. This is only possible if the AEF has more than one term inthe interval.
119
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
First-positive First-negative Fast-decayingstroke stroke
T1/T2 10/350 1/200 8/20
tm1 [µs] 31.428 3.552 15.141
Im1 [kA] 200 100 0.001
Q0 [C] 100 / /
W0 [MJ/Ω] 10 / /
t1 [µs] 14.5 1.47 6.34
β1,1 0.114 1.84 7.666
β1,2 2.17 9.99 2.626
β2,1 0.284 0.099 0.925
β2,2 0 0.127 2.420
η1,1 −0.197 1 0
η1,2 1.197 0 1
η2,1 1 0.401 0.2227
η2,2 0 0.599 0.7773
Table 3.1: AEF function’s parameters for some current waveshapes
Conclusions
This section investigated the possibility to approximate, in general, multi-peaked lightning currents using an AEF function. Furthermore, existence ofthe analytical solution for the derivative and the integral of such function hasbeen proven, which is needed in order to perform lightning electromagneticfield (LEMF) calculations based on it.
Two single-peak Standard IEC 62305-1 waveforms, and a fast-decayingone, have been represented using a variation of the proposed AEF function(113). The estimation of their parameters has been performed applying theMLS method using two pairs of data points for each function part (risingand decaying). The results show that there are several factors that need tobe taken into consideration to get the best possible approximation of a givenwaveform. The accuracy of the approximation varies with the chosen datapoints and the number of terms in the AEF. In several cases the two-termsum converged towards a single term sum. This can probably be improvedby choosing the number of terms and the number and placement of datapoints in other ways which the authors intend to examine further. Furtherexamples of fitted (single- and multi-peaked) waveforms can be found in [108]
120
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
Figure 3.8: AEF fitted to measurements from [152]. Here the peaks have been chosento correspond to local maxima in the measured data.
Figure 3.9: AEF fitted to measurements from [152]. Here the peaks have been chosento correspond to local maxima and minima in the measured data.
and [81].
3.3 Approximation of electrostatic discharge cur-rents
This section is based on Paper F
In this paper we analyse the applicability of the generalized multi-peakedAEF function to representation of ESD currents by interpolation of datapoints chosen according to a D-optimal design. This is illustrated throughtwo examples corresponding to modelling of the IEC Standard 61000-4-2waveshape, [72,73] and an experimentally measured ESD current from [87].
121
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
First-positive First-negative Fast-decayingstroke stroke
T1/T2 10/350 1/200 8/20
tm1 [µs] 31.428 3.552 15.141
Im1 [kA] 200 100 0.001
Q0 [C] 100 / /
W0 [MJ/Ω] 10 / /
t1 [µs] 14.5 1.47 6.34
β1,1 0.114 1.84 7.666
β1,2 2.17 9.99 2.626
β2,1 0.284 0.099 0.925
β2,2 0 0.127 2.420
η1,1 −0.197 1 0
η1,2 1.197 0 1
η2,1 1 0.401 0.2227
η2,2 0 0.599 0.7773
Table 3.1: AEF function’s parameters for some current waveshapes
Conclusions
This section investigated the possibility to approximate, in general, multi-peaked lightning currents using an AEF function. Furthermore, existence ofthe analytical solution for the derivative and the integral of such function hasbeen proven, which is needed in order to perform lightning electromagneticfield (LEMF) calculations based on it.
Two single-peak Standard IEC 62305-1 waveforms, and a fast-decayingone, have been represented using a variation of the proposed AEF function(113). The estimation of their parameters has been performed applying theMLS method using two pairs of data points for each function part (risingand decaying). The results show that there are several factors that need tobe taken into consideration to get the best possible approximation of a givenwaveform. The accuracy of the approximation varies with the chosen datapoints and the number of terms in the AEF. In several cases the two-termsum converged towards a single term sum. This can probably be improvedby choosing the number of terms and the number and placement of datapoints in other ways which the authors intend to examine further. Furtherexamples of fitted (single- and multi-peaked) waveforms can be found in [108]
120
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
Figure 3.8: AEF fitted to measurements from [152]. Here the peaks have been chosento correspond to local maxima in the measured data.
Figure 3.9: AEF fitted to measurements from [152]. Here the peaks have been chosento correspond to local maxima and minima in the measured data.
and [81].
3.3 Approximation of electrostatic discharge cur-rents
This section is based on Paper F
In this paper we analyse the applicability of the generalized multi-peakedAEF function to representation of ESD currents by interpolation of datapoints chosen according to a D-optimal design. This is illustrated throughtwo examples corresponding to modelling of the IEC Standard 61000-4-2waveshape, [72,73] and an experimentally measured ESD current from [87].
121
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Voltage [kV] Ipeak [A] tr [ns] I30 [A] I60 [A]
2 7.5± 15% 0.8± 25% 4.0± 30% 2.0± 30%
4 15.0± 15% 0.8± 25% 8.0± 30% 4.0± 30%
6 22.5± 15% 0.8± 25% 12.0± 30% 6.0± 30%
8 30.0± 15% 0.8± 25% 16.0± 30% 8.0± 30%
Table 3.2: IEC 61000-4-2 Standard ESD Current and its Key Parameters, [73].
3.3.1 IEC 61000-4-2 Standard current waveshape
ESD generators used in testing of the equipment and devices should be ableto reproduce the same ESD current waveshape each time. This repeata-bility feature is ensured if the design is carried out in compliance with therequirements defined in the IEC 61000-4-2 Standard [73].
Among other relevant issues, the Standard includes graphical represen-tation of the typical ESD current, figure 3.10, and also defines, for a giventest level voltage, required values of ESD current’s key parameters. Theseare listed in table 3.2 for the case of the contact discharge, where:
• Ipeak is the ESD current initial peak;
• tr is the rising time defined as the difference between time momentscorresponding to 10% and 90% of the current peak Ipeak, figure 3.10;
• I30 and I60 is the ESD current values calculated for time periods of 30and 60 ns, respectively, starting from the time point corresponding to10% of Ipeak, figure 3.10.
Important Features of ESD Currents
Various mathematical expressions have been introduced in the literaturethat can be used for representation of the ESD currents, either the IEC61000-4-2 Standard one [73], or experimentally measured ones, e.g. [47].These functions are to certain extent in accordance with the requirementsgiven in table 3.2. Furthermore, they have to satisfy the following:
• the value of the ESD current and its first derivative must be equal tozero at the moment t = 0, since neither the transient current nor the
122
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
61000-4-2 © IEC:1995+A1:1998 – 43 – +A2:2000
Values are given in table 2.
Figure 3 – Typical waveform of the output current of the ESD generator
tr
Figure 3.10: Illustration of the IEC 61000-4-2 Standard ESD current and its keyparameters, [73].
radiated field generated by the ESD current can change abruptly atthat moment.
• the ESD current function must be time-integrable in order to allownumerical calculation of the ESD radiated fields.
Multi-peaked analytically extended function
A so-called multi-peaked analytically extended function (AEF) has beenproposed and applied to lightning discharge current modelling in Section 3.1and [107]. Initial considerations on applying the function to ESD currentshave also been made in [108].
The AEF consists of scaled and translated power-exponential functions,
that is functions of the form x(β; t) =(te1−t
)β, see Definition 3.1.
Here we define the AEF with p peaks as
i(t) =
q−1∑k=1
Imk+ Imq
nq∑k=1
ηq,kxq,k(t), (114)
for tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p, and
p∑k=1
Imk
np+1∑k=1
ηp+1,kxp+1,k(t), (115)
for tmp ≤ t.
123
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Voltage [kV] Ipeak [A] tr [ns] I30 [A] I60 [A]
2 7.5± 15% 0.8± 25% 4.0± 30% 2.0± 30%
4 15.0± 15% 0.8± 25% 8.0± 30% 4.0± 30%
6 22.5± 15% 0.8± 25% 12.0± 30% 6.0± 30%
8 30.0± 15% 0.8± 25% 16.0± 30% 8.0± 30%
Table 3.2: IEC 61000-4-2 Standard ESD Current and its Key Parameters, [73].
3.3.1 IEC 61000-4-2 Standard current waveshape
ESD generators used in testing of the equipment and devices should be ableto reproduce the same ESD current waveshape each time. This repeata-bility feature is ensured if the design is carried out in compliance with therequirements defined in the IEC 61000-4-2 Standard [73].
Among other relevant issues, the Standard includes graphical represen-tation of the typical ESD current, figure 3.10, and also defines, for a giventest level voltage, required values of ESD current’s key parameters. Theseare listed in table 3.2 for the case of the contact discharge, where:
• Ipeak is the ESD current initial peak;
• tr is the rising time defined as the difference between time momentscorresponding to 10% and 90% of the current peak Ipeak, figure 3.10;
• I30 and I60 is the ESD current values calculated for time periods of 30and 60 ns, respectively, starting from the time point corresponding to10% of Ipeak, figure 3.10.
Important Features of ESD Currents
Various mathematical expressions have been introduced in the literaturethat can be used for representation of the ESD currents, either the IEC61000-4-2 Standard one [73], or experimentally measured ones, e.g. [47].These functions are to certain extent in accordance with the requirementsgiven in table 3.2. Furthermore, they have to satisfy the following:
• the value of the ESD current and its first derivative must be equal tozero at the moment t = 0, since neither the transient current nor the
122
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
61000-4-2 © IEC:1995+A1:1998 – 43 – +A2:2000
Values are given in table 2.
Figure 3 – Typical waveform of the output current of the ESD generator
tr
Figure 3.10: Illustration of the IEC 61000-4-2 Standard ESD current and its keyparameters, [73].
radiated field generated by the ESD current can change abruptly atthat moment.
• the ESD current function must be time-integrable in order to allownumerical calculation of the ESD radiated fields.
Multi-peaked analytically extended function
A so-called multi-peaked analytically extended function (AEF) has beenproposed and applied to lightning discharge current modelling in Section 3.1and [107]. Initial considerations on applying the function to ESD currentshave also been made in [108].
The AEF consists of scaled and translated power-exponential functions,
that is functions of the form x(β; t) =(te1−t
)β, see Definition 3.1.
Here we define the AEF with p peaks as
i(t) =
q−1∑k=1
Imk+ Imq
nq∑k=1
ηq,kxq,k(t), (114)
for tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p, and
p∑k=1
Imk
np+1∑k=1
ηp+1,kxp+1,k(t), (115)
for tmp ≤ t.
123
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The current value of the first peak is denoted by Im1 , the differencebetween each pair of subsequent peaks by Im2 , Im3 , . . . , Imp , and their cor-responding times by tm1 , tm2 , . . . , tmp . In each time interval q, with 1 ≤ q ≤p+ 1, the number of terms is given by nq, 0 < nq ∈ Z. Parameters ηq,k are
such that ηq,k ∈ R for q = 1, 2, . . . , p + 1, k = 1, 2, . . . , nq and
nq∑k=1
ηq,k = 1.
