Concis e Numerical

Concis e Numerica l Mathematic s

http://dx.doi.org/10.1090/gsm/057

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Concis e Numerica l Mathematic s

Robert Plato

Translated by Richard Le Borne Sabine Le Borne

Graduate Studies

in Mathematics

Volume 57

American Mathematical Society Providence, Rhode Island

^XDED"

Editorial Boar d

Walter Crai g Nikolai Ivano v

Steven G . Krant z David Saltma n (Chair )

Originally publishe d i n th e Germa n languag e b y Friedr . Viewe g Sz Soh n Verlagsgesellschaft mbH , D-6518 9 Wiesbaden , Germany , unde r th e t i t l e "Rober t P la to : Numerische Ma thema t i k kompakt . 1 . Auflag e (1s t edi t ion)" . © Friedr . Viewe g & Soh n

Verlagsgesellschaft mbH , Braunschweig/Wiesbaden , 2000 .

Trans la ted fro m th e origina l Ge rma n editio n b y Richar d L e Born e an d Sabin e L e Borne .

2000 Mathematics Subject Classification. P r i m a r y 65-00 , 65-01 , 65B05 , 65B15 , 65Dxx, 65Fxx , 65G50 , 65H05 , 65H10 , 65H17 , 65K10 , 65Lxx ,

65Q05, 65T40 , 65T50 , 65Y20 , 49M15 , 49M20 .

For addit iona l informatio n an d upda t e s o n thi s book , visi t w w w . a m s . o r g / b o o k p a g e s / g s m - 5 7

Library o f Congres s Cataloging-in-Publicatio n D a t a

Plato, Robert , 1962 -[Numerische Mathemati k kompakte . English ] Concise numerica l mathematic s / Rober t Plat o ; translate d b y Richar d L e Borne , Sabin e

Le Borne . p. cm . - (Graduat e studie s i n mathematics , ISS N 1065-733 9 ; v. 57 )

Includes bibliographica l reference s an d index . ISBN 0-8218-2953- X (acid-fre e paper ) ISB N 0-8218-3414- 2 (softcover ) 1. Numerica l analysis . I . Title . II . Series .

QA297.P56713 200 3 519.4-dc21 200203301 0

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Contents

Preface t o th e Englis h Editio n x i

Preface t o th e Germa n Editio n xii i

Chapter 1 . Interpolatio n b y Polynomial s 1

§1.1. Genera l prerequisite s an d Landa u symbol s 1

§1.2. Existenc e an d uniquenes s o f a n interpolatin g polynomia l 3

§1.3. Neville' s algorith m 6

§1.4. Newton' s interpolatio n formula , divide d difference s 8

§1.5. Th e interpolatio n erro r 1 1

§1.6. Chebyshe v polynomial s 1 4

Additional topic s an d literatur e 1 8

Exercises 1 9

Chapter 2 . Splin e Function s 2 3

§2.1. Introductor y remark s 2 3

§2.2. Interpolatin g linea r splin e function s 2 4

§2.3. Minimalit y propertie s o f cubic splin e function s 2 5

§2.4. Th e calculatio n o f interpolating cubi c splin e function s 2 7

§2.5. Erro r estimate s fo r interpolatin g cubi c spline s 3 3


Exercises 3 8

Chapter 3 . Th e Discret e Fourie r Transfor m an d It s Application s 4 1

§3.1. Discret e Fourie r transfor m 4 1

VI Contents

§3.2. Application s o f the discret e Fourie r transfor m 4 3

§3.3. Fas t Fourie r transfor m (FFT ) 4 9


Exercises 5 7

Chapter 4 . Solutio n o f Linear System s o f Equations 5 9

§4.1. Triangula r system s 5 9

§4.2. Gaussia n eliminatio n 6 1

§4.3. Th e factorizatio n PA = LR 6 6

§4.4. LR factorizatio n 7 4

§4.5. Cholesk y factorizatio n fo r positiv e definit e matrice s 7 6

§4.6. Bande d matrice s 7 9

§4.7. Norm s an d erro r estimate s 8 1

§4.8. Th e factorizatio n A = QS 9 1


Exercises 10 0

Chapter 5 . Nonlinea r System s o f Equations 10 5

§5.1. Preliminar y remark s 10 5

§5.2. Th e one-dimensiona l cas e (N = 1 ) 10 7

§5.3. Banach' s fixed poin t theore m 10 9

§5.4. Newton' s metho d 11 2


Exercises 12 1

Chapter 6 . Th e Numerica l Integratio n o f Functions 12 3

§6.1. Quadratur e b y interpolatio n formula s 12 4

§6.2. Specia l quadratur e b y interpolatio n formula s 12 5

§6.3. Th e erro r du e t o quadratur e b y interpolatio n 12 9

§6.4. Degre e o f exactnes s fo r th e close d Newton-Cote s formulas ,

n eve n 13 2

§6.5. Composit e Newton-Cote s formula s 13 7

§6.6. Asymptoti c for m o f the composit e trapezoida l rul e 14 1

§6.7. Extrapolatio n method s 14 2

§6.8. Gaussia n quadratur e 14 6

§6.9. Appendix : Proo f o f th e asymptoti c for m fo r th e composit e trapezoidal rul e 15 5

