Tutorial: Linear Algebra In LabVIEW

UniversityCollegeofSoutheastNorway

http://home.hit.no/~hansha

LinearAlgebrainLabVIEWHans-PetterHalvorsen,2016-10-31

PrefaceThisdocumentexplainsthebasicconceptsofLinearAlgebraandhowyoumayuseLabVIEWforcalculationoftheseproblems.

YoushouldhavesomebasicknowledgeaboutLabVIEW,e.g.,the“AnIntroductiontoLabVIEW”training.Thisdocumentisavailablefordownloadathttp://home.hit.no/~hansha/.

FormoreinformationaboutLabVIEW,visitmyBlog:http://home.hit.no/~hansha/

iii

TableofContentsPreface......................................................................................................................................2

TableofContents.....................................................................................................................iii

1 IntroductiontoLabVIEW...................................................................................................1

1.1 Dataflowprogramming...............................................................................................1

1.2 Graphicalprogramming..............................................................................................2

1.3 Benefits.......................................................................................................................2

1.4 LabVIEWMathScriptRTModule................................................................................3

2 IntroductiontoLinearAlgebra..........................................................................................4

2.1.1 Transpose............................................................................................................4

2.1.2 Diagonal...............................................................................................................4

2.1.3 MatrixMultiplication...........................................................................................5

2.1.4 MatrixAddition...................................................................................................5

2.1.5 Determinant........................................................................................................5

2.1.6 InverseMatrices..................................................................................................5

2.2 Eigenvalues.................................................................................................................6

2.3 SolvingLinearEquations.............................................................................................6

2.4 LUfactorization...........................................................................................................7

2.5 TheSingularValueDecomposition(SVD)...................................................................7

3 LinearAlgebraPaletteinLabVIEW....................................................................................8

3.1 Vectors........................................................................................................................9

3.2 Matrices....................................................................................................................10

3.2.1 Transpose..........................................................................................................10

iv TableofContents

Tutorial:LinearAlgebrainLabVIEW

3.2.2 Diagonal.............................................................................................................11

3.2.3 MatrixMultiplication.........................................................................................12

3.2.4 MatrixAddition.................................................................................................13

3.2.5 Determinant......................................................................................................14

3.2.6 InverseMatrices................................................................................................15

3.3 Eigenvalues...............................................................................................................16

3.4 SolvingLinearEquations...........................................................................................16

3.5 LUfactorization.........................................................................................................17

3.6 TheSingularValueDecomposition(SVD).................................................................19

4 LabVIEWMathScriptRTModule.....................................................................................20

5 LabVIEWMathScript.......................................................................................................21

5.1 Help...........................................................................................................................22

5.2 Examples...................................................................................................................22

5.3 Usefulcommands.....................................................................................................25

5.4 FlowControl.............................................................................................................25

5.4.1 If-elseStatement...............................................................................................25

5.4.2 SwitchandCaseStatement...............................................................................26

5.4.3 Forloop.............................................................................................................26

5.4.4 Whileloop.........................................................................................................26

5.5 Plotting.....................................................................................................................28

6 LinearAlgebraExamplesusingMathScript.....................................................................30

6.1 Vectors......................................................................................................................30

6.2 Matrices....................................................................................................................31

6.2.1 Transpose..........................................................................................................31

6.2.2 Diagonal.............................................................................................................32

6.2.3 Triangular..........................................................................................................32

v TableofContents


6.2.4 MatrixMultiplication.........................................................................................33

6.2.5 MatrixAddition.................................................................................................33

6.2.6 Determinant......................................................................................................34

6.2.7 InverseMatrices................................................................................................35

6.3 Eigenvalues...............................................................................................................36

6.4 SolvingLinearEquations...........................................................................................36

6.5 LUfactorization.........................................................................................................37

6.6 TheSingularValueDecomposition(SVD).................................................................38

6.7 Commands................................................................................................................39

7 MathScriptNode.............................................................................................................40

7.1 TransferringMathScriptNodesbetweenComputers...............................................42

7.2 Examples...................................................................................................................42

7.3 Exercises...................................................................................................................46

8 WhatsNext?....................................................................................................................47

8.1 MyBlog.....................................................................................................................47

8.2 Training.....................................................................................................................47

8.3 MathScriptFunctions................................................................................................47

QuickReference......................................................................................................................49

1

1 IntroductiontoLabVIEWLabVIEW(shortforLaboratoryVirtualInstrumentationEngineeringWorkbench)isaplatformanddevelopmentenvironmentforavisualprogramminglanguagefromNationalInstruments.Thegraphicallanguageisnamed"G".OriginallyreleasedfortheAppleMacintoshin1986,LabVIEWiscommonlyusedfordataacquisition,instrumentcontrol,andindustrialautomationonavarietyofplatformsincludingMicrosoftWindows,variousflavorsofUNIX,Linux,andMacOSX.VisitNationalInstrumentsatwww.ni.com.

Thecodefileshavetheextension“.vi”,whichisanabbreviationfor“VirtualInstrument”.LabVIEWofferslotsofadditionalAdd-OnsandToolkits.