Furthermore xq,k(t), 1 ≤ q ≤ p+ 1 is given by
xq,k(t) =
x(βq,k;
t−tmq−1
tmq−tmq−1
), 1 ≤ q ≤ p,
x(βq,k;
ttmq
), q = p+ 1.
(116)
Remark 3.2. When previously applying the AEF, see Section 3.1.1, allexponents (β-parameters) of the AEF were set to β2+1 in order to guaranteethat the derivative of the AEF is continuous. Here this condition will besatisfied in a different manner.
Since the AEF is a linear function of elementary functions its derivativeand integral can be found using standard methods. For explicit formulaeplease refer to Theorem 3.1-3.3.
Previously, the authors have fitted AEF functions to lightning dischargecurrents and ESD currents using the Marquardt least square method buthave noticed that the obtained result varies greatly depending on how thewaveforms are sampled. This is problematic, especially since the methodol-ogy becomes computationally demanding when applied to large amounts ofdata. Here we will try one way to minimize the data needed but still enoughto get an as good approximation as possible.
The method examined here will be based on D-optimality of a regressionmodel. A D-optimal design is found by choosing sample points such that thedeterminant of the Fischer information matrix of the model is minimized.For a standard linear regression model this is also equivalent, by the so-called Kiefer-Wolfowitz equivalence criterion, to G-optimality which meansthat the maximum of the prediction variance will be minimized. These arestandard results in the theory of optimal experiment design and a summarycan be found in for example [116].
Minimizing the prediction variance will in our case mean maximizing therobustness of the model. This does not guarantee a good approximation butit will increase the chances of the method working well when working withlimited precision and noisy data and thus improve the chances of finding agood approximation when it is possible.
124
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
3.3.2 D-Optimal approximation for exponents given by aclass of arithmetic sequences
It can be desirable to minimize the number of points used when constructingthe approximation. One way to do this is to choose the D-optimal samplingpoints.
In this section we will only consider the case where in each interval then exponents, β1, . . . , βn, are chosen according to
βm =k +m− 1
c
where k is a non-negative integer and c a positive real number. Note thatin order to guarantee continuity of the AEF derivative the condition is thatk > c.
Then in each interval we want an approximation of the formand by setting z(t) = (te1−t)
1c then
y(t) =
n∑i=1
ηiz(t)k+i−1.
If we have n sample points, ti, i = 1, . . . , n, then the Fischer informationmatrix, M, of this system is M = UU where
U =
z(t1)k z(t2)
k . . . z(tn)k
z(t1)k+1 z(t2)
k+1 . . . z(tn)k+1
......
. . ....
z(t1)k+n−1 z(t2)
k+n−1 . . . z(tn)k+n−1
.
Thus if we want to maximize det(M) = det(U)2 it is sufficient to maximize
or minimize the determinant det(U). Set z(ti) = (tie1−ti)
1c = xi then
un(t1, . . . , tn) = det(U)
=
(n∏
k=1
xk
) ∏
1≤i<j≤n
(xj − xi)
. (117)
To find ti we will use the Lambert W function. Formally the LambertW function is the function W that satisfies t = W (tet). Using W we caninvert z(t) in the following way
te1−t = xc ⇔ −te−t = −e−1xc
⇔ t = −W (−e−1xc). (118)
125
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The current value of the first peak is denoted by Im1 , the differencebetween each pair of subsequent peaks by Im2 , Im3 , . . . , Imp , and their cor-responding times by tm1 , tm2 , . . . , tmp . In each time interval q, with 1 ≤ q ≤p+ 1, the number of terms is given by nq, 0 < nq ∈ Z. Parameters ηq,k are
such that ηq,k ∈ R for q = 1, 2, . . . , p + 1, k = 1, 2, . . . , nq and
nq∑k=1
ηq,k = 1.
Furthermore xq,k(t), 1 ≤ q ≤ p+ 1 is given by
xq,k(t) =
x(βq,k;
t−tmq−1
tmq−tmq−1
), 1 ≤ q ≤ p,
x(βq,k;
ttmq
), q = p+ 1.
(116)
Remark 3.2. When previously applying the AEF, see Section 3.1.1, allexponents (β-parameters) of the AEF were set to β2+1 in order to guaranteethat the derivative of the AEF is continuous. Here this condition will besatisfied in a different manner.
Since the AEF is a linear function of elementary functions its derivativeand integral can be found using standard methods. For explicit formulaeplease refer to Theorem 3.1-3.3.
Previously, the authors have fitted AEF functions to lightning dischargecurrents and ESD currents using the Marquardt least square method buthave noticed that the obtained result varies greatly depending on how thewaveforms are sampled. This is problematic, especially since the methodol-ogy becomes computationally demanding when applied to large amounts ofdata. Here we will try one way to minimize the data needed but still enoughto get an as good approximation as possible.
The method examined here will be based on D-optimality of a regressionmodel. A D-optimal design is found by choosing sample points such that thedeterminant of the Fischer information matrix of the model is minimized.For a standard linear regression model this is also equivalent, by the so-called Kiefer-Wolfowitz equivalence criterion, to G-optimality which meansthat the maximum of the prediction variance will be minimized. These arestandard results in the theory of optimal experiment design and a summarycan be found in for example [116].
Minimizing the prediction variance will in our case mean maximizing therobustness of the model. This does not guarantee a good approximation butit will increase the chances of the method working well when working withlimited precision and noisy data and thus improve the chances of finding agood approximation when it is possible.
124
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
3.3.2 D-Optimal approximation for exponents given by aclass of arithmetic sequences
It can be desirable to minimize the number of points used when constructingthe approximation. One way to do this is to choose the D-optimal samplingpoints.
In this section we will only consider the case where in each interval then exponents, β1, . . . , βn, are chosen according to
βm =k +m− 1
c
where k is a non-negative integer and c a positive real number. Note thatin order to guarantee continuity of the AEF derivative the condition is thatk > c.
Then in each interval we want an approximation of the formand by setting z(t) = (te1−t)
1c then
y(t) =
n∑i=1
ηiz(t)k+i−1.
If we have n sample points, ti, i = 1, . . . , n, then the Fischer informationmatrix, M, of this system is M = UU where
U =
z(t1)k z(t2)
k . . . z(tn)k
z(t1)k+1 z(t2)
k+1 . . . z(tn)k+1
......
. . ....
z(t1)k+n−1 z(t2)
k+n−1 . . . z(tn)k+n−1
.
Thus if we want to maximize det(M) = det(U)2 it is sufficient to maximize
or minimize the determinant det(U). Set z(ti) = (tie1−ti)
1c = xi then
un(t1, . . . , tn) = det(U)
=
(n∏
k=1
xk
) ∏
1≤i<j≤n
(xj − xi)
. (117)
To find ti we will use the Lambert W function. Formally the LambertW function is the function W that satisfies t = W (tet). Using W we caninvert z(t) in the following way
te1−t = xc ⇔ −te−t = −e−1xc
⇔ t = −W (−e−1xc). (118)
125
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The Lambert W is multivalued but since we are only interested in real-valued solutions we are restricted to the main branches W0 and W−1. SinceW0 ≥ −1 and W−1 ≤ −1 the two branches correspond to the rising anddecaying parts of the AEF respectively. We will deal with the details offinding the correct points for the two parts separately.
D-Optimal interpolation on the rising part
To find the D-optimal points on the rising part we use Theorem 3.6.
Theorem 3.6. The determinant
un(k;x1, . . . , xn) =
(n∏
i=1
xki
) ∏
1≤i<j≤n
(xj − xi)
is maximized or minimized on the cube [0, 1]n when x1 < . . . < xn−1 areroots of the Jacobi polynomial
P(2k−1,0)n−1 (1− 2x) =
(2k)n−1
(n− 1)!
n−1∑i=0
(−1)n(n− 1
i
)(2k + n)i
(2k)ixi
and xn = 1, or some permutation thereof.Here ab is the rising factorial ab = a(a+ 1) · · · (a+ b− 1).
Proof. Without loss of generality we can assume that the nodes are ordered0 < x1 < x2 < . . . < xn−1 < xn ≤ 1. Fix all xi except xn. When xnincreases all factors of wn that contain xn will also increase, thus wn willreach its maximum value on the edge of the cube where xn = 1. Using themethod of Lagrange multipliers in the plane given by xn = 1 gives
∂un∂xj
= un(k;x1, . . . , xn)
k
xj+
n∑i=1i =j
1
xj − xi
= 0,
for j = 1, . . . , n− 1. By setting f(x) =n∏
i=1
(x− xi) we get
k
xj+
n∑i=1i =j
1
xj − xi= 0 ⇔ k
xj+
1
2
f ′′(xj)
f ′(xj)= 0
⇔ xjf′′(xj) + 2kf ′(xj) = 0 (119)
126
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
for j = 1, . . . , n− 1. Since f(x) is a polynomial of degree n that has x = 1as a root then equation (119) implies
xf ′′(x) + 2kf ′(x) = cf(x)
x− 1
where c is some constant. Set f(x) = (x−1)g(x) and the resulting differentialequation is
x(x− 1)g′′(x) + ((2k + 2)x− 2k)g′(x) + (2k − c)g(x) = 0.
The constant c can be found by examining the terms with degree n− 1 andis given by c = 2k + (n− 1)(2k + n), thus
x(1− x)g′′(x) + (2k − (2k + 2)x)g′(x)
+(n− 1)(2k + n)g(x) = 0. (120)
Comparing (120) with the standard form of the hypergeometric function [2]
x(1− x)g′′(x) + (c− (a+ b+ 1)x)g′(x)− abg(x) = 0
shows that g(x) can be expressed as follows
g(x) = C · 2F1(1− n, 2k + n; 2k, x)
= C · (2k)n−1
(n− 1)!
n−1∑i=0
(−1)i(n− 1
i
)(2k + n)i
(2k)ixi
where C is an arbitrary constant and since we are only interested in theroots of the polynomial we can chose C so that it gives the desired form ofthe expression. The connection to the Jacobi polynomial is given by [2]
2F1(−m, 1 + α+ β + n;α+ 1;x) =m!
(α+ 1)mP (α,β)m (1− 2x),
and α = 2k − 1, β = 0, m = n− 1 gives the expression in Theorem 3.6.
We can now find the D-optimal t-values using the upper branch of theLambert W function as described in equation (118),
ti = −W0(−e−1xmi ),
where xi are the roots of the Jacobi polynomial given in Theorem 3.6. Since−1 ≤ W0(x) ≤ 0 for −e−1 ≤ x < 0 this will always give 0 ≤ ti ≤ 1.
Remark 3.3. Note that xn = 1 means that tn = tq and also is equivalent
to the condition that
nq∑r=1
ηq,r = 1. In other words we are interpolating the
peak and p− 1 points inside each interval.
127
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
The Lambert W is multivalued but since we are only interested in real-valued solutions we are restricted to the main branches W0 and W−1. SinceW0 ≥ −1 and W−1 ≤ −1 the two branches correspond to the rising anddecaying parts of the AEF respectively. We will deal with the details offinding the correct points for the two parts separately.