Contents vn


Exercises 15 9

Chapter 7 . Explici t One-Ste p Method s fo r Initia l Valu e Problem s

in Ordinar y Differentia l Equation s 16 1

§7.1. A n existenc e an d uniquenes s theore m 16 2

§7.2. Theor y o f one-step method s 16 3

§7.3. One-Ste p method s 16 6

§7.4. Analysi s o f round-off erro r 17 0

§7.5. Asymptoti c expansio n o f the approximation s 17 2

§7.6. Extrapolatio n method s fo r one-ste p method s 17 8

§7.7. Ste p siz e contro l 18 2


Exercises 18 6

Chapter 8 . Multiste p Method s fo r Initia l Valu e Problem s

of Ordinar y Differentia l Equation s 18 9

§8.1. Fundamenta l term s 18 9

§8.2. Th e globa l discretizatio n erro r fo r multiste p method s 19 2

§8.3. Specifi c linea r multiste p method s - preparation s 20 1

§8.4. Adam s metho d 20 4

§8-5. Nystro m an d Milne-Simpso n method s 21 0

§8.6. BD F metho d 21 4

§8.7. Predictor-correcto r method s 21 6

§8.8. Linea r homogeneou s differenc e equation s 22 2

§8.9. Stif f differentia l equation s 23 2


Exercises 24 2

Chapter 9 . Boundar y Valu e Problem s fo r Ordinar y Differentia l

Equations 24 7

§9.1. Proble m setting , existence , uniquenes s 24 7

§9.2. Differenc e method s 25 0

§9.3. Galerki n method s 26 0

§9.4. Simpl e shootin g method s 27 4


Exercises 27 7

V l l l Contents

Chapter 10 . Jacobi , Gauss-Seide l an d Relaxatio n Method s fo r th e Solution o f Linear System s o f Equations 28 1

§10.1. Iteratio n method s fo r th e solutio n o f linea r system s o f

equations 28 1

§10.2. Linea r fixe d poin t iteratio n 28 2

§10.3. Som e specia l classe s o f matrices an d thei r propertie s 28 7

§10.4. Th e Jacob i metho d 28 9

§10.5. Th e Gauss-Seide l metho d 29 2

§10.6. Th e relaxatio n metho d an d firs t convergenc e result s 29 5

§10.7. Th e relaxatio n metho d fo r consistentl y ordere d matrice s 30 0


Exercises 30 5

Chapter 11 . Th e Conjugat e Gradien t an d GMRE S Method s 31 1

§11.1. Prerequisite s 31 1

§11.2. Th e orthogona l residua l approac h (11.2 ) fo r positiv e

definite matrice s 31 3

§11.3. Th e C G metho d fo r positiv e definit e matrice s 31 6

§11.4. Th e convergenc e rat e o f the C G metho d 31 9

§11.5. Th e C G metho d fo r th e norma l equation s 32 3

§11.6. Arnold i proces s 32 4

§11.7. Realizatio n o f GMRE S o n th e basi s o f the Arnold i proces s 32 8

§11.8. Convergenc e rat e o f the GMRE S metho d 33 3

§11.9. Appendi x 1 : Krylo v subspace s 33 4

§11.10. Appendi x 2 : Interactiv e progra m system s wit h

multifunctionality 33 5


Exercises 33 7

Chapter 12 . Eigenvalu e Problem s 33 9

§12.1. Introductio n 33 9

§12.2. Perturbatio n theor y fo r eigenvalu e problem s 33 9

§12.3. Localizatio n o f eigenvalues 34 3

§12.4. Variationa l formulatio n fo r eigenvalue s o f symmetri c matrices 34 6

§12.5. Perturbatio n result s fo r th e eigenvalue s o f symmetri c matrices 34 9

Contents IX

§12.6. Appendix : Factorizatio n o f matrices 35 0


Exercises 35 1

Chapter 13 . Numerica l Method s fo r Eigenvalu e Problem s 35 5


§13.2. Transformatio n t o Hessenber g for m 35 7

§13.3. Newton' s metho d fo r th e calculatio n o f th e eigenvalue s of Hessenberg matrice s 36 2