ThispaperispartofaserieswithLabVIEWpapers:

• IntroductiontoLabVIEW• LinearAlgebrainLabVIEW• DataAcquisitionandInstrumentControlinLabVIEW• ControlDesignandSimulationinLabVIEW• SignalProcessinginLabVIEW• DataloggingandSupervisoryControlinLabVIEW• SystemidentificationinLabVIEW• ModelbasedControlinLabVIEW• AdvancedTopicsinLabVIEW

Eachpapermaybeusedindependentlyofeachother.

1.1 DataflowprogrammingTheprogramminglanguageusedinLabVIEW,alsoreferredtoasG,isadataflowprogramminglanguage.Executionisdeterminedbythestructureofagraphicalblockdiagram(theLV-sourcecode)onwhichtheprogrammerconnectsdifferentfunction-nodesbydrawingwires.Thesewirespropagatevariablesandanynodecanexecuteassoonasallitsinputdatabecomeavailable.Sincethismightbethecaseformultiplenodessimultaneously,Gisinherentlycapableofparallelexecution.Multi-processingandmulti-threadinghardwareisautomaticallyexploitedbythebuilt-inscheduler,whichmultiplexesmultipleOSthreadsoverthenodesreadyforexecution.

2 IntroductiontoLabVIEW


1.2 GraphicalprogrammingLabVIEWtiesthecreationofuserinterfaces(calledfrontpanels)intothedevelopmentcycle.LabVIEWprograms/subroutinesarecalledvirtualinstruments(VIs).EachVIhasthreecomponents:ablockdiagram,afrontpanel,andaconnectorpanel.ThelastisusedtorepresenttheVIintheblockdiagramsofother,callingVIs.Controlsandindicatorsonthefrontpanelallowanoperatortoinputdataintoorextractdatafromarunningvirtualinstrument.However,thefrontpanelcanalsoserveasaprogrammaticinterface.Thusavirtualinstrumentcaneitherberunasaprogram,withthefrontpanelservingasauserinterface,or,whendroppedasanodeontotheblockdiagram,thefrontpaneldefinestheinputsandoutputsforthegivennodethroughtheconnectorpane.ThisimplieseachVIcanbeeasilytestedbeforebeingembeddedasasubroutineintoalargerprogram.

Thegraphicalapproachalsoallowsnon-programmerstobuildprogramssimplybydragginganddroppingvirtualrepresentationsoflabequipmentwithwhichtheyarealreadyfamiliar.TheLabVIEWprogrammingenvironment,withtheincludedexamplesandthedocumentation,makesitsimpletocreatesmallapplications.Thisisabenefitononeside,butthereisalsoacertaindangerofunderestimatingtheexpertiseneededforgoodquality"G"programming.Forcomplexalgorithmsorlarge-scalecode,itisimportantthattheprogrammerpossessanextensiveknowledgeofthespecialLabVIEWsyntaxandthetopologyofitsmemorymanagement.ThemostadvancedLabVIEWdevelopmentsystemsofferthepossibilityofbuildingstand-aloneapplications.Furthermore,itispossibletocreatedistributedapplications,whichcommunicatebyaclient/serverscheme,andarethereforeeasiertoimplementduetotheinherentlyparallelnatureofG-code.

1.3 BenefitsOnebenefitofLabVIEWoverotherdevelopmentenvironmentsistheextensivesupportforaccessinginstrumentationhardware.Driversandabstractionlayersformanydifferenttypesofinstrumentsandbusesareincludedorareavailableforinclusion.Thesepresentthemselvesasgraphicalnodes.Theabstractionlayersofferstandardsoftwareinterfacestocommunicatewithhardwaredevices.Theprovideddriverinterfacessaveprogramdevelopmenttime.ThesalespitchofNationalInstrumentsis,therefore,thatevenpeoplewithlimitedcodingexperiencecanwriteprogramsanddeploytestsolutionsinareducedtimeframewhencomparedtomoreconventionalorcompetingsystems.Anewhardwaredrivertopology(DAQmxBase),whichconsistsmainlyofG-codedcomponentswithonlyafewregistercallsthroughNIMeasurementHardwareDDK(DriverDevelopmentKit)functions,providesplatformindependenthardwareaccesstonumerousdataacquisition

3 IntroductiontoLabVIEW


andinstrumentationdevices.TheDAQmxBasedriverisavailableforLabVIEWonWindows,MacOSXandLinuxplatforms.

