Download Tutorial: Linear Algebra In LabVIEW

UniversityCollegeofSoutheastNorway
LinearAlgebrainLabVIEW
Hans-PetterHalvorsen,2016-10-31
http://home.hit.no/~hansha
Preface
ThisdocumentexplainsthebasicconceptsofLinearAlgebraandhowyoumayuseLabVIEW
forcalculationoftheseproblems.
YoushouldhavesomebasicknowledgeaboutLabVIEW,e.g.,the“AnIntroductionto
LabVIEW”training.Thisdocumentisavailablefordownloadathttp://home.hit.no/~hansha/.
FormoreinformationaboutLabVIEW,visitmyBlog:http://home.hit.no/~hansha/
TableofContents
Preface......................................................................................................................................2
TableofContents.....................................................................................................................iii
1
2
3
IntroductiontoLabVIEW...................................................................................................1
1.1
Dataflowprogramming...............................................................................................1
1.2
Graphicalprogramming..............................................................................................2
1.3
Benefits.......................................................................................................................2
1.4
LabVIEWMathScriptRTModule................................................................................3
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
LinearAlgebraPaletteinLabVIEW....................................................................................8
3.1
Vectors........................................................................................................................9
3.2
Matrices....................................................................................................................10
3.2.1
Transpose..........................................................................................................10
iii
iv
TableofContents
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
6
Plotting.....................................................................................................................28
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
Tutorial:LinearAlgebrainLabVIEW
v
7
8
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
MathScriptNode.............................................................................................................40
7.1
TransferringMathScriptNodesbetweenComputers...............................................42
7.2
Examples...................................................................................................................42
7.3
Exercises...................................................................................................................46
WhatsNext?....................................................................................................................47
8.1
MyBlog.....................................................................................................................47
8.2
Training.....................................................................................................................47
8.3
MathScriptFunctions................................................................................................47
QuickReference......................................................................................................................49
Tutorial:LinearAlgebrainLabVIEW
1 IntroductiontoLabVIEW
LabVIEW(shortforLaboratoryVirtualInstrumentationEngineeringWorkbench)isa
platformanddevelopmentenvironmentforavisualprogramminglanguagefromNational
Instruments.Thegraphicallanguageisnamed"G".OriginallyreleasedfortheApple
Macintoshin1986,LabVIEWiscommonlyusedfordataacquisition,instrumentcontrol,and
industrialautomationonavarietyofplatformsincludingMicrosoftWindows,variousflavors
ofUNIX,Linux,andMacOSX.VisitNationalInstrumentsatwww.ni.com.
Thecodefileshavetheextension“.vi”,whichisanabbreviationfor“VirtualInstrument”.
LabVIEWofferslotsofadditionalAdd-OnsandToolkits.
ThispaperispartofaserieswithLabVIEWpapers:
•
•
•
•
•
•
•
•
•
IntroductiontoLabVIEW
LinearAlgebrainLabVIEW
DataAcquisitionandInstrumentControlinLabVIEW
ControlDesignandSimulationinLabVIEW
SignalProcessinginLabVIEW
DataloggingandSupervisoryControlinLabVIEW
SystemidentificationinLabVIEW
ModelbasedControlinLabVIEW
AdvancedTopicsinLabVIEW
Eachpapermaybeusedindependentlyofeachother.
1.1 Dataflowprogramming
TheprogramminglanguageusedinLabVIEW,alsoreferredtoasG,isadataflow
programminglanguage.Executionisdeterminedbythestructureofagraphicalblock
diagram(theLV-sourcecode)onwhichtheprogrammerconnectsdifferentfunction-nodes
bydrawingwires.Thesewirespropagatevariablesandanynodecanexecuteassoonasall
itsinputdatabecomeavailable.Sincethismightbethecaseformultiplenodes
simultaneously,Gisinherentlycapableofparallelexecution.Multi-processingandmultithreadinghardwareisautomaticallyexploitedbythebuilt-inscheduler,whichmultiplexes
multipleOSthreadsoverthenodesreadyforexecution.