§13.4. Th e Jacob i metho d fo r th e off-diagona l elemen t

reduction fo r symmetri c matrice s 36 6

§13.5. Th e QR algorith m 37 3

§13.6. Th e LR algorith m 38 6

§13.7. Th e vecto r iteratio n 38 7


Exercises 39 0

Chapter 14 . Peano' s Erro r Representatio n 39 3


§14.2. Pean o kernel s 39 4

§14.3. Application s 39 7


Exercises 39 8

Chapter 15 . Approximatio n Theor y 40 1


§15.2. Existenc e o f a bes t approximatio n 40 2

§15.3. Uniquenes s o f a bes t approximatio n 40 4

§15.4. Approximatio n theor y i n space s wit h a scala r produc t 40 8

§15.5. Unifor m approximatio n o f continuou s function s b y

polynomials o f maximu m degre e n — 1 41 1

§15.6. Application s o f the alternatio n theore m 41 5

§15.7. Haa r spaces , Chebyshe v system s 41 7


Exercises 42 0

Chapter 16 . Compute r Arithmeti c 42 3

§16.1. Numbe r representation s 42 3

X Contents

§16.2. Genera l floating poin t numbe r system s 42 4

§16.3. Floatin g poin t numbe r system s i n practica l application s 42 9

§16.4. Rounding , truncatin g 43 2

§16.5. Arithmeti c i n floating poin t numbe r system s 43 6


Bibliography 44 3

Index 44 9

Preface t o th e Englis h Edition

This textbook i s a translation o f Numerische Mathematik kompakt publishe d by Viewe g Verlag , wit h onl y a fe w changes : mos t o f th e reference s t o Ger -man monographs appearing in the German edition are replaced by reference s to Englis h writte n textbooks , som e ne w publication s ar e added , an d a fe w errors hav e bee n corrected . A webpag e fo r thi s boo k i s maintaine d o n th e AMS website at :

http://www.ams.org/bookpages/gsm-57.

I woul d lik e t o than k Richar d C . L e Born e an d Sabin e L e Born e (the y are bot h a t Tennesse e Technologica l University , Cookeville ) fo r th e carefu l translation an d fo r a ver y efficien t an d pleasan t cooperation . I a m gratefu l to severa l colleague s an d student s fo r suggestion s o n th e Germa n editio n which ar e considere d t o som e exten t i n thi s edition . Moreover , I expres s thanks t o th e Christia n -Albrechts Universit y fo r trave l suppor t i n connec -tion wit h thi s translatio n an d fo r givin g me the opportunit y durin g th e las t two year s t o hol d introductor y course s o n numerica l mathematic s fo r stu -dents o f mathematics , engineering , an d compute r sciences . Thes e course s were base d o n th e Viewe g editio n o f thi s textbook . Additionally , I woul d like t o than k Edwar d Dunn e fro m th e America n Mathematica l Societ y fo r several helpfu l suggestion s an d hi s patience .

XI

Xl l Preface to the English Edition

Finally tw o comment s o n frequentl y use d notations . Th e symbo l A mark s the en d o f a n example , a remark , o r a n algorithm , an d th e symbo l ' — « — ' is used a s a ditt o mark .

Berlin, German y Octobe r 200 2 Rober t Plat o

Preface t o th e Germa n Edition

This textboo k emerge d fro m m y two-semeste r course s i n numerica l mathe -matics which I have repeatedly taught sinc e 1997 at the Technical Universit y of Berlin. Thes e lectures were attended primaril y b y students o f mathemat -ics and t o a smalle r exten t b y student s o f physics an d compute r sciences .

In its current for m this textbook addresse s students and graduates of mathe-matics, computer sciences, and natural and engineering sciences. I n a concise form severa l basi c topics o f numerical mathematic s wit h importan t applica -tions ar e treated :

• interpolation , fas t Fourie r transfor m an d integration ,

• direc t an d iterativ e solutio n o f linea r system s o f equations ,

• iterativ e method s fo r nonlinea r system s o f equations ,

• numerica l solutio n o f initia l an d boundar y valu e problem s for ordinar y differentia l equations ,

• eigenvalu e problem s fo r matrices ,

• approximatio n theor y an d compute r arithmetic .

For th e purpos e o f conciseness , th e numerica l solutio n o f partia l differentia l equations a s wel l a s nonlinea r optimizatio n i s not treate d here .