FormoreinformationaboutLabVIEW,visitmyBlog:http://home.hit.no/~hansha/

1.4 LabVIEWMathScriptRTModuleTheLabVIEWMathScriptRTModuleisanadd-onmoduletoLabVIEW.WithLabVIEWMathScriptRTModuleyoucan:

• Deployyourcustom.mfilestoNIreal-timehardware• ReusemanyofyourscriptscreatedwithTheMathWorks,Inc.MATLAB®softwareand

others• Developyour.mfileswithaninteractivecommand-lineinterface• EmbedyourscriptsintoyourLabVIEWapplicationsusingtheMathScriptNode

4

2 IntroductiontoLinearAlgebra

GivenamatrixA:

𝐴 =𝑎$$ ⋯ 𝑎$&⋮ ⋱ ⋮𝑎)$ ⋯ 𝑎)&

∈ 𝑅)-&

Example:

𝐴 = 0 1−2 −3

2.1.1 Transpose

TheTransposeofmatrixA:

𝐴3 = 0 −21 −3

2.1.2 Diagonal

TheDiagonalelementsofmatrixAisthevector

𝑑𝑖𝑎𝑔(𝐴) =

𝑎$$𝑎99⋮𝑎::

∈ 𝑅:;<=>(-,&)

TheDiagonalmatrixΛisgivenby:

Λ =

𝜆$ 0 ⋯ 00 𝜆9 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 ⋯ 𝜆)

∈ 𝑅)-)

GiventheIdentitymatrixI:



𝐼 =

1 0 ⋯ 00 1 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 ⋯ 1

∈ 𝑅)-&

2.1.3 MatrixMultiplication

Giventhematrices 𝐴 ∈ 𝑅)-& and 𝐵 ∈ 𝑅&-:,then

𝐶 = 𝐴𝐵 ∈ 𝑅)-:

2.1.4 MatrixAddition

Giventhematrices 𝐴 ∈ 𝑅)-& and 𝐵 ∈ 𝑅)-&,then

𝐶 = 𝐴 + 𝐵 ∈ 𝑅)-&

2.1.5 Determinant

Givenamatrix 𝐴 ∈ 𝑅)-),thentheDeterminantisgiven:

det 𝐴 = 𝐴

Givena2x2matrix

𝐴 =𝑎$$ 𝑎$9𝑎9$ 𝑎99 ∈ 𝑅9-9

Then

det 𝐴 = 𝐴 = 𝑎$$𝑎99 − 𝑎9$𝑎$9

Noticethat

det 𝐴𝐵 = det 𝐴 det 𝐵

and

det 𝐴3 = det(𝐴)

2.1.6 InverseMatrices

Theinverseofaquadraticmatrix 𝐴 ∈ 𝑅)-) isdefinedby:

𝐴I$



if

𝐴𝐴I$ = 𝐴I$𝐴 = 𝐼

Fora2x2matrixwehave:

𝐴 =𝑎$$ 𝑎$9𝑎9$ 𝑎99 ∈ 𝑅9-9

Theinverse 𝐴I$ isgivenby

𝐴I$ =1

det(𝐴)𝑎99 −𝑎$9−𝑎9$ 𝑎$$ ∈ 𝑅9-9

2.2 EigenvaluesGiven 𝐴 ∈ 𝑅)-),thentheEigenvaluesisdefinedas:

det 𝜆𝐼 − 𝐴 = 0

2.3 SolvingLinearEquationsGiventhelinearequation

𝐴𝑥 = 𝑏

withthesolution:

𝑥 = 𝐴I$𝑏

(AssumingthattheinverseofAexists)

Example:

Theequations

𝑥$ + 2𝑥9 = 53𝑥$ + 4𝑥9 = 6

maybewritten

𝐴𝑥 = 𝑏

1 23 4

𝑥$𝑥9 = 5

6

where



𝐴 = 1 23 4

𝑥 =𝑥$𝑥9

𝑏 = 56

2.4 LUfactorizationLUfactorizationof 𝐴 ∈ 𝑅)-& isgivenby

𝐴 = 𝐿𝑈

where

Lisalowertriangularmatrix

Uisauppertriangularmatrix

OrsometimesLUfactorizationof 𝐴 ∈ 𝑅)-& isgivenby

𝐴 = 𝐿𝑈 = 𝐿𝐷𝑈

where

Disadiagonalmatrix

2.5 TheSingularValueDecomposition(SVD)TheSingularvalueDecomposition(SVD)ofthematrix 𝐴 ∈ 𝑅)-& isgivenby

𝐴 = 𝑈𝑆𝑉3

where

Uisaorthogonalmatrix

Visaorthogonalmatrix

Sisadiagonalsingularmatrix

8

3 LinearAlgebraPaletteinLabVIEW

ForanIntroductiontoLabVIEW,seethetraining:“AnIntroductiontoLabVIEW”.YoumaydownloaditfrommyBlog:http://home.hit.no/~hansha/

UsetheLinearAlgebraPaletteinordertosolveLinearAlgebraproblemswiththeuseofGraphicalprogramming.

IntheMatrixSubPalettewehavethefollowingfunctions:



LabVIEWusesarraystorepresentsvectorsandmatrices.Avectorisrepresentedasaonedimensionalarray,whileamatrixisrepresentedasatwodimensionalarray.