1
2
IntroductiontoLabVIEW 1.2 Graphicalprogramming
LabVIEWtiesthecreationofuserinterfaces(calledfrontpanels)intothedevelopmentcycle.
LabVIEWprograms/subroutinesarecalledvirtualinstruments(VIs).EachVIhasthree
components:ablockdiagram,afrontpanel,andaconnectorpanel.Thelastisusedto
representtheVIintheblockdiagramsofother,callingVIs.Controlsandindicatorsonthe
frontpanelallowanoperatortoinputdataintoorextractdatafromarunningvirtual
instrument.However,thefrontpanelcanalsoserveasaprogrammaticinterface.Thusa
virtualinstrumentcaneitherberunasaprogram,withthefrontpanelservingasauser
interface,or,whendroppedasanodeontotheblockdiagram,thefrontpaneldefinesthe
inputsandoutputsforthegivennodethroughtheconnectorpane.ThisimplieseachVIcan
beeasilytestedbeforebeingembeddedasasubroutineintoalargerprogram.
Thegraphicalapproachalsoallowsnon-programmerstobuildprogramssimplybydragging
anddroppingvirtualrepresentationsoflabequipmentwithwhichtheyarealreadyfamiliar.
TheLabVIEWprogrammingenvironment,withtheincludedexamplesandthe
documentation,makesitsimpletocreatesmallapplications.Thisisabenefitononeside,
butthereisalsoacertaindangerofunderestimatingtheexpertiseneededforgoodquality
"G"programming.Forcomplexalgorithmsorlarge-scalecode,itisimportantthatthe
programmerpossessanextensiveknowledgeofthespecialLabVIEWsyntaxandthe
topologyofitsmemorymanagement.ThemostadvancedLabVIEWdevelopmentsystems
offerthepossibilityofbuildingstand-aloneapplications.Furthermore,itispossibletocreate
distributedapplications,whichcommunicatebyaclient/serverscheme,andaretherefore
easiertoimplementduetotheinherentlyparallelnatureofG-code.
1.3 Benefits
OnebenefitofLabVIEWoverotherdevelopmentenvironmentsistheextensivesupportfor
accessinginstrumentationhardware.Driversandabstractionlayersformanydifferenttypes
ofinstrumentsandbusesareincludedorareavailableforinclusion.Thesepresent
themselvesasgraphicalnodes.Theabstractionlayersofferstandardsoftwareinterfacesto
communicatewithhardwaredevices.Theprovideddriverinterfacessaveprogram
developmenttime.ThesalespitchofNationalInstrumentsis,therefore,thatevenpeople
withlimitedcodingexperiencecanwriteprogramsanddeploytestsolutionsinareduced
timeframewhencomparedtomoreconventionalorcompetingsystems.Anewhardware
drivertopology(DAQmxBase),whichconsistsmainlyofG-codedcomponentswithonlya
fewregistercallsthroughNIMeasurementHardwareDDK(DriverDevelopmentKit)
functions,providesplatformindependenthardwareaccesstonumerousdataacquisition
Tutorial:LinearAlgebrainLabVIEW
3
IntroductiontoLabVIEW andinstrumentationdevices.TheDAQmxBasedriverisavailableforLabVIEWonWindows,
MacOSXandLinuxplatforms.