The intentio n o f thi s textboo k i s t o conside r topic s i n a n elementar y an d clear form , thereb y onl y requirin g a basi c knowledg e o f calculu s an d linea r

xm

XIV Preface to the German Edition

algebra. Moreover , man y o f the presente d method s ar e illustrate d b y fig -ures. Fo r numerou s algorithms , practica l an d relevan t wor k estimate s ar e given and pseudo code s are provided tha t ca n be used fo r implementations . The approximatel y 12 0 exercises presente d wit h differen t level s of difficult y proved t o be good i n the courses .

I use d preliminar y version s o f thi s textboo k a s lectur e note s fo r m y two-semester course s i n numerical mathematics , eac h cours e meetin g fou r hour s per week . Th e firs t si x chapter s wer e alway s considere d i n th e firs t part , and Chapter s 7-1 3 were treate d i n the secon d par t o f these lectures . I n a one-semester cours e in numerical mathematics , an alternative is to consider numerical integration (Chapte r 6 ) only briefly, an d then to present the foun-dations of one-step method s for the numerical solution of initial value prob-lems for ordinary differentia l equation s (Chapte r 7 ) and relaxation method s for th e iterative solutio n o f linear system s o f equations (Chapte r 10) .

For thi s textboo k ther e exist s a n onlin e servic e whic h i s availabl e unde r http:/ /www.math.tu-berlin.de/numerik/plato/viewegbuch. I t contains hints fo r th e presente d exercise s an d MATLA B routine s fo r som e o f th e pseudo code s presente d i n the textbook . Moreover , thi s onlin e servic e wil l contain a list o f possibly necessar y corrections . Comment s o n this textboo k can b e sent t o my email address , [email protected] .

I would like to thank my colleagues Prof. Dr. R. D. Grigorieff an d Dipl. Math. Etienne Emmric h fo r severa l suggestion s whic h ar e considered i n this text -book t o a larg e extent . I a m indebte d t o Til l Tanta u an d Olivie r Pfeiffe r and som e other participant s o f the lectures fo r several mino r bu t importan t improvements. Finall y I woul d lik e t o than k Prof . Dr . Chuc k Groetsch , Prof. Dr . Marti n Hanke-Bourgeoi s an d Prof . Dr . Hans-Jiirge n Reinhard t for thei r suppor t o f thi s boo k projec t an d Ulrik e Schmickler-Hirzebruc h from th e publisher Viewe g fo r her pleasant cooperation .

Berlin, Ma y 2000 Robert Plat o

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Index

3/8-rule , 12 7 A H , 4 2 Ak - > A, 37 4 C[a,b], 1 9 Cr[a,b], 1 9

L2 ~scalar produc t (u,v)2 — Ja u(x)v(x) dx, 261

C\ [a , b] , space of piecewise continuously dif -ferent iable functions , 26 2

CS(D,RN) fo r D C I M , 17 3 0 , 2 A/"(L), nul l spac e L , 22 3 5 (B) , 36 7 X M , characteristi c functio n wit h respec t t o

a give n se t M , 19 7 C - ^ a , ^ ] , 39 3

^ G 5 , 29 2 Ti j , 29 0 H(u), 29 5 J ( / ) , 12 3 2n ( / ) , 12 3 0, 2

M, i n £ x , 34 6 /imax, 16 5 Xmax, larges t elemen t i n th e floatin g poin t

number syste m F , 42 6 #mim smalles t positiv e normalize d elemen t

in th e floatin g poin t numbe r syste m F , 426

^min5 smalles t positiv e denormalize d float -ing poin t numbe r i n F , 42 8

D y f(t,u), Dyijj(t,u; • ) , Jacobia n matri x with respec t t o spatia l variables , 17 5

B(x^S), 10 5

B{ x\ r), close d bal l centered a t x wit h radiu s r, 40 4

Vkgu, backwar d differences , 20 1 No, se t o f natura l number s > 0 , 6 I • ||p , l < p < o o , 8 3 — « —, x i

R, se t o f rea l numbers , 1 n ( A ) , 1 2 cr(B), spectru m o f matri x B , 8 4 ra(B), spectra l radiu s o f matri x B, 8 4 s ( K ) , spac e o f sequences , 22 3 | i t | 2 fo r a continuou s functio n u : [a , 6] —*•

K, 2 5 I u J 00 > maximu m nor m o f a continuous func -

tion u, 2 5 A, xi

A-conjugate vectors , 31 5 A -norm, 31 3 a posterior i erro r estimate , 11 1 a prior i erro r estimate , 11 1 Adams - Bashfor t method , 20 4 Adams - Moulto n method , 20 8 algebro-differential equations , system s of ,