IntheArray,Matrix&ClusterPaletteavailablefromtheFrontPanel,wehavethebasicarrayandmatrixcontrols:

3.1 VectorsGivenavectorx

𝑥 =

𝑥$𝑥9⋮𝑥)

∈ 𝑅)



Example:Vectors

𝑥 =123

ImplementingavectorintheFrontPanel:

3.2 MatricesGivenamatrixA:

𝐴 =𝑎$$ ⋯ 𝑎$&⋮ ⋱ ⋮𝑎)$ ⋯ 𝑎)&

∈ 𝑅)-&

Example:Matrices

𝐴 = 0 1−2 −3

FrontPanel:

3.2.1 Transpose


𝐴3 =𝑎$$ ⋯ 𝑎)$⋮ ⋱ ⋮

𝑎$& ⋯ 𝑎)&∈ 𝑅&-)

Example:Transpose



𝐴3 = 0 1−2 −3

3= 0 −2

1 −3

FrontPanel:

BlockDiagram:

3.2.2 Diagonal



𝑎$$𝑎99⋮𝑎::

∈ 𝑅:;<=>(-,&)

Example:Diagonal

FrontPanel:

BlockDiagram:




Λ =

𝜆$ 0 ⋯ 00 𝜆9 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 ⋯ 𝜆)

∈ 𝑅)-)


𝐼 =

1 0 ⋯ 00 1 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 ⋯ 1

∈ 𝑅)-&

Example:IdentityMatrix

FrontPanel:

BlockDiagram:



𝐶 = 𝐴𝐵 ∈ 𝑅)-:

where

𝑐UV = 𝑎UW𝑏WV

)

W;$

Example:MatrixMultiplication

FrontPanel:



BlockDiagram:

Note!

𝐴𝐵 ≠ 𝐵𝐴

𝐴 𝐵𝐶 = 𝐴𝐵 𝐶

𝐴 + 𝐵 𝐶 = 𝐴𝐶 + 𝐵𝐶

𝐶 𝐴 + 𝐵 = 𝐶𝐴 + 𝐶𝐵

→ProvethisinLabVIEW



𝐶 = 𝐴 + 𝐵 ∈ 𝑅)-&

Example:MatrixAddition

FrontPanel:

BlockDiagram:



Note!Thereisnospecialfunctionformatrixaddition,justusethestandardaddfunctionintheNumericpalette.

3.2.5 Determinant


det 𝐴 = 𝐴

Givena2x2matrix

𝐴 =𝑎$$ 𝑎$9𝑎9$ 𝑎99 ∈ 𝑅9-9

Then

det 𝐴 = 𝐴 = 𝑎$$𝑎99 − 𝑎9$𝑎$9

Example:Determinant

FrontPanel:

BlockDiagram:

Noticethat




and





𝐴I$

if



𝐴 =𝑎$$ 𝑎$9𝑎9$ 𝑎99 ∈ 𝑅9-9


𝐴I$ =1

det(𝐴)𝑎99 −𝑎$9−𝑎9$ 𝑎$$ ∈ 𝑅9-9

Example:Inverse

FrontPanel:

BlockDiagram:

Noticethat:







Example:Eigenvalues

FrontPanel:

BlockDiagram:


𝐴𝑥 = 𝑏

withthesolution:

𝑥 = 𝐴I$𝑏


Example:SolvingLinearEquations

Theequations

𝑥$ + 2𝑥9 = 53𝑥$ + 4𝑥9 = 6



maybewritten

𝐴𝑥 = 𝑏

1 23 4

𝑥$𝑥9 = 5

6

where

𝐴 = 1 23 4

𝑥 =𝑥$𝑥9

𝑏 = 56

Thesolutionis:

FrontPanel:

BlockDiagram:

Or:


𝐴 = 𝐿𝑈

where





Example:LUFactorization

FrontPanel:

BlockDiagram:



where

Disadiagonalmatrix


FrontPanel:

BlockDiagram:



3.6 TheSingularValueDecomposition(SVD)TheSingularvalueDecomposition(SVD)ofthematrix 𝐴 ∈ 𝑅)-& isgivenby


where




Example:SVDDecomposition

FrontPanel:

BlockDiagram:

20

4 LabVIEWMathScriptRTModule

YoucanworkwithLabVIEWMathScriptthrougheitheroftwointerfaces:the“LabVIEWMathScriptInteractiveWindow”orthe“MathScriptNode”.

YoucanworkwithLabVIEWMathScriptRTModulethroughbothinteractiveandprogrammaticinterfaces.Foraninteractiveinterfaceinwhichyoucanload,save,design,andexecuteyour.mfilescripts,youcanworkwiththe“MathScriptInteractiveWindow”.Todeployyour.mfilescriptsaspartofaLabVIEWapplicationandcombinegraphicalandtextualprogramming,youcanworkwiththe“MathScriptNode”.