FormoreinformationaboutLabVIEW,visitmyBlog:http://home.hit.no/~hansha/
1.4 LabVIEWMathScriptRTModule
TheLabVIEWMathScriptRTModuleisanadd-onmoduletoLabVIEW.WithLabVIEW
MathScriptRTModuleyoucan:
•
•
•
•
Deployyourcustom.mfilestoNIreal-timehardware
ReusemanyofyourscriptscreatedwithTheMathWorks,Inc.MATLAB®softwareand
others
Developyour.mfileswithaninteractivecommand-lineinterface
EmbedyourscriptsintoyourLabVIEWapplicationsusingtheMathScriptNode
Tutorial:LinearAlgebrainLabVIEW
2 IntroductiontoLinear
Algebra
GivenamatrixA:
𝑎$$
𝐴= ⋮
𝑎)$
⋯
⋱
⋯
𝑎$&
⋮ ∈ 𝑅)-& 𝑎)&
Example:
𝐴=
2.1.1
0
1
−2 −3
Transpose
TheTransposeofmatrixA:
𝐴3 =
2.1.2
0 −2
1 −3
Diagonal
TheDiagonalelementsofmatrixAisthevector
𝑎$$
𝑎99
𝑑𝑖𝑎𝑔(𝐴) = ⋮ ∈ 𝑅:;<=>(-,&) 𝑎::
TheDiagonalmatrixΛisgivenby:
𝜆$
0
Λ=
⋮
0
0
𝜆9
⋮
0
⋯
⋯
⋱
⋯
GiventheIdentitymatrixI:
4
0
0
∈ 𝑅)-) ⋮
𝜆)
5
1 0 ⋯
0 1 ⋯
𝐼=
⋮ ⋮ ⋱
0 0 ⋯
2.1.3
IntroductiontoLinearAlgebra 0
0
∈ 𝑅)-& ⋮
1
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$$
𝐴= 𝑎
9$
𝑎$9
9-9
𝑎99 ∈ 𝑅 Then
det 𝐴 = 𝐴 = 𝑎$$ 𝑎99 − 𝑎9$ 𝑎$9 Noticethat
det 𝐴𝐵 = det 𝐴 det 𝐵 and
det 𝐴3 = det(𝐴)
2.1.6
InverseMatrices
Theinverseofaquadraticmatrix 𝐴 ∈ 𝑅)-) isdefinedby:
𝐴I$ Tutorial:LinearAlgebrainLabVIEW
6
IntroductiontoLinearAlgebra if
𝐴𝐴I$ = 𝐴I$ 𝐴 = 𝐼
Fora2x2matrixwehave:
𝑎$$
𝐴= 𝑎
9$
𝑎$9
9-9
𝑎99 ∈ 𝑅 Theinverse 𝐴I$ isgivenby
𝐴I$ =
1
𝑎99
det(𝐴) −𝑎9$
−𝑎$9
9-9
𝑎$$ ∈ 𝑅 2.2 Eigenvalues
Given 𝐴 ∈ 𝑅)-) ,thentheEigenvaluesisdefinedas:
det 𝜆𝐼 − 𝐴 = 0
2.3 SolvingLinearEquations
Giventhelinearequation
𝐴𝑥 = 𝑏
withthesolution:
𝑥 = 𝐴I$ 𝑏
(AssumingthattheinverseofAexists)
Example:
Theequations
𝑥$ + 2𝑥9 = 5
3𝑥$ + 4𝑥9 = 6
maybewritten
𝐴𝑥 = 𝑏
1 2 𝑥$
5
=
3 4 𝑥9
6
where
Tutorial:LinearAlgebrainLabVIEW
7
𝐴=
IntroductiontoLinearAlgebra 1 2
3 4
𝑥$
𝑥= 𝑥 9
𝑏=
5
6
2.4 LUfactorization
LUfactorizationof 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 = 𝑈𝑆𝑉 3 where
Uisaorthogonalmatrix
Visaorthogonalmatrix
Sisadiagonalsingularmatrix
Tutorial:LinearAlgebrainLabVIEW
3 LinearAlgebraPalettein
LabVIEW
ForanIntroductiontoLabVIEW,seethetraining:“AnIntroductiontoLabVIEW”.Youmay
downloaditfrommyBlog:http://home.hit.no/~hansha/
UsetheLinearAlgebraPaletteinordertosolveLinearAlgebraproblemswiththeuseof
Graphicalprogramming.
IntheMatrixSubPalettewehavethefollowingfunctions:
8
9
LinearAlgebraPaletteinLabVIEW LabVIEWusesarraystorepresentsvectorsandmatrices.Avectorisrepresentedasaone
dimensionalarray,whileamatrixisrepresentedasatwodimensionalarray.