186 algorithm o f Strassen , 10 0 alternant, alternatio n theorem , 41 2 arithmetic operation , 4 Arnoldi process , 32 5 asymptotic expansio n o f th e erro r fo r o n e -

step methods , 17 2 Aubin-Nitsche trick , 27 2

backward difference , 25 0 banded matrix , 7 9

449

450 Index

basis, 42 3 BDF method , 21 4 Bernoulli numbers , 15 7 best approximation , 40 1 bias notation , 42 9 bit reverse , 5 2 Bk(x), Bernoull i polynomials , 15 5 block relaxatio n method , 30 5 bounded bilinea r form , 26 8 Bulirsch sequence , 14 5

Cauchy-Schwarz inequality , 40 8 central differenc e o f firs t order , 25 0 central differenc e o f secon d order , 25 0 CG method , 31 6 CGNR method , 32 3 characteristic polynomia l o f a difference equa -

tion, 22 4 Chebyshev polynomial s

of th e secon d kin d U n, 2 0 of th e firs t kin d T n , 1 4 optimality property , 1 5

Chebyshev system , 41 8 Cholesky factorization , 7 7 circulant, 33 7 classical Runge-Kut t a method , 16 9 closed subse t o f a norme d space , 40 4 coercive bilinea r form , 26 8 column norm , 8 3 composite quadratur e formulas , 13 7

asymptotics o f th e composit e trapezoida l rule, 14 1

composite rectangl e rules , 13 7 composite Simpson' s rule , 14 0 composite trapezoida l rule , 13 9

condition numbe r o f a matrix , 8 8 conjugate gradien t method , 31 6 connecting chain , 30 6 consistency condition , 18 7 consistency orde r o f multiste p methods , 19 1 consistency orde r o f one-s te p methods , 16 5 consistent matri x norm , 8 4 contraction, 11 0 convergence orde r o f one-s tep methods , 16 5 convergent stationar y iterativ e method , 28 3 convex subse t o f a norme d space , 40 4 cubic splin e function , 2 4 cubic B - splines, 27 0

Dahlquist's roo t condition , 19 2 data compression , 4 4 deflation, 11 8 denormalized floating poin t number , 42 5 diagonalizable matrix , 35 0 digital dat a transfer , 4 4 discrete Fourie r transform , 4 1 discrete invers e Fourie r transform , 4 3

discrete maximu m principle , 27 8 dissipative differentia l equation , 23 3 divergent stationar y iterativ e method , 28 3 divided differences , 9 duality trick , 27 2

eigenvalue problem , 33 9 elementary permutation , 6 8 elementary permutatio n matrix , 6 8 energy functional , 27 3 energy norm , 26 5 eps , precisio n o f th e floating poin t numbe r

system F , 42 7 error constant , 20 1 Euclidian norm , 8 2 Euler's method , 16 6 Euler - Maclauri n summatio n formula , 157 exchange metho d o f Remez , 41 7 explicit ra-step method , 19 0 extended format , additiona l floating poin t

number systems , 42 9 extrapolation methods , 17 8

F( b, t, emin , em a x ) , system of normalized float-ing poin t numbers , 42 5

F( b, t, emin , em ax ), extended syste m o f float-ing poin t numbers , 42 5

FFT, Fas t Fourie r Transform , 4 9 finite elemen t method , 26 9 five poin t formula , five poin t star , 28 6 fixed point , 10 6 fixed poin t iteration , 106 , 11 0

convergent o f orde r p > 1 , 10 6 exactly convergen t o f orde r p, 10 6 linear convergence , 10 6 quadratic convergence , 10 6

floating poin t numbers , 42 5 forward difference , 25 0 Fourier coefficient s

complex, 4 3 real, 4 3

Friedrich's inequality , 26 3 Frobenius companio n matrix , 11 8 Frobenius matri x o f inde x /c , 69 Frobenius norm , 8 3 function o f a constan t sign , 12 9

Galerkin approximatio n 's 6 <S , 266 Galerkin method , 26 6

fully discrete , 26 9 quasi-optimality, 26 7

Gauss-Seidel method , 29 2 Gaussian elimination , 6 1

with (column ) pivoting , 6 6 with tota l pivoting , 10 1

Gaussian quadratur e formulas , 15 2 generalized, wea k solution , 265 , 27 4

Index 451

generating polynomial , 19 2 Gerschgorin discs , 34 3 global iteratio n erro r o f one-s te p methods ,

164 Gram matrix , 41 1 Gram-Schmidt process , 9 3 grid functio n Uh(t), 17 2 Gronwall's lemma , 19 5

discrete version , 19 7

Haar space , 41 8 Hadamard's estimat e fo r determinants , 10 2 Hamming's method , 24 4 harmonic sequence , 14 5 hat functions , 26 9 Hermite interpolation , 1 9 Hessenberg matrix , 328 , 35 5

lower Hessenber g matrix , 35 6 upper Hessenber g matrix , 35 6

Heun's method , 16 8 homogeneous differenc e equation , 22 3 Horner scheme , 8 Householder transformation , 9 5