D-Optimal interpolation on the rising part
To find the D-optimal points on the rising part we use Theorem 3.6.
Theorem 3.6. The determinant
un(k;x1, . . . , xn) =
(n∏
i=1
xki
) ∏
1≤i<j≤n
(xj − xi)
is maximized or minimized on the cube [0, 1]n when x1 < . . . < xn−1 areroots of the Jacobi polynomial
P(2k−1,0)n−1 (1− 2x) =
(2k)n−1
(n− 1)!
n−1∑i=0
(−1)n(n− 1
i
)(2k + n)i
(2k)ixi
and xn = 1, or some permutation thereof.Here ab is the rising factorial ab = a(a+ 1) · · · (a+ b− 1).
Proof. Without loss of generality we can assume that the nodes are ordered0 < x1 < x2 < . . . < xn−1 < xn ≤ 1. Fix all xi except xn. When xnincreases all factors of wn that contain xn will also increase, thus wn willreach its maximum value on the edge of the cube where xn = 1. Using themethod of Lagrange multipliers in the plane given by xn = 1 gives
∂un∂xj
= un(k;x1, . . . , xn)
k
xj+
n∑i=1i =j
1
xj − xi
= 0,
for j = 1, . . . , n− 1. By setting f(x) =n∏
i=1
(x− xi) we get
k
xj+
n∑i=1i =j
1
xj − xi= 0 ⇔ k
xj+
1
2
f ′′(xj)
f ′(xj)= 0
⇔ xjf′′(xj) + 2kf ′(xj) = 0 (119)
126
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
for j = 1, . . . , n− 1. Since f(x) is a polynomial of degree n that has x = 1as a root then equation (119) implies
xf ′′(x) + 2kf ′(x) = cf(x)
x− 1
where c is some constant. Set f(x) = (x−1)g(x) and the resulting differentialequation is
x(x− 1)g′′(x) + ((2k + 2)x− 2k)g′(x) + (2k − c)g(x) = 0.
The constant c can be found by examining the terms with degree n− 1 andis given by c = 2k + (n− 1)(2k + n), thus
x(1− x)g′′(x) + (2k − (2k + 2)x)g′(x)
+(n− 1)(2k + n)g(x) = 0. (120)
Comparing (120) with the standard form of the hypergeometric function [2]
x(1− x)g′′(x) + (c− (a+ b+ 1)x)g′(x)− abg(x) = 0
shows that g(x) can be expressed as follows
g(x) = C · 2F1(1− n, 2k + n; 2k, x)
= C · (2k)n−1
(n− 1)!
n−1∑i=0
(−1)i(n− 1
i
)(2k + n)i
(2k)ixi
where C is an arbitrary constant and since we are only interested in theroots of the polynomial we can chose C so that it gives the desired form ofthe expression. The connection to the Jacobi polynomial is given by [2]
2F1(−m, 1 + α+ β + n;α+ 1;x) =m!
(α+ 1)mP (α,β)m (1− 2x),
and α = 2k − 1, β = 0, m = n− 1 gives the expression in Theorem 3.6.
We can now find the D-optimal t-values using the upper branch of theLambert W function as described in equation (118),
ti = −W0(−e−1xmi ),
where xi are the roots of the Jacobi polynomial given in Theorem 3.6. Since−1 ≤ W0(x) ≤ 0 for −e−1 ≤ x < 0 this will always give 0 ≤ ti ≤ 1.
Remark 3.3. Note that xn = 1 means that tn = tq and also is equivalent
to the condition that
nq∑r=1
ηq,r = 1. In other words we are interpolating the
peak and p− 1 points inside each interval.
127
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
D-Optimal interpolation on the decaying part
Finding theD-optimal points for the decaying part is more complicated thanit is for the rising part. Suppose we denote the largest value for time that canreasonably be used (for computational or experimental reasons) with tmax.
This corresponds to some value xmax = (tmax exp(1 − tmax))1c . Ideally we
would want a corresponding theorem to Theorem 3.6 over [1, xmax]n instead
of [0, 1]n. It is easy to see that if xi = 0 or xi = 1 for some 1 ≤ xi ≤ n − 1then wn(k;x1, . . . , xn) = 0 and thus there must exist some local extremepoint such that 0 < x1 < x2 < . . . < xn−1 < 1. This is no longer guaranteedwhen considering the volume [1, xmax]
n instead. Therefore we will insteadextend Theorem 3.6 to the volume [0, xmax]
n and give an extra constrainton the parameter k that guarantees 1 < x1 < x2 < . . . < xn−1 < xn = xmax.
Theorem 3.7. Let y1 < y2 < . . . < yn−1 be the roots of the Jacobi poly-
nomial P(2k−1,0)n−1 (1 − 2y). If k is chosen such that 1 < xmax · y1 then the
determinant un(k;x1, . . . , xn) given in Theorem 3.6 is maximized or min-imized on the cube [1, xmax]
n (where xmax > 1) when xi = xmax · yi andxn = xmax, or some permutation thereof.
Proof. This theorem follows from Theorem 3.6 combined with the fact thatun(k;x1, . . . , xn) is a homogeneous polynomial. Since
un(k; b · x1, . . . , c · xn) = bk+n(n−1)
2 · un(k;x1, . . . , xn)
if (x1, . . . , xn) is an extreme point in [0, 1]n then (b·x1, . . . , b·xn) is an extremepoint in [0, b]n. Thus by Theorem 3.6 the points given by xi = xmax · yi willmaximize or minimize wn(k;x1, . . . , xn) on [0, xmax]
n.
Remark 3.4. It is in many cases possible to ensure the condition 1 <
xmax · y1 without actually calculating the roots of P(2k−1,0)n−1 (1− 2y). In the
literature on orthogonal polynomials there are many expressions for upperand lower bounds of the roots of the Jacobi polynomials. For instance in [38]an upper bound on the largest root of a Jacobi polynomial is given that inour case can be rewritten as
y1 > 1− 3
4k2 + 2kn+ n2 − k − 2n+ 1
and thus
1− 3
4k2 + 2kn+ n2 − k − 2n+ 1>
1
xmax
128
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
guarantees that 1 < xmax · y1. If a more precise condition is needed thereare expressions that give tighter bounds of the largest root of the Jacobipolynomials, see [104].
We can now find the D-optimal t-values using the lower branch of theLambert W function as described in equation (118),
ti = −W−1(−e−1xci ),
where xi are the roots of the Jacobi polynomial given in Theorem 3.6. Since−1 ≤ W−1(x) < −∞ for −e−1 ≤ x ≤ 0 then 1 ≤ ti < tmax with tmax =−W−1(−e−1xmax) so xmax is given by the highest feasible t.
Remark 3.5. Note that here just like in the rising part t1 = tp whichmeans that we will interpolate to the final peak as well as p − 1 points inthe decaying part.
3.3.3 Examples of models from applications and experiments
In this section some results of applying the described scheme to two dif-ferent waveforms will be presented. The two waveforms are the StandardESD current given in IEC 61000-4-2 [73] and a waveform from experimentalmeasurements from [87].
The values of n, k and c have been chosen by manual experimentationand since both waveforms are given as data rather than explicit functionsthe D-optimal points have been calculated and then the closest availabledata points have been chosen.
Note that the quality of the results can vary greatly depending on howthe k andm parameters are chosen before this type of approximation schemeis applied, and in practice a strategy for choosing the values effectively shouldbe devised. In many cases increasing the number of interpolation points, n,improves the results but there are many cases where the interpolation is notstable.
Interpolated AEF representing the IEC 61000-4-2 Standard cur-rent
In this section we present the results of fitting 2- and 3-peak AEF to theStandard ESD current given in IEC 61000-4-2. Data points which are usedin the optimization procedure are manually sampled from the graphicallygiven Standard [73] current function. The peak currents and correspondingtimes are also extracted, and the results of D-optimal interpolation with 2
129
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
D-Optimal interpolation on the decaying part
Finding theD-optimal points for the decaying part is more complicated thanit is for the rising part. Suppose we denote the largest value for time that canreasonably be used (for computational or experimental reasons) with tmax.
This corresponds to some value xmax = (tmax exp(1 − tmax))1c . Ideally we
would want a corresponding theorem to Theorem 3.6 over [1, xmax]n instead
of [0, 1]n. It is easy to see that if xi = 0 or xi = 1 for some 1 ≤ xi ≤ n − 1then wn(k;x1, . . . , xn) = 0 and thus there must exist some local extremepoint such that 0 < x1 < x2 < . . . < xn−1 < 1. This is no longer guaranteedwhen considering the volume [1, xmax]
n instead. Therefore we will insteadextend Theorem 3.6 to the volume [0, xmax]
n and give an extra constrainton the parameter k that guarantees 1 < x1 < x2 < . . . < xn−1 < xn = xmax.
Theorem 3.7. Let y1 < y2 < . . . < yn−1 be the roots of the Jacobi poly-
nomial P(2k−1,0)n−1 (1 − 2y). If k is chosen such that 1 < xmax · y1 then the
determinant un(k;x1, . . . , xn) given in Theorem 3.6 is maximized or min-imized on the cube [1, xmax]
n (where xmax > 1) when xi = xmax · yi andxn = xmax, or some permutation thereof.
Proof. This theorem follows from Theorem 3.6 combined with the fact thatun(k;x1, . . . , xn) is a homogeneous polynomial. Since
un(k; b · x1, . . . , c · xn) = bk+n(n−1)
2 · un(k;x1, . . . , xn)
if (x1, . . . , xn) is an extreme point in [0, 1]n then (b·x1, . . . , b·xn) is an extremepoint in [0, b]n. Thus by Theorem 3.6 the points given by xi = xmax · yi willmaximize or minimize wn(k;x1, . . . , xn) on [0, xmax]
n.
Remark 3.4. It is in many cases possible to ensure the condition 1 <
xmax · y1 without actually calculating the roots of P(2k−1,0)n−1 (1− 2y). In the
literature on orthogonal polynomials there are many expressions for upperand lower bounds of the roots of the Jacobi polynomials. For instance in [38]an upper bound on the largest root of a Jacobi polynomial is given that inour case can be rewritten as
y1 > 1− 3
4k2 + 2kn+ n2 − k − 2n+ 1
and thus
1− 3
4k2 + 2kn+ n2 − k − 2n+ 1>
1
xmax
128
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
guarantees that 1 < xmax · y1. If a more precise condition is needed thereare expressions that give tighter bounds of the largest root of the Jacobipolynomials, see [104].
We can now find the D-optimal t-values using the lower branch of theLambert W function as described in equation (118),
ti = −W−1(−e−1xci ),
where xi are the roots of the Jacobi polynomial given in Theorem 3.6. Since−1 ≤ W−1(x) < −∞ for −e−1 ≤ x ≤ 0 then 1 ≤ ti < tmax with tmax =−W−1(−e−1xmax) so xmax is given by the highest feasible t.