IntheArray,Matrix&ClusterPaletteavailablefromtheFrontPanel,wehavethebasicarray
andmatrixcontrols:
3.1 Vectors
Givenavectorx
𝑥$
𝑥9
𝑥 = ⋮ ∈ 𝑅) 𝑥)
Tutorial:LinearAlgebrainLabVIEW
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LinearAlgebraPaletteinLabVIEW Example:Vectors
1
𝑥= 2
3
ImplementingavectorintheFrontPanel:
3.2 Matrices
GivenamatrixA:
𝑎$$
𝐴= ⋮
𝑎)$
⋯
⋱
⋯
𝑎$&
⋮ ∈ 𝑅)-& 𝑎)&
Example:Matrices
𝐴=
0
1
−2 −3
FrontPanel:
3.2.1
Transpose
TheTransposeofmatrixA:
𝑎$$
𝐴 = ⋮
𝑎$&
3
⋯
⋱
⋯
𝑎)$
⋮ ∈ 𝑅&-) 𝑎)&
Example:Transpose
Tutorial:LinearAlgebrainLabVIEW
11
𝐴3 =
0
1
−2 −3
LinearAlgebraPaletteinLabVIEW 3
=
0 −2
1 −3
FrontPanel:
BlockDiagram:
3.2.2
Diagonal
TheDiagonalelementsofmatrixAisthevector
𝑎$$
𝑎99
𝑑𝑖𝑎𝑔(𝐴) = ⋮ ∈ 𝑅:;<=>(-,&) 𝑎::
Example:Diagonal
FrontPanel:
BlockDiagram:
TheDiagonalmatrixΛisgivenby:
Tutorial:LinearAlgebrainLabVIEW
12
𝜆$
0
Λ=
⋮
0
0
𝜆9
⋮
0
⋯
⋯
⋱
⋯
LinearAlgebraPaletteinLabVIEW 0
0
∈ 𝑅)-) ⋮
𝜆)
GiventheIdentitymatrixI:
1 0 ⋯
0 1 ⋯
𝐼=
⋮ ⋮ ⋱
0 0 ⋯
0
0
∈ 𝑅)-& ⋮
1
Example:IdentityMatrix
FrontPanel:
BlockDiagram:
3.2.3
MatrixMultiplication
Giventhematrices 𝐴 ∈ 𝑅)-& and 𝐵 ∈ 𝑅&-: ,then
𝐶 = 𝐴𝐵 ∈ 𝑅)-: where
)
𝑎UW 𝑏WV 𝑐UV =
W;$
Example:MatrixMultiplication
FrontPanel:
Tutorial:LinearAlgebrainLabVIEW
13
LinearAlgebraPaletteinLabVIEW BlockDiagram:
Note!
𝐴𝐵 ≠ 𝐵𝐴
𝐴 𝐵𝐶 = 𝐴𝐵 𝐶
𝐴 + 𝐵 𝐶 = 𝐴𝐶 + 𝐵𝐶
𝐶 𝐴 + 𝐵 = 𝐶𝐴 + 𝐶𝐵
→ProvethisinLabVIEW
3.2.4
MatrixAddition
Giventhematrices 𝐴 ∈ 𝑅)-& and 𝐵 ∈ 𝑅)-& ,then
𝐶 = 𝐴 + 𝐵 ∈ 𝑅)-& Example:MatrixAddition
FrontPanel:
BlockDiagram:
Tutorial:LinearAlgebrainLabVIEW
14
LinearAlgebraPaletteinLabVIEW Note!Thereisnospecialfunctionformatrixaddition,justusethestandardaddfunctionin
theNumericpalette.