IEEE, Institut e o f Electrica l an d Electronic s Engineers, 42 9

ill-conditioned linea r syste m o f equations , 90

implicit Eule r method , 20 9 implicit m -step method , 19 0 induced matri x norm , 8 4 initial valu e problem , existenc e an d unique -

ness, 16 2 inner product , 40 8 integration ste p for o n e- an d multiste p method ,

214 inverse monotonicity , 27 8 inverse triangl e inequality , 8 1 inverse Wieland t iteration , 38 9 involutory matrix , 9 4 irreducible matrix , 28 7 isometric, 9 2 iteration function , 105 , 16 3 iteration matrix , 28 3 iteration method , 28 1

Jacobi metho d fo r eigenvalu e problems , 36 7 classical, 37 1 cyclic, 37 2

Jacobi metho d fo r linea r system s o f equa -tions, 29 0

KANT, 33 6 Krylov subspac e methods , 31 2

Lagrange interpolatio n formula , 4 Lagrange polynomials , 3

Lanczos process , 32 6 Landau symbols , 2 leading coefficient , 1 1 least square s method , 27 9 least square s problem , 9 9 Lemma o f Cea , 26 8 lexicographical ordering , 28 6 linear differenc e operator , 22 3 linear elementar y divisors , 19 8 linear fixed poin t iteration , 28 3 linear ra~step method , 19 0 linear multiste p methods , 19 0 linear splin e function , 2 4 LLT factorization , 7 8 local iteratio n erro r o f one-s te p methods ,

164 local truncatio n erro r o f multiste p methods ,

191 logarithmic norm , 24 1 LR algorithm , 38 6 LR factorization , 7 5

M-mat r ix , 29 8 m - s t e p methods , 19 0 MACSYMA, 33 6 mantissa, 42 5 mantissa length , 42 5 Maple, 33 6 Mathematica, 33 6 MATLAB, xiv , 33 6 matrix equilibration , 10 0

row equilibrate d matrix , 10 2 matrix norm , 8 1 maximum norm , 8 2 method o f Hyman , 36 2 method o f lines , 24 0 midpoint rule , 128 , 190 , 21 2 Milne's method , 213 , 24 4 Milne's rule , 12 7 Milne-Simpson method , 21 2 minimal distance , 40 1 minimal residua l approach , 31 1 minimizing sequence , 40 3 MINRES, 32 4 modified Eule r method , 16 8 moments o f a splin e function , 2 9 multigrid method , 30 5 multiple shootin g methods , 27 7 multistep methods , 189 , 19 0 MuPAD, Mult i Processin g Algebra Data Tool ,

336

n-digit decima l arithmetic , 42 7 NaN, No t a Number , 43 0 Neumann series , 25 6 Neville's algorithm , 7 Newton polynomials , 8

452 Index

Newton's interpolatio n formula , 9 Newton's method , on e - dimensiona l case , 10 8 Newton's method, multidimensiona l case , 11 4 Newton-Cotes formulas , 12 5 Newton-Cotes formulas , closed , 12 5 nonlinear optimization , xiii , 40 2 nonnegative matrix , 25 3 norm, 8 1 normal equation s A 1Ax — A Tb, 32 3 normal equation s fo r bes t approximations ,

411 numerical solution of partial differentia l equa -

tions, xii i Nystrom method , 21 0

Octave, 33 6 one-s tep methods , 16 3 online servic e t o thi s book , xi , xi v open subse t o f a norme d space , 40 4 order o f convergenc e o f multiste p methods ,

191 orthogonal complement o f sets (i n vector space s

with inne r products) , 40 9 orthogonal polynomials , 14 7 orthogonal residua l approach , 31 1 over-relaxation, 29 6 overflow, underflow , 43 5

parallel- an d vecto r computers , 10 0 parallelogram equation , 40 9 partitioning accordin g t o Crout , 7 5 P ( E C ) M E method , 22 1 P ( E C ) M method , 22 2 Peano kernel , 39 4 permutation, 6 6 permutation matrix , 6 6 n n , 3 n £ , 20 2 11^, orthogona l complemen t o f I l n C n , 14 7 piecewise continuously differentiat e function ,

262 pivot, 6 6 Poisson equation , 28 4 polynomial interpolation , 3

error, 11 , 1 3 existence an d uniqueness , 4

positive definit e matrix , 7 6 positive homogeneity , 8 1 power boundednes s o f a matrix , 19 4 predictor-corrector method , 21 6 preprocessing phas e fo r multiste p method ,