Remark 3.5. Note that here just like in the rising part t1 = tp whichmeans that we will interpolate to the final peak as well as p − 1 points inthe decaying part.
3.3.3 Examples of models from applications and experiments
In this section some results of applying the described scheme to two dif-ferent waveforms will be presented. The two waveforms are the StandardESD current given in IEC 61000-4-2 [73] and a waveform from experimentalmeasurements from [87].
The values of n, k and c have been chosen by manual experimentationand since both waveforms are given as data rather than explicit functionsthe D-optimal points have been calculated and then the closest availabledata points have been chosen.
Note that the quality of the results can vary greatly depending on howthe k andm parameters are chosen before this type of approximation schemeis applied, and in practice a strategy for choosing the values effectively shouldbe devised. In many cases increasing the number of interpolation points, n,improves the results but there are many cases where the interpolation is notstable.
Interpolated AEF representing the IEC 61000-4-2 Standard cur-rent
In this section we present the results of fitting 2- and 3-peak AEF to theStandard ESD current given in IEC 61000-4-2. Data points which are usedin the optimization procedure are manually sampled from the graphicallygiven Standard [73] current function. The peak currents and correspondingtimes are also extracted, and the results of D-optimal interpolation with 2
129
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and 3 peaks are illustrated, see figure 3.11 (a) and 3.11 (b). The parametersare listed in table 3.3. In the illustrated examples a fairly good fit is foundbut typically areas with steep rise and the decay part are somewhat moredifficult to fit with good accuracy than the other parts of the waveform.
t [s] #10-80 2 4 6 8
i(t) [
A]
0
0.2
0.4
0.6
0.8
1IEC 61000-4-2PeaksInterpolated points2-peaked AEF
(a)
t [s] #10-80 2 4 6 8
i(t) [
A]
0
0.2
0.4
0.6
0.8
1IEC 61000-4-2PeaksInterpolated points3-peaked AEF
(b)
Figure 3.11: AEF representing the IEC 61000-4-2 Standard ESD current waveshapefor 4kV with (a) 2 peaks, (b) 3 peaks. For parameters see Table 3.3.
3-peaked AEF representing measured data
In this section we present the results of fitting a 1-, 2- and a 3-peaked AEFto a waveform from experimental measurements from [87]. The result is alsocompared to a common type of function used for modelling ESD current,also from [87].
In figures 3.12 (a), 3.12 (b) and 3.13 the results of the interpolationof D-optimal points for certain parameters are shown together with themeasured data, as well as a sum of two Heidler functions that was fitted tothe experimental data in [87]. This function is given by
i(t) = I1
(tτ1
)nH
1 +(
tτ1
)nHe− t
τ2 + I2
(tτ3
)nH
1 +(
tτ3
)nHe− t
τ4 , (121)
130
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
Local maxima and minima and corresponding timesextracted from the IEC 61000-4-2, [73]
Imax1 = 15 [A] Imin1 = 7.1484 [A] Imax2 = 9.0921 [A]tmax1 = 6.89 [ns] tmin1 = 12.85 [ns] tmax2 = 25.54 [ns]
Parameters of interpolated AEF shown in figure 3.11 (a)
Interval n k c
0 ≤ t ≤ tmax1 3 35 1tmax1 ≤ t ≤ tmax2 3 3 2
tmax2 < t 5 3 1
Parameters of interpolated AEF shown in figure 3.11 (b)
Interval n k c
0 ≤ t ≤ tmax1 3 35 1tmax1 ≤ t ≤ tmin1 3 3 1tmin1 ≤ t ≤ tmax2 3 4 1
tmax2 < t 5 3 1
Table 3.3: Parameters’ values of the multi-peaked AEFs representing the IEC 61000-4-2 Standard waveshape.
with
I1 = 31.365 A, I2 = 6.854 A, nH = 4.036,
τ1 = 1.226 ns, τ2 = 1.359 ns, τ3 = 3.982 ns, τ4 = 28.817 ns.
Note that this function does not reproduce the second local minimumbut that all three AEF functions can reproduce all local minima and maxima(to a modest degree of accuracy) when suitable values for the n, k and mparameters are chosen.
3.3.4 Summary of ESD modelling
Here we examined a mathematical model for representation of ESD currents,either from the IEC 61000-4-2 Standard [73], or experimentally measuredones. The model has been proposed and successfully applied to lightningcurrent modelling in Section 3.2 and [107] and named the multi-peakedanalytically extended function (AEF).
It conforms to the requirements for the ESD current and its first deriva-tive, which are imposed by the Standard [73] stating that they must be equalto zero at moment t = 0. Furthermore, the AEF function is time-integrable,see Section 3.1.1, which is necessary for numerical calculation of radiatedfields originating from the ESD current.
131
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
and 3 peaks are illustrated, see figure 3.11 (a) and 3.11 (b). The parametersare listed in table 3.3. In the illustrated examples a fairly good fit is foundbut typically areas with steep rise and the decay part are somewhat moredifficult to fit with good accuracy than the other parts of the waveform.
t [s] #10-80 2 4 6 8
i(t) [
A]
0
0.2
0.4
0.6
0.8
1IEC 61000-4-2PeaksInterpolated points2-peaked AEF
(a)
t [s] #10-80 2 4 6 8
i(t) [
A]
0
0.2
0.4
0.6
0.8
1IEC 61000-4-2PeaksInterpolated points3-peaked AEF
(b)
Figure 3.11: AEF representing the IEC 61000-4-2 Standard ESD current waveshapefor 4kV with (a) 2 peaks, (b) 3 peaks. For parameters see Table 3.3.
3-peaked AEF representing measured data
In this section we present the results of fitting a 1-, 2- and a 3-peaked AEFto a waveform from experimental measurements from [87]. The result is alsocompared to a common type of function used for modelling ESD current,also from [87].
In figures 3.12 (a), 3.12 (b) and 3.13 the results of the interpolationof D-optimal points for certain parameters are shown together with themeasured data, as well as a sum of two Heidler functions that was fitted tothe experimental data in [87]. This function is given by
i(t) = I1
(tτ1
)nH
1 +(
tτ1
)nHe− t
τ2 + I2
(tτ3
)nH
1 +(
tτ3
)nHe− t
τ4 , (121)
130
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
Local maxima and minima and corresponding timesextracted from the IEC 61000-4-2, [73]
Imax1 = 15 [A] Imin1 = 7.1484 [A] Imax2 = 9.0921 [A]tmax1 = 6.89 [ns] tmin1 = 12.85 [ns] tmax2 = 25.54 [ns]
Parameters of interpolated AEF shown in figure 3.11 (a)
Interval n k c
0 ≤ t ≤ tmax1 3 35 1tmax1 ≤ t ≤ tmax2 3 3 2
tmax2 < t 5 3 1
Parameters of interpolated AEF shown in figure 3.11 (b)
Interval n k c
0 ≤ t ≤ tmax1 3 35 1tmax1 ≤ t ≤ tmin1 3 3 1tmin1 ≤ t ≤ tmax2 3 4 1
tmax2 < t 5 3 1
Table 3.3: Parameters’ values of the multi-peaked AEFs representing the IEC 61000-4-2 Standard waveshape.
with
I1 = 31.365 A, I2 = 6.854 A, nH = 4.036,
τ1 = 1.226 ns, τ2 = 1.359 ns, τ3 = 3.982 ns, τ4 = 28.817 ns.
Note that this function does not reproduce the second local minimumbut that all three AEF functions can reproduce all local minima and maxima(to a modest degree of accuracy) when suitable values for the n, k and mparameters are chosen.
3.3.4 Summary of ESD modelling
Here we examined a mathematical model for representation of ESD currents,either from the IEC 61000-4-2 Standard [73], or experimentally measuredones. The model has been proposed and successfully applied to lightningcurrent modelling in Section 3.2 and [107] and named the multi-peakedanalytically extended function (AEF).
It conforms to the requirements for the ESD current and its first deriva-tive, which are imposed by the Standard [73] stating that they must be equalto zero at moment t = 0. Furthermore, the AEF function is time-integrable,see Section 3.1.1, which is necessary for numerical calculation of radiatedfields originating from the ESD current.
131
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Local maxima and corresponding times extracted from [87, figure 3]
Imax1 = 7.37 [A] Imax2 = 5.02 [A] Imax3 = 3.82 [A]tmax1 = 1.23 [ns] tmax2 = 6.39 [ns] tmax3 = 15.5 [ns]
Parameters of interpolated AEF shown in figure 3.12 (a)
Interval n k c
0 ≤ t ≤ tmax3 12 8 0.8tmax3 < t 6 2 1
Parameters of interpolated AEF shown in figure 3.12 (b)
Interval n k c
0 ≤ t ≤ tmax1 5 40 1tmax1 ≤ t ≤ tmax3 7 4 1
tmax3 < t 6 2 1
Parameters of interpolated AEF shown in figure 3.13
Interval n k c
0 ≤ t ≤ tmax1 5 40 1tmax1 ≤ t ≤ tmax2 3 3 1tmax2 ≤ t ≤ tmax3 4 4 1
tmax3 < t 6 2 1
Table 3.4: Parameters’ values of multi-peaked AEFs representing experimental data.
Here we consider how the model can be fitted to a waveform using D-optimal interpolation and the resulting methodology is illustrated on theIEC 61000-4-2 Standard waveform [73] and experimental data from [87].
The resulting methodology can give fairly accurate results even with amodest number of interpolated points but strategies for choosing some ofthe involved parameters should be further investigated.
132
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
t [s] #10-82 4 6 8
i(t)
0
2
4
6
Measured dataTwo HeidlerPeaksInterpolated points1-peak AEF
(a)
t [s] #10-82 4 6 8
i(t)
0
2
4
6
Measured dataTwo HeidlerPeaksInterpolated points2-peaked AEF
(b)
Figure 3.12: AEF interpolated to D-optimal points chosen from measured ESD cur-rents from figure 3 in [87] with (a) 1 peak (b) 2 peaks. Parameters aregiven in Table 3.4.
t [s] #10-82 4 6 8
i(t)
0
2
4
6
Measured dataTwo HeidlerPeaksInterpolated points3-peaked AEF
Figure 3.13: 3-peaked AEF interpolated to D-optimal points chosen from measuredESD current from [87, figure 3]. Parameters are given in Table 3.4.
133
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
Local maxima and corresponding times extracted from [87, figure 3]
Imax1 = 7.37 [A] Imax2 = 5.02 [A] Imax3 = 3.82 [A]tmax1 = 1.23 [ns] tmax2 = 6.39 [ns] tmax3 = 15.5 [ns]
Parameters of interpolated AEF shown in figure 3.12 (a)
Interval n k c
0 ≤ t ≤ tmax3 12 8 0.8tmax3 < t 6 2 1
Parameters of interpolated AEF shown in figure 3.12 (b)
Interval n k c
0 ≤ t ≤ tmax1 5 40 1tmax1 ≤ t ≤ tmax3 7 4 1
tmax3 < t 6 2 1
Parameters of interpolated AEF shown in figure 3.13
Interval n k c
0 ≤ t ≤ tmax1 5 40 1tmax1 ≤ t ≤ tmax2 3 3 1tmax2 ≤ t ≤ tmax3 4 4 1
tmax3 < t 6 2 1
Table 3.4: Parameters’ values of multi-peaked AEFs representing experimental data.