3.2.5
Determinant
Givenamatrix 𝐴 ∈ 𝑅)-) ,thentheDeterminantisgiven:
det 𝐴 = 𝐴 Givena2x2matrix
𝑎$$
𝐴= 𝑎
9$
𝑎$9
9-9
𝑎99 ∈ 𝑅 Then
det 𝐴 = 𝐴 = 𝑎$$ 𝑎99 − 𝑎9$ 𝑎$9 Example:Determinant
FrontPanel:
BlockDiagram:
Noticethat
Tutorial:LinearAlgebrainLabVIEW
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LinearAlgebraPaletteinLabVIEW det 𝐴𝐵 = det 𝐴 det 𝐵 and
det 𝐴3 = det(𝐴)
→ProvethisinLabVIEW
3.2.6
InverseMatrices
Theinverseofaquadraticmatrix 𝐴 ∈ 𝑅)-) isdefinedby:
𝐴I$ if
𝐴𝐴I$ = 𝐴I$ 𝐴 = 𝐼
Fora2x2matrixwehave:
𝑎$$
𝐴= 𝑎
9$
𝑎$9
9-9
𝑎99 ∈ 𝑅 Theinverse 𝐴I$ isgivenby
𝐴I$ =
1
𝑎99
det(𝐴) −𝑎9$
−𝑎$9
9-9
𝑎$$ ∈ 𝑅 Example:Inverse
FrontPanel:
BlockDiagram:
Noticethat:
𝐴𝐴I$ = 𝐴I$ 𝐴 = 𝐼
Tutorial:LinearAlgebrainLabVIEW
16
LinearAlgebraPaletteinLabVIEW →ProvethisinLabVIEW
3.3 Eigenvalues
Given 𝐴 ∈ 𝑅)-) ,thentheEigenvaluesisdefinedas:
det 𝜆𝐼 − 𝐴 = 0
Example:Eigenvalues
FrontPanel:
BlockDiagram:
3.4 SolvingLinearEquations
Giventhelinearequation
𝐴𝑥 = 𝑏
withthesolution:
𝑥 = 𝐴I$ 𝑏
(AssumingthattheinverseofAexists)
Example:SolvingLinearEquations
Theequations
𝑥$ + 2𝑥9 = 5
3𝑥$ + 4𝑥9 = 6
Tutorial:LinearAlgebrainLabVIEW
17
LinearAlgebraPaletteinLabVIEW maybewritten
𝐴𝑥 = 𝑏
1 2 𝑥$
5
=
3 4 𝑥9
6
where
𝐴=
1 2
3 4
𝑥$
𝑥= 𝑥 9
𝑏=
5
6
Thesolutionis:
FrontPanel:
BlockDiagram:
Or:
3.5 LUfactorization
LUfactorizationof 𝐴 ∈ 𝑅)-& isgivenby
𝐴 = 𝐿𝑈
where
Tutorial:LinearAlgebrainLabVIEW
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LinearAlgebraPaletteinLabVIEW Lisalowertriangularmatrix
Uisauppertriangularmatrix
Example:LUFactorization
FrontPanel:
BlockDiagram:
OrsometimesLUfactorizationof 𝐴 ∈ 𝑅)-& isgivenby
𝐴 = 𝐿𝑈 = 𝐿𝐷𝑈
where
Disadiagonalmatrix
Example:LUFactorization
FrontPanel:
BlockDiagram:
Tutorial:LinearAlgebrainLabVIEW
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LinearAlgebraPaletteinLabVIEW 3.6 TheSingularValueDecomposition(SVD)
TheSingularvalueDecomposition(SVD)ofthematrix 𝐴 ∈ 𝑅)-& isgivenby
𝐴 = 𝑈𝑆𝑉 3 where
Uisaorthogonalmatrix
Visaorthogonalmatrix
Sisadiagonalsingularmatrix
Example:SVDDecomposition
FrontPanel:
BlockDiagram:
Tutorial:LinearAlgebrainLabVIEW
4 LabVIEWMathScriptRT
Module
YoucanworkwithLabVIEWMathScriptthrougheitheroftwointerfaces:the“LabVIEW
MathScriptInteractiveWindow”orthe“MathScriptNode”.