TheLabVIEWMathScriptRTModulecomplementstraditionalLabVIEWgraphicalprogrammingforsuchtasksasalgorithmdevelopment,signalprocessing,andanalysis.TheLabVIEWMathScriptRTModulespeedsuptheseandothertasksbygivingusersasingleenvironmentinwhichtheycanchoosethemosteffectivesyntax,whethertextual,graphical,oracombinationofthetwo.Inaddition,youcanexploitthebestofLabVIEWandthousandsofpubliclyavailable.mfilescriptsfromtheweb,textbooks,oryourownexistingm-scriptapplications.LabVIEWMathScriptRTModuleisabletoprocessyourfilescreatedusingthecurrentMathScriptsyntaxand,forbackwardscompatibility,filescreatedusinglegacyMathScriptsyntaxes.LabVIEWMathScriptRTModulecanalsoprocesscertainofyourfilesutilizingothertext-basedsyntaxes,suchasfilesyoucreatedusingMATLABsoftware.BecausetheMathScriptRTengineisusedtoprocessscriptscontainedinaMathScriptWindowsorMathScriptNode,andbecausetheMathScriptRTenginedoesnotsupportallsyntaxes,notallexistingtext-basedscriptsaresupported.

LabVIEWMathScriptRTModulesupportsmostofthefunctionalityavailableinMATLAB,thesyntaxisalsosimilar.

Formoredetails,seehttp://zone.ni.com/devzone/cda/tut/p/id/3257

21

5 LabVIEWMathScriptRequires:MathScriptRTModule

The“LabVIEWMathScriptWindow”isaninteractiveinterfaceinwhichyoucanenter.mfilescriptcommandsandseeimmediateresults,variablesandcommandshistory.Thewindowincludesacommand-lineinterfacewhereyoucanentercommandsone-by-oneforquickcalculations,scriptdebuggingorlearning.Alternatively,youcanenterandexecutegroupsofcommandsthroughascripteditorwindow.

Asyouwork,avariabledisplayupdatestoshowthegraphical/textualresultsandahistorywindowtracksyourcommands.Thehistoryviewfacilitatesalgorithmdevelopmentbyallowingyoutousetheclipboardtoreuseyourpreviouslyexecutedcommands.

Youcanusethe“LabVIEWMathScriptWindow”toentercommandsoneattime.Youalsocanenterbatchscriptsinasimpletexteditorwindow,loadedfromatextfile,orimportedfromaseparatetexteditor.The“LabVIEWMathScriptWindow”providesimmediatefeedbackinavarietyofforms,suchasgraphsandtext.

Example:

22 LabVIEWMathScript


5.1 HelpYoumayalsotypehelpinyourcommandwindow

>>help

Ormorespecific,e.g.,

>>help plot

5.2 ExamplesIadviseyoutotestalltheexamplesinthistextinLabVIEWMathScriptinordertogetfamiliarwiththeprogramanditssyntax.Allexamplesinthetextareoutlinedinaframelikethis:

>> …



ThisiscommandsyoushouldwriteintheCommandWindow.

YoutypeallyourcommandsintheCommandWindow.Iwillusethesymbol“>>”toillustratethatthecommandsshouldbewrittenintheCommandWindow.

Example:Matrices

Definingthefollowingmatrix

𝐴 = 1 20 3

Thesyntaxisasfollows:

>> A = [1 2;0 3]

Or

>> A = [1,2;0,3]

Ifyou,foranexample,wanttofindtheanswerto

𝑎 + 𝑏,𝑤ℎ𝑒𝑟𝑒𝑎 = 4, 𝑏 = 3

>>a=4 >>b=3 >>a+b

MathScriptthenresponds:

ans = 7

MathScriptprovidesasimplewaytodefinesimplearraysusingthesyntax:“init:increment:terminator”.Forinstance:

>> array = 1:2:9 array = 1 3 5 7 9

definesavariablenamedarray(orassignsanewvaluetoanexistingvariablewiththenamearray)whichisanarrayconsistingofthevalues1,3,5,7,and9.Thatis,thearraystartsat1(theinitvalue),incrementswitheachstepfromthepreviousvalueby2(theincrementvalue),andstopsonceitreaches(ortoavoidexceeding)9(theterminatorvalue).

Theincrementvaluecanactuallybeleftoutofthissyntax(alongwithoneofthecolons),touseadefaultvalueof1.

>> ari = 1:5 ari =



1 2 3 4 5

assignstothevariablenamedarianarraywiththevalues1,2,3,4,and5,sincethedefaultvalueof1isusedastheincrementer.

Notethattheindexingisone-based,whichistheusualconventionformatricesinmathematics.Thisisatypicalforprogramminglanguages,whosearraysmoreoftenstartwithzero.

Matricescanbedefinedbyseparatingtheelementsofarowwithblankspaceorcommaandusingasemicolontoterminateeachrow.Thelistofelementsshouldbesurroundedbysquarebrackets:[].Parentheses:()areusedtoaccesselementsandsubarrays(theyarealsousedtodenoteafunctionargumentlist).

>> A = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1] A = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 >> A(2,3) ans = 11

Setsofindicescanbespecifiedbyexpressionssuchas"2:4",whichevaluatesto[2,3,4].Forexample,asubmatrixtakenfromrows2through4andcolumns3through4canbewrittenas:

>> A(2:4,3:4) ans = 11 8 7 12 14 1

Asquareidentitymatrixofsizencanbegeneratedusingthefunctioneye,andmatricesofanysizewithzerosoronescanbegeneratedwiththefunctionszerosandones,respectively.