190 principal submatrices , 76 , 10 1 program system s with multifunctionality , 33 5

computer algebr a functionality , 33 5 graphical functionality , 33 6 numerical functionality , 33 5

QR algorithm , 37 3 quadratic splin e function , 2 4 quadrature formula , 12 3

degree o f exactness , 12 3 by interpolation , 12 4

rational interpolation , 1 8 Rayleigh quotient , 348 , 38 8 rectangle rules , 12 8 REDUCE, 33 6 reducible matrix , 28 7 regression line , 9 8 regular decompositio n A = B — P o f a

matrix, 30 5 regularization methods , 10 0 relaxation method , 29 5 residual, 31 2 Ritz method , 26 6 Romberg integration , 14 2 Romberg sequence , 14 5 rounding t o t digits , 43 3 row norm , 8 3

scalar product , 40 8 Schulz's method , 12 2 Schur factorization , 35 1 Scilab, 33 6 $>A,£i 2 3 search direction , 31 6 secant method , 12 1 Sherman-Morrison formula , 10 3 sign matrix , 37 3 simple Kutt a rule , 18 8 simple shootin g method , 27 4 Simpson's rule , 12 7 singular valu e decompositio n o f a matrix ,

101 singular value s o f a matrix , 10 1 smoothing, 4 4 sparse matrix , 28 2 spectral norm , 8 6 spectral radiu s o f a matrix , 8 4 spectrum o f a matrix , 8 4 spline function , spline , 2 3 square roo t method , 7 8 stability o f one-s te p methods , 16 5 step length , 31 6 step siz e control , 18 2 stiff differentia l equations , 23 2 stiffness matrix , 26 8 Stormer's method , 24 3 strictly conve x set , 40 4 strictly diagonall y dominant , 3 2 strictly norme d space , 40 5 Sturm-Liouville's boundar y valu e problem ,

249 sum norm , 8 2

Index 453

support abscissas , 1 support ordinates , 9 support points , 3 symbolic computation , 33 6 symmetric matrix , 35 0 system matrix , 26 8

Taylor's method , 18 7 Theorem o f

Bauer/Fike, 34 0 Courant/Fischer, 34 7 Faber, 1 3 Kahan, 29 6 Kusmin, 12 8 Ostrowski/Reich, 29 7 Perron, 25 9 Picard/Lindelof, 16 2 Rayleigh/Ritz, 34 8

th ree- te rm recursio n fo r orthogonal polyno -mials, 14 8

trapezoidal rule , 126 , 20 9 triangle inequality , 8 1 triangular linea r systems of equations, uppe r

and lower , 5 9 triangular matrix , 5 9 trigonometric polynomial s (complex) , 4 4 trigonometric polynomial s (real) , 4 8 truncation t o t digits , 43 5 two gri d iteration , 30 5

under-relaxation, 29 6 unitary matrix , 34 1 upper Lipschit z condition , 23 3

van de r Pol' s differentia l equation , 18 7 variational equality , 26 5 vector iteration , 38 7 vector norm , 8 1 vertices, 2 3 von Mise s iteration , 38 9

weight function , 14 6

zero-stability, 19 2

Titles i n Thi s Serie s

57 Rober t P lato , Concis e numerica l mathematics , 200 3

56 E . B . Vinberg , A cours e i n algebra , 200 3

55 C . Herber t Clemens , A scrapboo k o f comple x curv e theory , secon d edition , 200 3

54 Alexande r Barvinok , A cours e i n convexity , 200 2

53 Henry k Iwaniec , Spectra l method s o f automorphi c forms , 200 2

52 Ilk a Agricol a an d Thoma s Friedrich , Globa l analysis : Differentia l form s i n analysis , geometry an d physics , 200 2