YoucanworkwithLabVIEWMathScriptRTModulethroughbothinteractiveand
programmaticinterfaces.Foraninteractiveinterfaceinwhichyoucanload,save,design,
andexecuteyour.mfilescripts,youcanworkwiththe“MathScriptInteractiveWindow”.To
deployyour.mfilescriptsaspartofaLabVIEWapplicationandcombinegraphicaland
textualprogramming,youcanworkwiththe“MathScriptNode”.
TheLabVIEWMathScriptRTModulecomplementstraditionalLabVIEWgraphical
programmingforsuchtasksasalgorithmdevelopment,signalprocessing,andanalysis.The
LabVIEWMathScriptRTModulespeedsuptheseandothertasksbygivingusersasingle
environmentinwhichtheycanchoosethemosteffectivesyntax,whethertextual,graphical,
oracombinationofthetwo.Inaddition,youcanexploitthebestofLabVIEWandthousands
ofpubliclyavailable.mfilescriptsfromtheweb,textbooks,oryourownexistingm-script
applications.LabVIEWMathScriptRTModuleisabletoprocessyourfilescreatedusingthe
currentMathScriptsyntaxand,forbackwardscompatibility,filescreatedusinglegacy
MathScriptsyntaxes.LabVIEWMathScriptRTModulecanalsoprocesscertainofyourfiles
utilizingothertext-basedsyntaxes,suchasfilesyoucreatedusingMATLABsoftware.
BecausetheMathScriptRTengineisusedtoprocessscriptscontainedinaMathScript
WindowsorMathScriptNode,andbecausetheMathScriptRTenginedoesnotsupportall
syntaxes,notallexistingtext-basedscriptsaresupported.
LabVIEWMathScriptRTModulesupportsmostofthefunctionalityavailableinMATLAB,the
syntaxisalsosimilar.
Formoredetails,seehttp://zone.ni.com/devzone/cda/tut/p/id/3257
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5 LabVIEWMathScript
Requires:MathScriptRTModule
The“LabVIEWMathScriptWindow”isaninteractiveinterfaceinwhichyoucanenter.mfile
scriptcommandsandseeimmediateresults,variablesandcommandshistory.Thewindow
includesacommand-lineinterfacewhereyoucanentercommandsone-by-oneforquick
calculations,scriptdebuggingorlearning.Alternatively,youcanenterandexecutegroupsof
commandsthroughascripteditorwindow.
Asyouwork,avariabledisplayupdatestoshowthegraphical/textualresultsandahistory
windowtracksyourcommands.Thehistoryviewfacilitatesalgorithmdevelopmentby
allowingyoutousetheclipboardtoreuseyourpreviouslyexecutedcommands.
Youcanusethe“LabVIEWMathScriptWindow”toentercommandsoneattime.Youalso
canenterbatchscriptsinasimpletexteditorwindow,loadedfromatextfile,orimported
fromaseparatetexteditor.The“LabVIEWMathScriptWindow”providesimmediate
feedbackinavarietyofforms,suchasgraphsandtext.
Example:
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LabVIEWMathScript 5.1 Help
Youmayalsotypehelpinyourcommandwindow
>>help
Ormorespecific,e.g.,
>>help plot
5.2 Examples
IadviseyoutotestalltheexamplesinthistextinLabVIEWMathScriptinordertoget
familiarwiththeprogramanditssyntax.Allexamplesinthetextareoutlinedinaframelike
this:
>>
…
Tutorial:LinearAlgebrainLabVIEW
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LabVIEWMathScript ThisiscommandsyoushouldwriteintheCommandWindow.
YoutypeallyourcommandsintheCommandWindow.Iwillusethesymbol“>>”to
illustratethatthecommandsshouldbewrittenintheCommandWindow.
Example:Matrices
Definingthefollowingmatrix
𝐴=
1 2
0 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(orassignsanewvaluetoanexistingvariablewiththename
array)whichisanarrayconsistingofthevalues1,3,5,7,and9.Thatis,thearraystartsat1
(theinitvalue),incrementswitheachstepfromthepreviousvalueby2(theincrement
value),andstopsonceitreaches(ortoavoidexceeding)9(theterminatorvalue).