>> eye(3) ans = 1 0 0 0 1 0 0 0 1 >> zeros(2,3) ans = 0 0 0 0 0 0 >> ones(2,3) ans = 1 1 1



1 1 1

5.3 UsefulcommandsHerearesomeusefulcommands:

Command Description

eye(x), eye(x,y) Identitymatrixoforderx

ones(x), ones(x,y) Amatrixwithonlyones

zeros(x), zeros(x,y) Amatrixwithonlyzeros

diag([x y z]) Diagonalmatrix

size(A) DimensionofmatrixA

A’ InverseofmatrixA

5.4 FlowControlThischapterexplainsthebasicconceptsofflowcontrolinMathScript.

Thetopicsareasfollows:

• If-elsestatement• Switchandcasestatement• Forloop• Whileloop

5.4.1 If-elseStatement

Theifstatementevaluatesalogicalexpressionandexecutesagroupofstatementswhentheexpressionistrue.Theoptionalelseifandelsekeywordsprovidefortheexecutionofalternategroupsofstatements.Anendkeyword,whichmatchestheif,terminatesthelastgroupofstatements.Thegroupsofstatementsaredelineatedbythefourkeywords—nobracesorbracketsareinvolved.

Example:If-ElseStatement

Testthefollowingcode:



n=5 if n > 2 M = eye(n) elseif n < 2 M = zeros(n) else M = ones(n) end

5.4.2 SwitchandCaseStatement

Theswitchstatementexecutesgroupsofstatementsbasedonthevalueofavariableorexpression.Thekeywordscaseandotherwisedelineatethegroups.Onlythefirstmatchingcaseisexecuted.Theremustalwaysbeanendtomatchtheswitch.

Example:SwitchandCaseStatement


n=2 switch(n) case 1 M = eye(n) case 2 M = zeros(n) case 3 M = ones(n) end

5.4.3 Forloop

Theforlooprepeatsagroupofstatementsafixed,predeterminednumberoftimes.Amatchingenddelineatesthestatements.

Example:ForLoop


m=5 for n = 1:m r(n) = rank(magic(n)); end r

5.4.4 Whileloop

Thewhilelooprepeatsagroupofstatementsanindefinitenumberoftimesundercontrolofalogicalcondition.Amatchingenddelineatesthestatements.



Example:WhileLoop


m=5; while m > 1 m = m - 1; zeros(m) end



5.5 PlottingThischapterexplainsthebasicconceptsofcreatingplotsinMathScript.

Topics:

• BasicPlotcommands

Example:Plotting

Functionplotcanbeusedtoproduceagraphfromtwovectorsxandy.Thecode:

x = 0:pi/100:2*pi; y = sin(x); plot(x,y)

producesthefollowingfigureofthesinefunction:

Example:Plotting

Three-dimensionalgraphicscanbeproducedusingthefunctionssurf,plot3ormesh.

[X,Y] = meshgrid(-10:0.25:10,-10:0.25:10); f = sinc(sqrt((X/pi).^2+(Y/pi).^2)); mesh(X,Y,f); axis([-10 10 -10 10 -0.3 1]) xlabel('{\bfx}') ylabel('{\bfy}') zlabel('{\bfsinc} ({\bfR})')



hidden off

Thiscodeproducesthefollowing3Dplot:

30 LinearAlgebraExamplesusingMathScript



Requires:MathScriptRTModule

Linearalgebraisabranchofmathematicsconcernedwiththestudyofmatrices,vectors,vectorspaces(alsocalledlinearspaces),linearmaps(alsocalledlineartransformations),andsystemsoflinearequations.

MathScriptarewellsuitedforLinearAlgebra.

6.1 VectorsGivenavectorx

𝑥 =

𝑥$𝑥9⋮𝑥)

∈ 𝑅)

Example:Vectors

Giventhefollowingvector

𝑥 =123

>> x=[1; 2; 3] x = 1 2 3

TheTransposeofvectorx:

𝑥3 = 𝑥$ 𝑥9 ⋯ 𝑥) ∈ 𝑅$-)

>> x' ans = 1 2 3

TheLengthofvectorx:



𝑥 = 𝑥3𝑥 = 𝑥$9 + 𝑥99 + ⋯+ 𝑥)9

Orthogonality:

𝑥3𝑦 = 0

6.2 MatricesGivenamatrixA:

𝐴 =𝑎$$ ⋯ 𝑎$&⋮ ⋱ ⋮𝑎)$ ⋯ 𝑎)&

∈ 𝑅)-&

Example:Matrices

Giventhefollowingmatrix:

𝐴 = 0 1−2 −3

>> A=[0 1;-2 -3] A = 0 1 -2 -3

6.2.1 Transpose


𝐴3 =𝑎$$ ⋯ 𝑎)$⋮ ⋱ ⋮

𝑎$& ⋯ 𝑎)&∈ 𝑅&-)

Example:Transpose

Giventhematrix:

𝐴3 = 0 1−2 −3

3= 0 −2

1 −3

>> A' ans = 0 -2 1 -3



6.2.2 Diagonal



𝑎$$𝑎99⋮𝑎::

∈ 𝑅:;<=>(-,&)

Example:Diagonal

FindthediagonalelementsofmatrixA:

>> diag(A) ans = 0 -3


Λ =

𝜆$ 0 ⋯ 00 𝜆9 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 ⋯ 𝜆)

∈ 𝑅)-)


𝐼 =

1 0 ⋯ 00 1 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 ⋯ 1

∈ 𝑅)-&

Example:IdentityMatrix

Getthe3x3Identitymatrix:

>> eye(3) ans = 1 0 0 0 1 0 0 0 1

6.2.3 Triangular

LowerTriangularmatrixL:

𝐿 =. 0 0⋮ ⋱ 0. ⋯ .



UpperTriangularmatrixU:

𝑈 =. ⋯ .0 ⋱ ⋮0 0 .



𝐶 = 𝐴𝐵 ∈ 𝑅)-:

where

𝑐UV = 𝑎UW𝑏WV

)

W;$

Example:MatrixMultiplication

Matrixmultiplication:

>> A=[0 1;-2 -3] A = 0 1 -2 -3 >> B=[1 0;3 -2] B = 1 0 3 -2 >> A*B ans = 3 -2 -11 6

Note!

𝐴𝐵 ≠ 𝐵𝐴

𝐴 𝐵𝐶 = 𝐴𝐵 𝐶

𝐴 + 𝐵 𝐶 = 𝐴𝐶 + 𝐵𝐶

𝐶 𝐴 + 𝐵 = 𝐶𝐴 + 𝐶𝐵



𝐶 = 𝐴 + 𝐵 ∈ 𝑅)-&



Example:MatrixAddition

Matrixaddition:

>> A=[0 1;-2 -3] >> B=[1 0;3 -2] >> A+B ans = 1 1 1 -5

6.2.6 Determinant


det 𝐴 = 𝐴

Givena2x2matrix

𝐴 =𝑎$$ 𝑎$9𝑎9$ 𝑎99 ∈ 𝑅9-9

Then

det 𝐴 = 𝐴 = 𝑎$$𝑎99 − 𝑎9$𝑎$9

Example:Determinant

Findthedeterminant:

A = 0 1 -2 -3 >> det(A) ans = 2

Noticethat


and




Example:Determinant

Determinants:

>> det(A*B) ans = -4 >> det(A)*det(B) ans = -4 >> det(A') ans = 2 >> det(A) ans = 2



𝐴I$

if



𝐴 =𝑎$$ 𝑎$9𝑎9$ 𝑎99 ∈ 𝑅9-9


𝐴I$ =1

det(𝐴)𝑎99 −𝑎$9−𝑎9$ 𝑎$$ ∈ 𝑅9-9

Example:InverseMatrices

Inversematrix:

A = 0 1 -2 -3 >> inv(A) ans = -1.5000 -0.5000 1.0000 0

Noticethat:




→ProvethisinMathScript



Example:Eigenvalues

FindtheEigenvalues:

A = 0 1 -2 -3 >> eig(A) ans = -1 -2


𝐴𝑥 = 𝑏

withthesolution:

𝑥 = 𝐴I$𝑏



Solvingthefollowingequation:

Theequations

𝑥$ + 2𝑥9 = 53𝑥$ + 4𝑥9 = 6

maybewritten

𝐴𝑥 = 𝑏



1 23 4

𝑥$𝑥9 = 5

6

where

𝐴 = 1 23 4

𝑥 =𝑥$𝑥9

𝑏 = 56

Thesolutionis:

A = 1 2 3 4 >> b=[5;6] b = 5 6 >> x=inv(A)*b x = -4.0000 4.5000

InMathScriptyoucouldalsowrite“x=A\b”,whichshouldgivethesameanswer.ThissyntaxcanalsobeusedwhentheinverseofAdon’texists.


Illegaloperation:

>> A=[1 2;3 4;7 8] >> x=inv(A)*b ??? Error using ==> inv Matrix must be square. >> x=A\b x = -3.5000 4.1786


𝐴 = 𝐿𝑈

where





TheMathScriptsyntaxis[L,U]=lu(A)


FindLandU:

>> A=[1 2;3 4] >> [L,U]=lu(A) L = 0.3333 1.0000 1.0000 0 U = 3.0000 4.0000 0 0.6667



where

Disadiagonalmatrix

TheMathScriptsyntaxis[L,U,P]=lu(A)


FindL,UandP:

>> A=[1 2;3 4] A = 1 2 3 4 >> [L,U,P]=lu(A) L = 1.0000 0 0.3333 1.0000 U = 3.0000 4.0000 0 0.6667 P = 0 1 1 0

6.6 TheSingularValueDecomposition(SVD)



TheSingularvalueDecomposition(SVD)ofthematrix 𝐴 ∈ 𝑅)-& isgivenby


where




Example:SVDDecomposition

FindS,VandD:

>> A=[1 2;3 4]; >> [U,S,V] = svd(A) U = -0.4046 -0.9145 -0.9145 0.4046 S = 5.4650 0 0 0.3660 V = -0.5760 0.8174 -0.8174 -0.5760

6.7 Commands

Command Description

[L,U]=lu(A)

[L,U,P]=lu(A)LUFactorization

[U,S,V] = svd(A) SingularValueDecomposition(SVD)

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7 MathScriptNodeThe“MathScriptNode”offersanintuitivemeansofcombininggraphicalandtextualcodewithinLabVIEW.Thefigurebelowshowsthe“MathScriptNode”ontheblockdiagram,representedbythebluerectangle.Using“MathScriptNodes”,youcanenter.mfilescripttextdirectlyorimportitfromatextfile.