Here we consider how the model can be fitted to a waveform using D-optimal interpolation and the resulting methodology is illustrated on theIEC 61000-4-2 Standard waveform [73] and experimental data from [87].
The resulting methodology can give fairly accurate results even with amodest number of interpolated points but strategies for choosing some ofthe involved parameters should be further investigated.
132
3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS
t [s] #10-82 4 6 8
i(t)
0
2
4
6
Measured dataTwo HeidlerPeaksInterpolated points1-peak AEF
(a)
t [s] #10-82 4 6 8
i(t)
0
2
4
6
Measured dataTwo HeidlerPeaksInterpolated points2-peaked AEF
(b)
Figure 3.12: AEF interpolated to D-optimal points chosen from measured ESD cur-rents from figure 3 in [87] with (a) 1 peak (b) 2 peaks. Parameters aregiven in Table 3.4.
t [s] #10-82 4 6 8
i(t)
0
2
4
6
Measured dataTwo HeidlerPeaksInterpolated points3-peaked AEF
Figure 3.13: 3-peaked AEF interpolated to D-optimal points chosen from measuredESD current from [87, figure 3]. Parameters are given in Table 3.4.
133
REFERENCES
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[2] Milton Abramowitz and Irene Stegun. Handbook of MathematicalFunctions with Formulas, Graphs, and Mathematical Tables. Dover,New York, 1964.
[3] Alexander Craig Aitken. Determinants and Matrices. Intersciencepublishers, Inc., 3rd edition, 1944.
[4] Richard Askey. Orthogonal Polynomials and Special Functions. Soci-ety for Industrial and Applied Mathematics, 1975.
[5] Sheldon Axler. Linear Algebra Done Right. Springer InternationalPublishing, 3rd edition, 2015.
[6] Tom Backstrom. Vandermonde factorization of Toeplitz matrices andapplications in filtering and warping. IEEE Transactions on SignalProcessing, 61(24):6257–6263, 2013.
[7] Tom Backstrom, Johannes Fischer, and Daniel Boley. Implementationand evaluation of the Vandermonde transform. In 22nd EuropeanSignal Processing Conference (EUSIPCO), pages 71–75, 2014.
[8] Michael Fielding Barnsley. Fractal functions and interpolation. Con-structive Approximation, 2(1):303–329, 1986.
[9] Michael Fielding Barnsley. Fractals Everywhere. Academic Press, Inc.,1988.
[10] Richard Bellman. Introduction to Matrix Analysis. McGraw-Hill BookCompany, New York, 2nd edition, 1970.
[11] Arthur T. Benjamin and Gregory P. Dresden. A combinatorial proofof Vandermonde’s determinant. The American Mathematical Monthly,114(4):338–341, April 2007.
[12] Jean-Paul Berrut and Lloyd N. Trefethen. Barycentric Lagrange in-terpolation. SIAM Review, 46(3):501–517, 2004.
[13] Ake Bjorck and Victor Pereyra. Solution of Vandermonde systems ofequations. Mathematics of Computation, 24(112):893–903, 1970.
135
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
[14] Maxime Bocher. Certain cases in which the vanishing of the Wronskianis a sufficient condition for linear dependence. Transactions of theAmerican Mathematical Society, 2(2):139–149, April 1900.
[15] Maxime Bocher. On linear dependence of functions of one variable.Bulletin of the American Mathematical Society, pages 120–121, De-cember 1900.
[16] Maxime Bocher. The theory of linear dependence. Annals of Mathe-matics, 2(1):81–96, January 1900.
[17] Nicholas Bonello, Sheng Chen, and Lajos Hanzo. Construction ofregular quasi-cyclic protograph LDPC codes based on Vandermondematrices. IEEE transactions on vehicular technology, 57(8):2583–2588,July 2008.
[18] Ray Chandra Bose and Dwijendra Kumar Ray-Chaudhuri. On a classof error correcting binary group codes. Information and Control, 1960.
[19] Alin Bostan and Phillippe Dumas. Wronskians and linear dependence.American Mathematical Monthly, 117(8):722–727, 2010.
[20] William E. Boyce and Richard C. DiPrima. Elementary DifferentialEquations and Boundary Value Problems. John Wiley & Sons, Inc.,7th edition, 2001.
[21] David Marius Bressoud. Proofs and Confirmations: The Story of theAlternating Sign Matrix Conjecture. Spectrum. Cambridge UniversityPress, 1999.
[22] Charles Edward Rhodes Bruce and R. H. Golde. The lightning dis-charge. The Journal of the Institution of Electrical Engineers - PartII: Power Engineering, 88(6):487 – 505, December 1941.
[23] Marshall W. Buck, Raymond A. Coley, and David P. Robbins. Ageneralized Vandermonde determinant. Journal of Algebraic Combi-natorics, 1:105–109, 1992.
[24] Augustin-Louis Cauchy. Memoire sur les fonctions qui ne peuventobtenir que deux valeurs egales et de signes contraires par suite destranspositions operees entre les variables qu’elles renferment. Jour-nal de l’Ecole Polytechnique, 10(17):29–112, 1815. Reprinted in Eu-vres completes d’Augustin Cauchy series 2, Volume 1, pp. 91–161,Gauthier-Villars, Paris (1899).
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[27] Paul A. Chatterton and Michael A. Houlden. EMC ElectromagneticTheory to Practical Design. John Wiley & Sons, Inc., 1992.
[28] Young-Min Chen, Hsuan-Chu Li, and Eng-Tjioe Tan. An explicit fac-torization of totally positive generalized Vandermonde matrices avoid-ing Schur functions. Applied Mathematics E-Notes, 8:138–147, 2008.
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[33] Stefano De Marchi. Polynomials arising in factoring generalized Van-dermonde determinants: an algorithm for computing their coefficients.Mathematical and Computer Modelling, 34(3):271–281, 2001.
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137
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
[14] Maxime Bocher. Certain cases in which the vanishing of the Wronskianis a sufficient condition for linear dependence. Transactions of theAmerican Mathematical Society, 2(2):139–149, April 1900.
[15] Maxime Bocher. On linear dependence of functions of one variable.Bulletin of the American Mathematical Society, pages 120–121, De-cember 1900.
[16] Maxime Bocher. The theory of linear dependence. Annals of Mathe-matics, 2(1):81–96, January 1900.
[17] Nicholas Bonello, Sheng Chen, and Lajos Hanzo. Construction ofregular quasi-cyclic protograph LDPC codes based on Vandermondematrices. IEEE transactions on vehicular technology, 57(8):2583–2588,July 2008.
[18] Ray Chandra Bose and Dwijendra Kumar Ray-Chaudhuri. On a classof error correcting binary group codes. Information and Control, 1960.
[19] Alin Bostan and Phillippe Dumas. Wronskians and linear dependence.American Mathematical Monthly, 117(8):722–727, 2010.
[20] William E. Boyce and Richard C. DiPrima. Elementary DifferentialEquations and Boundary Value Problems. John Wiley & Sons, Inc.,7th edition, 2001.
[21] David Marius Bressoud. Proofs and Confirmations: The Story of theAlternating Sign Matrix Conjecture. Spectrum. Cambridge UniversityPress, 1999.
[22] Charles Edward Rhodes Bruce and R. H. Golde. The lightning dis-charge. The Journal of the Institution of Electrical Engineers - PartII: Power Engineering, 88(6):487 – 505, December 1941.
[23] Marshall W. Buck, Raymond A. Coley, and David P. Robbins. Ageneralized Vandermonde determinant. Journal of Algebraic Combi-natorics, 1:105–109, 1992.
[24] Augustin-Louis Cauchy. Memoire sur les fonctions qui ne peuventobtenir que deux valeurs egales et de signes contraires par suite destranspositions operees entre les variables qu’elles renferment. Jour-nal de l’Ecole Polytechnique, 10(17):29–112, 1815. Reprinted in Eu-vres completes d’Augustin Cauchy series 2, Volume 1, pp. 91–161,Gauthier-Villars, Paris (1899).
136
REFERENCES
[25] Arthur Cayley. A memoir on the theory of matrices. PhilosophicalTransactions of the Royal Society of London, 148:17–37, 1858.
[26] Graziano Cerri, Roberto De Leo, and Valter Mariani Primian. ESDindirect coupling modelling. IEEE Transactions on ElectromagneticCompatibility, 38(3):274–281, 1996.
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[133] Giuseppe Peano. Sur le determinant Wronskien. Mathesis, 9:75–76,1889.
[134] Giuseppe Peano. Sur le Wronskiens. Mathesis, 9:110–112, 1889.
[135] Lennart Persson. Handbook of Fish Biology and Fisheries, volume 1,chapter 15 Community Ecology of Freshwater Fishes, pages 321–340.Blackwell Publishing, 2002.
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[124] Thomas Muir and William Henry Metzler. A Treatise on the Theoryof Determinants. Dover Publications Inc., New York, 1966.
[125] Gary L. Mullen and Daniel Panario. Handbook of Finite Fields. CRCPress, 2013.
[126] Isidor Pavlovich Natanson. Constructive Function Theory, Volume 1:Uniform Approximation. Frederick Ungar Publishing Co., Inc., 1964.
[127] Joseph Needham and Wang Ling. Science and Civilisation in China,Volume 3: Mathematics and the Sciences of the Heavens and theEarth. Cambridge University Press, 1959.
[128] Øystein Ore. On a special class of polynomials. Transactions of theAmerican Mathematical Society, 35(3):559–584, July 1933.
[129] Halil Oruc. LU factorization of the Vandermonde matrix and its ap-plications. Applied Mathematics Letters, 20:982–987, 2007.
[130] Halil Oruc and Hakan K. Akmaz. Symmetric functions and the Van-dermonde matrix. Journal of Computational and Applied Mathemat-ics, pages 49–64, 2004.
[131] Alexander Ostrowski. Uber ein Analogon der Wronskischen Deter-minante bei Funktionen mehrerer Veranderlicher. MathematischeZeitschrift, 4(3):223–230, September 1919.
[132] Clayton R. Paul. Introduction to Electromagnetic Compatibility. JohnWiley & Sons, Inc., 1992.
[133] Giuseppe Peano. Sur le determinant Wronskien. Mathesis, 9:75–76,1889.
[134] Giuseppe Peano. Sur le Wronskiens. Mathesis, 9:110–112, 1889.
[135] Lennart Persson. Handbook of Fish Biology and Fisheries, volume 1,chapter 15 Community Ecology of Freshwater Fishes, pages 321–340.Blackwell Publishing, 2002.