51 Y . A . Abramovic h an d C D . Aliprantis , Problem s i n operato r theory , 200 2

50 Y . A . Abramovic h an d C D . Aliprantis , A n invitatio n t o operato r theory , 200 2

49 Joh n R . Harper , Secondar y cohomolog y operations , 200 2

48 Y . Eliashber g an d N . Mishachev , Introductio n t o th e /i-principle , 200 2

47 A . Yu . Kitaev , A . H . Shen , an d M . N . Vyalyi , Classica l an d quantu m computation , 2002

46 Josep h L . Taylor , Severa l comple x variable s wit h connection s t o algebrai c geometr y an d Lie groups , 200 2

45 Inde r K . Rana , A n introductio n t o measur e an d integration , secon d edition , 200 2

44 J i m Agle r an d Joh n E . M c C a r t h y , Pic k interpolatio n an d Hilber t functio n spaces , 200 2

43 N . V . Krylov , Introductio n t o th e theor y o f rando m processes , 200 2

42 Ji n Hon g an d Seok-Ji n Kang , Introductio n t o quantu m group s an d crysta l bases , 200 2

41 Georg i V . Smirnov , Introductio n t o th e theor y o f differentia l inclusions , 200 2

40 Rober t E . Green e an d Steve n G . Krantz , Functio n theor y o f on e comple x variable , 2002

39 Larr y C Grove , Classica l group s an d geometri c algebra , 200 2

38 Elto n P . Hsu , Stochasti c analysi s o n manifolds , 200 2

37 Hershe l M . Farka s an d Irwi n Kra , Thet a constants , Rieman n surface s an d th e modula r group, 200 1

36 Marti n Schechter , Principle s o f functiona l analysis , secon d edition , 200 2

35 Jame s F . Davi s an d Pau l Kirk , Lectur e note s i n algebrai c topology , 200 1

34 Sigurdu r Helgason , Differentia l geometry , Li e groups , an d symmetri c spaces , 200 1

33 Dmitr i Burago , Yur i Burago , an d Serge i Ivanov , A cours e i n metri c geometry , 200 1

32 Rober t G . Bartle , A moder n theor y o f integration , 200 1

31 Ral f Kor n an d Elk e Korn , Optio n pricin g an d portfoli o optimization : Moder n method s

of financia l mathematics , 200 1

30 J . C McConnel l an d J . C Robson , Noncommutativ e Noetheria n rings , 200 1

29 Javie r Duoandikoetxea , Fourie r analysis , 200 1

28 Livi u I . Nicolaescu , Note s o n Seiberg-Witte n theory , 200 0

27 Thierr y Aubin , A cours e i n differentia l geometry , 200 1

26 Rol f Berndt , A n introductio n t o symplecti c geometry , 200 1

25 Thoma s Friedrich , Dira c operator s i n Riemannia n geometry , 200 0

24 Helmu t Koch , Numbe r theory : Algebrai c number s an d functions , 200 0

23 Albert o Cande l an d Lawrenc e Conlon , Foliation s I , 200 0 22 Giinte r R . Kraus e an d Thoma s H . Lenagan , Growt h o f algebra s an d Gelfand-Kirillo v

dimension, 200 0

21 Joh n B . Conway , A cours e i n operato r theory , 200 0

20 Rober t E . Gomp f an d Andra s I . Stipsicz , 4-manifold s an d Kirb y calculus , 199 9

19 Lawrenc e C Evans , Partia l differentia l equations , 199 8

TITLES I N THI S SERIE S

18 Winfrie d Jus t an d Marti n Weese , Discoverin g moder n se t theory . II : Set-theoreti c

tools fo r ever y mathematician , 199 7

17 Henry k Iwaniec , Topic s i n classica l automorphi c forms , 199 7

16 Richar d V . Kadiso n an d Joh n R . Ringrose , Fundamental s o f th e theor y o f operato r

algebras. Volum e II : Advance d theory , 199 7

15 Richar d V . Kadiso n an d Joh n R . Ringrose , Fundamental s o f th e theor y o f operato r

algebras. Volum e I : Elementar y theory , 199 7

14 Elliot t H . Lie b an d Michae l Loss , Analysis , 199 7

13 Pau l C . Shields , Th e ergodi c theor y o f discret e sampl e paths , 199 6

12 N . V . Krylov , Lecture s o n ellipti c an d paraboli c equation s i n Holde r spaces , 199 6

11 Jacque s Dixmier , Envelopin g algebras , 199 6 Printin g

10 Barr y Simon , Representation s o f finite an d compac t groups , 199 6

9 D in o Lorenzini , A n invitatio n t o arithmeti c geometry , 199 6

8 Winfrie d Jus t an d Marti n Weese , Discoverin g moder n se t theory . I : Th e basics , 199 6

7 Geral d J . Janusz , Algebrai c numbe r fields , secon d edition , 199 6

6 Jen s Carste n Jantzen , Lecture s o n quantu m groups , 199 6

5 Ric k Miranda , Algebrai c curve s an d Rieman n surfaces , 199 5

4 Russel l A . Gordon , Th e integral s o f Lebesgue , Denjoy , Perron , an d Henstock , 199 4

3 Wil l ia m W . A d a m s an d Phi l ipp e Loustaunau , A n introductio n t o Grobne r bases ,

1994

2 Jac k Graver , Brigit t e Servatius , an d Herma n Servatius , Combinatoria l rigidity ,

1993

1 Etha n Akin , Th e genera l topolog y o f dynamica l systems , 199 3

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