YoucandefinenamedinputsandoutputsontheMathScriptNodebordertospecifythedatatotransferbetweenthegraphicalLabVIEWenvironmentandthetextualMathScriptcode.

Youcanassociate.mfilescriptvariableswithLabVIEWgraphicalprogramming,bywiringNodeinputsandoutputs.Thenyoucantransferdatabetween.mfilescriptswithyourgraphicalLabVIEWprogramming.Thetextual.mfilescriptscannowaccessfeaturesfromtraditionalLabVIEWgraphicalprogramming.

TheMathScriptNodeisavailablefromLabVIEWfromtheFunctionsPalette:Mathematics→Scripts&Formulas

41 MathScriptNode


IfyouclickCtrl+HyougethelpabouttheMathScriptNode:

Click“Detailedhelp”inordertogetmoreinformationabouttheMathScriptNode.

UsetheNIExampleFinderinordertofindexamples:

42 MathScriptNode


7.1 TransferringMathScriptNodesbetweenComputers

IfascriptinaMathScriptNodecallsauser-definedfunction,LabVIEWusesthedefaultsearchpathlisttolinkthefunctioncalltothespecified.mfile.AfteryouconfigurethedefaultsearchpathlistandsavetheVIthatcontainstheMathScriptNode,youdonotneedtoreconfiguretheMathScriptsearchpathlistwhenyouopentheVIonadifferentcomputerbecauseLabVIEWlooksforthe.mfileinthedirectorywherethe.mfilewaslocatedwhenyoulastsavedtheVI.However,youmustmaintainthesamerelativepathbetweentheVIandthe.mfile.

7.2 ExamplesExample:UsingtheMathScriptNode

HereisanexampleofhowyouusetheMathScriptNode.Ontheleftborderyouconnectinputvariablestothescript,ontherightborderyouhaveoutputvariables.Right-clickontheborderandselect“AddInput”or“AddOutput”.

43 MathScriptNode


Example:CallingaWindowsDLL:

Example:Usingm-filesintheMathScriptNode:

UsetheLabVIEWMathScripttocreateam-filescript(oryoumayuseMATLABtocreatethesamescript):

44 MathScriptNode


Right-clickontheborderoftheMathScriptNodeandselect“Import”,andthenselectthem-fileyouwanttoimportintotheNode.

45 MathScriptNode


Right-clickontherightborderandselect“AddOutput”.Thenright-clickontheoutputvariableandselect“CreateIndicator”.

BlockDiagram:

Theresultisasfollows(clicktheRunbutton):

Ifyou,e.g.,addthefollowingcommandintheMathScriptNode:plot(x),thefollowingwindowappears:

46 MathScriptNode


7.3 ExercisesUsetheMathScriptNodeandtestthesameexamplesyoudidinthepreviouschapter(Chapter6-“LinearAlgebraExamplesusingMathScript”)

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8 WhatsNext?

8.1 MyBlogFormoreinformationaboutLabVIEW,visitmyBlog:http://home.hit.no/~hansha/

8.2 TrainingThisTrainingisapartofaserieswithotherTrainingKitsIhavemade,suchas:

• IntroductiontoLabVIEW• DataAcquisitioninLabVIEW• ControlandSimulationinLabVIEW• LabVIEWMathScript• LinearAlgebrainLabVIEW• DataloggingandSupervisoryControlinLabVIEW• WirelessDataAcquisitioninLabVIEW• IntermediateTopicsinLabVIEW• AdvancedTopicsinLabVIEW

TheseTrainingKitsareavailablefordownloadfrommyblog:http://home.hit.no/~hansha

8.3 MathScriptFunctionsIntheHelpsystemthereisdetailedinformationaboutalltheMathScriptfunctionsavailable.InadditiontotheMathScriptRTModulefunctions,differentadd-onmodulesandtoolkitsinstallsadditionalfunctions.TheLabVIEWControlDesignandSimulationModuleandLabVIEWDigitalFilterDesignToolkitinstallsalotsofadditionalfunctions.

48 MathScriptNode


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QuickReference

50 QuickReference


Hans-PetterHalvorsen,M.Sc.

E-mail:[email protected]

Blog:http://home.hit.no/~hansha/

UniversityCollegeofSoutheastNorway

www.usn.no

Tutorial: Linear Algebra In LabVIEW

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