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Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
[136] Lennart Persson, Kjell Leonardsson, Andre M. de Roos, Mats Gyllen-berg, and Bent Christensen. Ontogenetic scaling of foraging rates andthe dynamics of a size-structured consumer-resource model. Theoret-ical Population Biology, 54:270–293, 1998.
[137] Dragan Poljak. Advanced Modeling in Computational ElectromagneticCompatibility. John Wiley & Sons, Inc., 2007.
[138] William H. Press, Saul A. Teukolsky, William T. Vetterling, andBrian P. Flannery. Numerical Recipes in C: The Art of ScientificComputing. Cambridge University Press, 3rd edition, 2007.
[139] Anatoli Prudnikov, Jurij Aleksandrovic Bryckov, and Oleg IgorevicMaricev. Integrals and Series: More Special Functions, volume 3.Gordon and Breach Science Publishers, 1990.
[140] Jozef H. Przytycki. History of the knot theory from Vandermonde toJones. In XXIVth National Congress of the Mexican MathematicalSociety (Spanish) (Oaxtepec, 1991), pages 173–185, 1991.
[141] Jennifer J. Quinn. Visualizing Vandermonde’s determinant throughnonintersecting lattice paths. Journal of Statistical Planning and In-ference, 140(8):2346–2350, 2010.
[142] Kamisetti Ramamohan Rao, Do Nyeon Kim, and Jae-Jong Hwang.Fast Fourier Transform - Algorithms and Applications. Springer, 2010.
[143] Irving Stoy Reed and Gustave Solomon. Polynomial codes over cer-tain finite fields. Journal of the Society for Industrial and AppliedMathematics, 8(2):300–304, 1960.
[144] Ralph Tyrell Rockafellar. Lagrange multipliers and optimality. SIAMReview, 35(2):183–238, 1993.
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[148] Thomas Scharf, Jean-Yves Thibon, and Brian Garner. Wybourne.Powers of the Vandermonde determinant and the quantum Hall ef-fect. Journal of Physics A: General Physics, 27(12):4211–4219, 1994.
[149] Hans Peter Schlickewei and Carlo Viola. Generalized Vandermondedeterminants. Acta Arithmetica, XCV(2):123–137, 2000.
[150] George Arthur Frederick Seber and Chris J. Wild. Nonlinear Regres-sion. John Wiley & Sons, Inc., 2003.
[151] Lewis B. Sheiner and Stuart L. Beal. Evaluation of methods forestimating population pharmacokinetic parameters II. biexponentialmodel and experimental pharmacokinetic data. Journal of Pharma-cokinetics and Biopharmaceutics, 9(5):635–651, 1981.
[152] Takatoshi Shindo, Toru Miki, Mikihisa Saito, Daiki Tanaka, AkiraAsakawa, Hideki Motoyama, Masaru Ishii, Takeo Sonehara, YusukeSuzuhigashi, and Hiroshi Taguchi. Lightning observations at TokyoSkytree: Observation systems and observation results in 2012 and2013. In Proceedings of the 2014 International Symposium on Elec-tromagnetic Compatibility (EMC Europe 2014), Gothenburg, Sweden,pages 583–588, 2014.
[153] David Eugene Smith. Leibniz on determinants. In A Source Book inMathematics, volume 1. Dover Publications Inc., New York, 1959.
[154] Kirstine Smith. On the standard deviations of adjusted and interpo-lated values of an observed polynomial function and its constants andthe guidance they give towards a proper choice of the distribution ofthe observations. Biometrika, 12(1/2):1–85, 1918.
[155] Garrett Sobczyk. Generalized Vandermonde determinants and appli-cations. Aportaciones Matematicas, 30:41–53, 2002.
[156] S. Songlin, B. Zengjun, T. Minghong, and L. Shange. A new analyticalexpression of current waveform in standard IEC 61000-4-20. HighPower Laser and Particle Beams, 5:464–466, 2003.
[157] James Joseph Sylvester. Additions to the articles in the Septembernumber of this journal, “on a new class of theorems,” and on Pascal’stheorem. Philosophical Magazine Series 3, 37(251):363–370, 1850.
[158] Gabor Szego. Orthogonal Polynomials. American Mathematics Soci-ety, 1975.
147
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
[136] Lennart Persson, Kjell Leonardsson, Andre M. de Roos, Mats Gyllen-berg, and Bent Christensen. Ontogenetic scaling of foraging rates andthe dynamics of a size-structured consumer-resource model. Theoret-ical Population Biology, 54:270–293, 1998.
[137] Dragan Poljak. Advanced Modeling in Computational ElectromagneticCompatibility. John Wiley & Sons, Inc., 2007.
[138] William H. Press, Saul A. Teukolsky, William T. Vetterling, andBrian P. Flannery. Numerical Recipes in C: The Art of ScientificComputing. Cambridge University Press, 3rd edition, 2007.
[139] Anatoli Prudnikov, Jurij Aleksandrovic Bryckov, and Oleg IgorevicMaricev. Integrals and Series: More Special Functions, volume 3.Gordon and Breach Science Publishers, 1990.
[140] Jozef H. Przytycki. History of the knot theory from Vandermonde toJones. In XXIVth National Congress of the Mexican MathematicalSociety (Spanish) (Oaxtepec, 1991), pages 173–185, 1991.
[141] Jennifer J. Quinn. Visualizing Vandermonde’s determinant throughnonintersecting lattice paths. Journal of Statistical Planning and In-ference, 140(8):2346–2350, 2010.
[142] Kamisetti Ramamohan Rao, Do Nyeon Kim, and Jae-Jong Hwang.Fast Fourier Transform - Algorithms and Applications. Springer, 2010.
[143] Irving Stoy Reed and Gustave Solomon. Polynomial codes over cer-tain finite fields. Journal of the Society for Industrial and AppliedMathematics, 8(2):300–304, 1960.
[144] Ralph Tyrell Rockafellar. Lagrange multipliers and optimality. SIAMReview, 35(2):183–238, 1993.
[145] Klaus Friedrich Roth. Rational approximations to algebraic numbers.Mathematika, 2(1):1–20, 1955.
[146] Walter Rudin. Real and Complex Analysis. WCB/McGraw-Hill BookCompany, 3rd edition, 1987.
[147] Andrzej Ruszczynski. Nonlinear Optimization. Princeton UniversityPress, 2006.
146
REFERENCES
[148] Thomas Scharf, Jean-Yves Thibon, and Brian Garner. Wybourne.Powers of the Vandermonde determinant and the quantum Hall ef-fect. Journal of Physics A: General Physics, 27(12):4211–4219, 1994.
[149] Hans Peter Schlickewei and Carlo Viola. Generalized Vandermondedeterminants. Acta Arithmetica, XCV(2):123–137, 2000.
[150] George Arthur Frederick Seber and Chris J. Wild. Nonlinear Regres-sion. John Wiley & Sons, Inc., 2003.
[151] Lewis B. Sheiner and Stuart L. Beal. Evaluation of methods forestimating population pharmacokinetic parameters II. biexponentialmodel and experimental pharmacokinetic data. Journal of Pharma-cokinetics and Biopharmaceutics, 9(5):635–651, 1981.
[152] Takatoshi Shindo, Toru Miki, Mikihisa Saito, Daiki Tanaka, AkiraAsakawa, Hideki Motoyama, Masaru Ishii, Takeo Sonehara, YusukeSuzuhigashi, and Hiroshi Taguchi. Lightning observations at TokyoSkytree: Observation systems and observation results in 2012 and2013. In Proceedings of the 2014 International Symposium on Elec-tromagnetic Compatibility (EMC Europe 2014), Gothenburg, Sweden,pages 583–588, 2014.
[153] David Eugene Smith. Leibniz on determinants. In A Source Book inMathematics, volume 1. Dover Publications Inc., New York, 1959.
[154] Kirstine Smith. On the standard deviations of adjusted and interpo-lated values of an observed polynomial function and its constants andthe guidance they give towards a proper choice of the distribution ofthe observations. Biometrika, 12(1/2):1–85, 1918.
[155] Garrett Sobczyk. Generalized Vandermonde determinants and appli-cations. Aportaciones Matematicas, 30:41–53, 2002.
[156] S. Songlin, B. Zengjun, T. Minghong, and L. Shange. A new analyticalexpression of current waveform in standard IEC 61000-4-20. HighPower Laser and Particle Beams, 5:464–466, 2003.
[157] James Joseph Sylvester. Additions to the articles in the Septembernumber of this journal, “on a new class of theorems,” and on Pascal’stheorem. Philosophical Magazine Series 3, 37(251):363–370, 1850.
[158] Gabor Szego. Orthogonal Polynomials. American Mathematics Soci-ety, 1975.
147
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
[159] Gabriel Tellez and Peter J. Forrester. Exact finite-size study of the 2DOCP at Γ = 4 and Γ = 6. Journal of Statistical Physics, 97(3):489–521, November 1999.
[160] Joseph F. Traub. Associated polynomials and uniform methods forthe solution of linear problems. SIAM Review, 8(3):277–301, 1966.
[161] Herbert Westren Turnbull and Alexander Craig Aitken. An Introduc-tion to the Theory of Canonical Matrices. Dover Publications, Inc.,1961.
[162] L. Richard Turner. Inverse of the Vandermonde matrix with applica-tions. Technical report, National Aeronautics and Space Administra-tion, Lewis Research Center, Cleveland Ohio, 1966.
[163] Steve Van den Berghe and Daniel De Zutter. Study of ESD signalentry through coaxial cable shields. Journal of Electrostatics, 44(3–4):135–148, September 1998.
[164] Alexandre-Theophile Vandermonde. Memoire sur la resolution desequations. Histoire de l’Academie royale des sciences avec lesmemoires de mathematiques et de physique pour la meme annee tiresdes registres de cette academie. Annee MDCCLXXI, pages 365–416,1774.
[165] Alexandre-Theophile Vandermonde. Remarques sur des problemes desituation. Histoire de l’Academie royale des sciences avec les memoiresde mathematiques et de physique pour la meme annee tires des reg-istres de cette academie. Annee MDCCLXXI, pages 566–574, 1774.
[166] Alexandre-Theophile Vandermonde. Memoire sur des irrationnelles dedifferents ordres avec une application au cercle. Histoire de l’Academieroyale des sciences avec les memoires de mathematiques et de physiquepour la meme annee tires des registres de cette academie. Annee MD-CCLXXII Premiere Partie, pages 489–498, 1775.
[167] Alexandre-Theophile Vandermonde. Memoire sur l’elimination.Histoire de l’Academie royale des sciences avec les memoires demathematiques et de physique pour la meme annee tires des registresde cette academie. Annee MDCCLXXII Seconde Partie, pages 516–532, 1776.
[168] Robert Vein and Paul Dale. Determinants and Their Applications inMathematical Physics. Springer-Verlag New York, 1999.
148
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[169] Abraham Wald. On the efficient design of statistical investigations.The Annals of Mathematical Statistics, 14(2):134–140, June 1943.
[170] Kai Wang, D. Pommerenke, R. Chundru, T. Van Doren, J. L. Drew-niak, and A. Shashindranath. Numerical modeling of electrostaticdischarge generators. IEEE Transactions on Electromagnetic Com-patibility, 45(2):258–271, 2003.
[171] Ke Wang, Jinshan Wang, and Xiaodong Wang. Four order electro-static discharge circuit model and its simulation. TELKOMNIKA,10(8):2006–2012, 2012.
[172] Edward Waring. Problems concerning interpolations. PhilosophicalTransactions of the Royal Society of London, 69:59–67, 1779.
[173] Tim Williams. EMC for Product Designers. Newnes, 3rd edition,2001.
[174] Kenneth Wolsson. A condition equivalent to linear dependence forfunctions with vanishing Wronskian. Linear Algebra and its Applica-tions, 116:1–8, 1989.
[175] Kenneth Wolsson. Linear dependence of a function set of m variableswith vanishing generalized Wronskians. Linear Algebra and its Appli-cations, 117:73–80, 1989.
[176] Sebastian Xambo-Descamps. Block Error-Correcting Codes. Springer-Verlag Berlin Heidelberg, 1st edition, 2003.
[177] Shang-Jun Yang, Hua-Zhang Wu, and Quan-Bing Zhang. Generaliza-tion of Vandermonde determinants. Linear Algebra and its Applica-tions, 336:201–204, October 2001.
[178] Chen Yazhou, Liu Shanghe, Wu Xiaorong, and Zhang Feizhou. A newkind of channel-base current function. In 3rd International symposiumon Electromagnetic Compatibility, pages 304–646, May 2002.
[179] Bernard Ycart. A case of mathematical eponymy: the Vandermondedeterminant. Revue d’Histoire des Mathematiques, 9(1):43–77, 2013.
[180] Zhiyong Yuan, Tun Li, Jinliang He, Shuiming Chen, and Rong Zeng.New mathematical descriptions of ESD current waveform based on thepolynomial of pulse function. IEEE Transactions on ElectromagneticCompatibility, 48(3):589–591, 2006.
149
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
[159] Gabriel Tellez and Peter J. Forrester. Exact finite-size study of the 2DOCP at Γ = 4 and Γ = 6. Journal of Statistical Physics, 97(3):489–521, November 1999.
[160] Joseph F. Traub. Associated polynomials and uniform methods forthe solution of linear problems. SIAM Review, 8(3):277–301, 1966.
[161] Herbert Westren Turnbull and Alexander Craig Aitken. An Introduc-tion to the Theory of Canonical Matrices. Dover Publications, Inc.,1961.
[162] L. Richard Turner. Inverse of the Vandermonde matrix with applica-tions. Technical report, National Aeronautics and Space Administra-tion, Lewis Research Center, Cleveland Ohio, 1966.
[163] Steve Van den Berghe and Daniel De Zutter. Study of ESD signalentry through coaxial cable shields. Journal of Electrostatics, 44(3–4):135–148, September 1998.
[164] Alexandre-Theophile Vandermonde. Memoire sur la resolution desequations. Histoire de l’Academie royale des sciences avec lesmemoires de mathematiques et de physique pour la meme annee tiresdes registres de cette academie. Annee MDCCLXXI, pages 365–416,1774.
[165] Alexandre-Theophile Vandermonde. Remarques sur des problemes desituation. Histoire de l’Academie royale des sciences avec les memoiresde mathematiques et de physique pour la meme annee tires des reg-istres de cette academie. Annee MDCCLXXI, pages 566–574, 1774.
[166] Alexandre-Theophile Vandermonde. Memoire sur des irrationnelles dedifferents ordres avec une application au cercle. Histoire de l’Academieroyale des sciences avec les memoires de mathematiques et de physiquepour la meme annee tires des registres de cette academie. Annee MD-CCLXXII Premiere Partie, pages 489–498, 1775.
[167] Alexandre-Theophile Vandermonde. Memoire sur l’elimination.Histoire de l’Academie royale des sciences avec les memoires demathematiques et de physique pour la meme annee tires des registresde cette academie. Annee MDCCLXXII Seconde Partie, pages 516–532, 1776.
[168] Robert Vein and Paul Dale. Determinants and Their Applications inMathematical Physics. Springer-Verlag New York, 1999.
148
REFERENCES
[169] Abraham Wald. On the efficient design of statistical investigations.The Annals of Mathematical Statistics, 14(2):134–140, June 1943.
[170] Kai Wang, D. Pommerenke, R. Chundru, T. Van Doren, J. L. Drew-niak, and A. Shashindranath. Numerical modeling of electrostaticdischarge generators. IEEE Transactions on Electromagnetic Com-patibility, 45(2):258–271, 2003.
[171] Ke Wang, Jinshan Wang, and Xiaodong Wang. Four order electro-static discharge circuit model and its simulation. TELKOMNIKA,10(8):2006–2012, 2012.
[172] Edward Waring. Problems concerning interpolations. PhilosophicalTransactions of the Royal Society of London, 69:59–67, 1779.
[173] Tim Williams. EMC for Product Designers. Newnes, 3rd edition,2001.
[174] Kenneth Wolsson. A condition equivalent to linear dependence forfunctions with vanishing Wronskian. Linear Algebra and its Applica-tions, 116:1–8, 1989.
[175] Kenneth Wolsson. Linear dependence of a function set of m variableswith vanishing generalized Wronskians. Linear Algebra and its Appli-cations, 117:73–80, 1989.
[176] Sebastian Xambo-Descamps. Block Error-Correcting Codes. Springer-Verlag Berlin Heidelberg, 1st edition, 2003.
[177] Shang-Jun Yang, Hua-Zhang Wu, and Quan-Bing Zhang. Generaliza-tion of Vandermonde determinants. Linear Algebra and its Applica-tions, 336:201–204, October 2001.
[178] Chen Yazhou, Liu Shanghe, Wu Xiaorong, and Zhang Feizhou. A newkind of channel-base current function. In 3rd International symposiumon Electromagnetic Compatibility, pages 304–646, May 2002.
[179] Bernard Ycart. A case of mathematical eponymy: the Vandermondedeterminant. Revue d’Histoire des Mathematiques, 9(1):43–77, 2013.
[180] Zhiyong Yuan, Tun Li, Jinliang He, Shuiming Chen, and Rong Zeng.New mathematical descriptions of ESD current waveform based on thepolynomial of pulse function. IEEE Transactions on ElectromagneticCompatibility, 48(3):589–591, 2006.
149
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
[181] Changqing Zhu, Sanghe Liu, and Ming Wei. Analytic expressionand numerical solution of ESD current. High Voltage Engineering,31(7):22–24, 2005. in Chinese.
150
Index
AEF, see analytically extended func-tion
alternant matrix, 25analytically extended function, 100,
108, 114
D-optimal design, 42, 119, 123determinant, 19digamma function, 113
electromagnetic compatibility, 44electromagnetic disturbance, 44electromagnetic interference, see elec-
tromagnetic disturbanceelectrostatic discharge, 45, 119EMC, see electromagnetic compati-
bilityESD, see electrostatic discharge
Fischer information matrix, 41, 123
G-optimal design, 41Gegenbauer polynomials, 94
Heidler function, 47Hermite polynomial, 77hypergeometric function, 125
interpolation, 30Hermite, 34polynomial, 31
Jacobi polynomial, 124Jacobian matrix, 26, 109
Lagrange interpolation, 31Lambert W function, 124least-squares method, 35lightning discharge, 45, 115linear model, 30
Marquardt least-squares method, 37,108, 114
Meijer G-function, 113MLSM, see Marquardt least-squares
method
orthogonal polynomialGegenbauer, 94Hermite, 77Jacobi, 124
power exponential function, 99
regression, 35Runge’s phenomenon, 32, 43
Schur polynomials, 29
VandermondeAlexandre Theophile, 17determinant, 19, 21matrix, 17generalized, 28, 123inverse, 24, 32
Wronskian matrix, 26
151
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
[181] Changqing Zhu, Sanghe Liu, and Ming Wei. Analytic expressionand numerical solution of ESD current. High Voltage Engineering,31(7):22–24, 2005. in Chinese.
150
Index
AEF, see analytically extended func-tion
alternant matrix, 25analytically extended function, 100,
108, 114
D-optimal design, 42, 119, 123determinant, 19digamma function, 113
electromagnetic compatibility, 44electromagnetic disturbance, 44electromagnetic interference, see elec-
tromagnetic disturbanceelectrostatic discharge, 45, 119EMC, see electromagnetic compati-
bilityESD, see electrostatic discharge
Fischer information matrix, 41, 123
G-optimal design, 41Gegenbauer polynomials, 94
Heidler function, 47Hermite polynomial, 77hypergeometric function, 125
interpolation, 30Hermite, 34polynomial, 31
Jacobi polynomial, 124Jacobian matrix, 26, 109
Lagrange interpolation, 31Lambert W function, 124least-squares method, 35lightning discharge, 45, 115linear model, 30
Marquardt least-squares method, 37,108, 114
Meijer G-function, 113MLSM, see Marquardt least-squares
method
orthogonal polynomialGegenbauer, 94Hermite, 77Jacobi, 124
power exponential function, 99
regression, 35Runge’s phenomenon, 32, 43
Schur polynomials, 29
VandermondeAlexandre Theophile, 17determinant, 19, 21matrix, 17generalized, 28, 123inverse, 24, 32
Wronskian matrix, 26
151
List of Figures
1.1 Relations between sections of the thesis. Arrows indicate thatthe target section uses some definition or theorem from thesource section. Dashed lines indicates a tangential or concep-tual relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2 Reference that demonstrates short routes to the different chap-ters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Some examples of different interpolating curves. The set ofred points are interpolated by a polynomial (left), a self-affinefractal (middle) and a Lissajous curve (right). . . . . . . . . . 32
1.4 Illustration of Lagrange interpolation of 4 data points. The
red dots are the data set and p(x) =4∑
k=1
ykp(xk) is the inter-
polating polynomial. . . . . . . . . . . . . . . . . . . . . . . . 34
1.5 Illustration of Runge’s phenomenon. Here we attempt to ap-proximate a function (dashed line) by polynomial interpola-tion (solid line). With 7 equidistant sample points (left figure)the approximation is poor near the edges of the interval andincreasing the number of sample points to 14 (center) and 19(right) clearly reduces accuracy at the edges further. . . . . . 35
1.6 The basic iteration step of the Marquardt least-squares method,definitions of computed quantities are given in (12), (13) and(14). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.7 Functions representing the Standard ESD current waveshapefor 4kV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Plot of v3(x3) over a proper torus (r1 = 1, r2 = 3), 3D-plot with curve marked (left), parametrised plot with curvemarked (center), values of v3(x(α)) along the curve (right). . 63
153
List of Figures
1.1 Relations between sections of the thesis. Arrows indicate thatthe target section uses some definition or theorem from thesource section. Dashed lines indicates a tangential or concep-tual relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2 Reference that demonstrates short routes to the different chap-ters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Some examples of different interpolating curves. The set ofred points are interpolated by a polynomial (left), a self-affinefractal (middle) and a Lissajous curve (right). . . . . . . . . . 32
1.4 Illustration of Lagrange interpolation of 4 data points. The
red dots are the data set and p(x) =4∑
k=1
ykp(xk) is the inter-
polating polynomial. . . . . . . . . . . . . . . . . . . . . . . . 34
1.5 Illustration of Runge’s phenomenon. Here we attempt to ap-proximate a function (dashed line) by polynomial interpola-tion (solid line). With 7 equidistant sample points (left figure)the approximation is poor near the edges of the interval andincreasing the number of sample points to 14 (center) and 19(right) clearly reduces accuracy at the edges further. . . . . . 35
1.6 The basic iteration step of the Marquardt least-squares method,definitions of computed quantities are given in (12), (13) and(14). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.7 Functions representing the Standard ESD current waveshapefor 4kV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Plot of v3(x3) over a proper torus (r1 = 1, r2 = 3), 3D-plot with curve marked (left), parametrised plot with curvemarked (center), values of v3(x(α)) along the curve (right). . 63
153
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
2.6 Plot of v3(x3) over a horn torus (r1 = 1, r2 = 1), 3D-plot withcurve marked (left), parametrised plot with curve marked(center), values of v3(x(α)) along the curve (right). . . . . . . 63
2.7 Plot of v3(x3) over a spindle torus (r1 = 3, r2 = 1), 3D-plot with curve marked (left), parametrised plot with curvemarked (center), values of v3(x(α)) along the curve (right). . 64
2.8 Illustration of the ellipsoid defined byx2
9+
y2
4+ z2 = 0 with
the extreme points of the Vandermonde determinant marked.Displayed in Cartesian coordinates on the right and in ellip-soidal coordinates on the left. . . . . . . . . . . . . . . . . . . 68
2.9 Illustration of the cylinder defined by y2 +16
25z2 = 1 with
the extreme points of the Vandermonde determinant marked.Displayed in Cartesian coordinates on the right and in cylin-drical coordinates on the left. . . . . . . . . . . . . . . . . . . 70
2.10 Illustration of the ellipsoid defined by (38) with the extremepoints of the Vandermonde determinant marked. Displayedin Cartesian coordinates on the right and in ellipsoidal coor-dinates on the left. . . . . . . . . . . . . . . . . . . . . . . . . 73
2.15 Value of v3(x3) over: S22 (left), S2
4 (middle left), S28 (middle
right) and S2∞ (right). . . . . . . . . . . . . . . . . . . . . . . 89
3.1 An illustration of how the steepness of the power exponentialfunction varies with β. . . . . . . . . . . . . . . . . . . . . . . 101
3.2 Illustration of the AEF (solid line) and its derivative (dashedline) with different βq,k-parameters but the same Imq and tmq .(a) 0 < βq,k < 1, (b) 4 < βq,k < 5, (c)12 < βq,k < 13, (d) a mixture of large and small βq,k-parameters104
3.3 An example of a two-peaked AEF where some of the ηq,k-parameters are negative, so that it has points where the firstderivative changes sign between two peaks. The solid line isthe AEF and the dashed lines is the derivative of the AEF. . 106
3.4 Schematic description of the parameter estimation algorithm 112
3.5 First-positive stroke represented by the AEF function. Hereit is fitted with respect to both the data points as well as Q0
and W0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.6 First-negative stroke represented by the AEF function. Hereit is fitted with the extra constraint 0 ≤ η ≤ 1 for all η-parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
154
LIST OF FIGURES
3.7 Fast-decaying waveshape represented by the AEF function.Here it is fitted with the extra constraint 0 ≤ η ≤ 1 for allη-parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.8 AEF fitted to measurements from [152]. Here the peaks havebeen chosen to correspond to local maxima in the measureddata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.9 AEF fitted to measurements from [152]. Here the peaks havebeen chosen to correspond to local maxima and minima inthe measured data. . . . . . . . . . . . . . . . . . . . . . . . . 121
3.10 Illustration of the IEC 61000-4-2 Standard ESD current andits key parameters, [73]. . . . . . . . . . . . . . . . . . . . . . 123
3.11 AEF representing the IEC 61000-4-2 Standard ESD currentwaveshape for 4kV with (a) 2 peaks, (b) 3 peaks. For param-eters see Table 3.3. . . . . . . . . . . . . . . . . . . . . . . . . 130
3.12 AEF interpolated to D-optimal points chosen from measuredESD currents from figure 3 in [87] with (a) 1 peak (b) 2 peaks.Parameters are given in Table 3.4. . . . . . . . . . . . . . . . 133
3.13 3-peaked AEF interpolated to D-optimal points chosen frommeasured ESD current from [87, figure 3]. Parameters aregiven in Table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . 133
155
Generalized Vandermonde matrices and determinants inelectromagnetic compatibility
2.6 Plot of v3(x3) over a horn torus (r1 = 1, r2 = 1), 3D-plot withcurve marked (left), parametrised plot with curve marked(center), values of v3(x(α)) along the curve (right). . . . . . . 63
2.7 Plot of v3(x3) over a spindle torus (r1 = 3, r2 = 1), 3D-plot with curve marked (left), parametrised plot with curvemarked (center), values of v3(x(α)) along the curve (right). . 64
2.8 Illustration of the ellipsoid defined byx2
9+
y2
4+ z2 = 0 with
the extreme points of the Vandermonde determinant marked.Displayed in Cartesian coordinates on the right and in ellip-soidal coordinates on the left. . . . . . . . . . . . . . . . . . . 68
2.9 Illustration of the cylinder defined by y2 +16
25z2 = 1 with
the extreme points of the Vandermonde determinant marked.Displayed in Cartesian coordinates on the right and in cylin-drical coordinates on the left. . . . . . . . . . . . . . . . . . . 70
2.10 Illustration of the ellipsoid defined by (38) with the extremepoints of the Vandermonde determinant marked. Displayedin Cartesian coordinates on the right and in ellipsoidal coor-dinates on the left. . . . . . . . . . . . . . . . . . . . . . . . . 73
2.15 Value of v3(x3) over: S22 (left), S2
4 (middle left), S28 (middle
right) and S2∞ (right). . . . . . . . . . . . . . . . . . . . . . . 89
3.1 An illustration of how the steepness of the power exponentialfunction varies with β. . . . . . . . . . . . . . . . . . . . . . . 101
3.2 Illustration of the AEF (solid line) and its derivative (dashedline) with different βq,k-parameters but the same Imq and tmq .(a) 0 < βq,k < 1, (b) 4 < βq,k < 5, (c)12 < βq,k < 13, (d) a mixture of large and small βq,k-parameters104
3.3 An example of a two-peaked AEF where some of the ηq,k-parameters are negative, so that it has points where the firstderivative changes sign between two peaks. The solid line isthe AEF and the dashed lines is the derivative of the AEF. . 106
3.4 Schematic description of the parameter estimation algorithm 112
3.5 First-positive stroke represented by the AEF function. Hereit is fitted with respect to both the data points as well as Q0
and W0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.6 First-negative stroke represented by the AEF function. Hereit is fitted with the extra constraint 0 ≤ η ≤ 1 for all η-parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
154
LIST OF FIGURES
3.7 Fast-decaying waveshape represented by the AEF function.Here it is fitted with the extra constraint 0 ≤ η ≤ 1 for allη-parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.8 AEF fitted to measurements from [152]. Here the peaks havebeen chosen to correspond to local maxima in the measureddata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.9 AEF fitted to measurements from [152]. Here the peaks havebeen chosen to correspond to local maxima and minima inthe measured data. . . . . . . . . . . . . . . . . . . . . . . . . 121
3.10 Illustration of the IEC 61000-4-2 Standard ESD current andits key parameters, [73]. . . . . . . . . . . . . . . . . . . . . . 123
3.11 AEF representing the IEC 61000-4-2 Standard ESD currentwaveshape for 4kV with (a) 2 peaks, (b) 3 peaks. For param-eters see Table 3.3. . . . . . . . . . . . . . . . . . . . . . . . . 130
3.12 AEF interpolated to D-optimal points chosen from measuredESD currents from figure 3 in [87] with (a) 1 peak (b) 2 peaks.Parameters are given in Table 3.4. . . . . . . . . . . . . . . . 133
3.13 3-peaked AEF interpolated to D-optimal points chosen frommeasured ESD current from [87, figure 3]. Parameters aregiven in Table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . 133
155
List of Tables
2.1 Table of some determinants of generalized Vandermonde ma-trices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1 AEF function’s parameters for some current waveshapes . . . 1203.2 IEC 61000-4-2 Standard ESD Current and its Key Parame-
ters, [73]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.3 Parameters’ values of the multi-peaked AEFs representing the
IEC 61000-4-2 Standard waveshape. . . . . . . . . . . . . . . 1313.4 Parameters’ values of multi-peaked AEFs representing exper-
imental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
List of Definitions
Definition 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Definition 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Definition 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Definition 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Definition 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Definition 1.6 (The G-optimality criterion) . . . . . . . . . . . . 43Definition 1.7 (The D-optimality criterion) . . . . . . . . . . . . 44
Definition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Definition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Definition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
156
List of Theorems
Theorem 1.1 (Leibniz formula for determinants) . . . . . . . . . 22Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Theorem 1.5 (Kiefer-Wolfowitz equivalence theorem) . . . . . . 44
Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Theorem 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Theorem 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Theorem 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
List of Lemmas
Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Lemma 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Lemma 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Lemma 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Lemma 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Lemma 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Lemma 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
157
List of Tables
2.1 Table of some determinants of generalized Vandermonde ma-trices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1 AEF function’s parameters for some current waveshapes . . . 1203.2 IEC 61000-4-2 Standard ESD Current and its Key Parame-
ters, [73]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.3 Parameters’ values of the multi-peaked AEFs representing the
IEC 61000-4-2 Standard waveshape. . . . . . . . . . . . . . . 1313.4 Parameters’ values of multi-peaked AEFs representing exper-
imental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
List of Definitions
Definition 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Definition 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Definition 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Definition 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Definition 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Definition 1.6 (The G-optimality criterion) . . . . . . . . . . . . 43Definition 1.7 (The D-optimality criterion) . . . . . . . . . . . . 44
Definition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Definition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Definition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
156
List of Theorems
Theorem 1.1 (Leibniz formula for determinants) . . . . . . . . . 22Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Theorem 1.5 (Kiefer-Wolfowitz equivalence theorem) . . . . . . 44
Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Theorem 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Theorem 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Theorem 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
List of Lemmas
Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Lemma 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Lemma 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Lemma 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Lemma 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Lemma 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Lemma 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
157