Theincrementvaluecanactuallybeleftoutofthissyntax(alongwithoneofthecolons),to
useadefaultvalueof1.
>> ari = 1:5
ari =
Tutorial:LinearAlgebrainLabVIEW
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LabVIEWMathScript 1 2 3 4 5
assignstothevariablenamedarianarraywiththevalues1,2,3,4,and5,sincethedefault
valueof1isusedastheincrementer.
Notethattheindexingisone-based,whichistheusualconventionformatricesin
mathematics.Thisisatypicalforprogramminglanguages,whosearraysmoreoftenstart
withzero.
Matricescanbedefinedbyseparatingtheelementsofarowwithblankspaceorcommaand
usingasemicolontoterminateeachrow.Thelistofelementsshouldbesurroundedby
squarebrackets:[].Parentheses:()areusedtoaccesselementsandsubarrays(theyarealso
usedtodenoteafunctionargumentlist).
>> 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].For
example,asubmatrixtakenfromrows2through4andcolumns3through4canbewritten
as:
>> A(2:4,3:4)
ans =
11 8
7 12
14 1
Asquareidentitymatrixofsizencanbegeneratedusingthefunctioneye,andmatricesof
anysizewithzerosoronescanbegeneratedwiththefunctionszerosandones,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
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25
LabVIEWMathScript 1 1 1
5.3 Usefulcommands
Herearesomeusefulcommands:
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 FlowControl
ThischapterexplainsthebasicconceptsofflowcontrolinMathScript.
Thetopicsareasfollows:
•
•
•
•
If-elsestatement
Switchandcasestatement
Forloop
Whileloop
5.4.1
If-elseStatement
Theifstatementevaluatesalogicalexpressionandexecutesagroupofstatementswhenthe
expressionistrue.Theoptionalelseifandelsekeywordsprovidefortheexecutionof
alternategroupsofstatements.Anendkeyword,whichmatchestheif,terminatesthelast
groupofstatements.Thegroupsofstatementsaredelineatedbythefourkeywords—no
bracesorbracketsareinvolved.
Example:If-ElseStatement
Testthefollowingcode:
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26
LabVIEWMathScript n=5
if n > 2
M = eye(n)
elseif n < 2
M = zeros(n)
else
M = ones(n)
end
5.4.2
SwitchandCaseStatement
Theswitchstatementexecutesgroupsofstatementsbasedonthevalueofavariableor
expression.Thekeywordscaseandotherwisedelineatethegroups.Onlythefirstmatching
caseisexecuted.Theremustalwaysbeanendtomatchtheswitch.
Example:SwitchandCaseStatement
Testthefollowingcode:
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.A
matchingenddelineatesthestatements.
Example:ForLoop
Testthefollowingcode:
m=5
for n = 1:m
r(n) = rank(magic(n));
end
r
5.4.4
Whileloop
Thewhilelooprepeatsagroupofstatementsanindefinitenumberoftimesundercontrolof
alogicalcondition.Amatchingenddelineatesthestatements.
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27
Example:WhileLoop
Testthefollowingcode:
m=5;
while m > 1
m = m - 1;
zeros(m)
end
Tutorial:LinearAlgebrainLabVIEW
LabVIEWMathScript 28
LabVIEWMathScript 5.5 Plotting
ThischapterexplainsthebasicconceptsofcreatingplotsinMathScript.
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})')
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LabVIEWMathScript hidden off
Thiscodeproducesthefollowing3Dplot:
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30
LinearAlgebraExamplesusingMathScript 6 LinearAlgebraExamples
usingMathScript
Requires:MathScriptRTModule
Linearalgebraisabranchofmathematicsconcernedwiththestudyofmatrices,vectors,
vectorspaces(alsocalledlinearspaces),linearmaps(alsocalledlineartransformations),and
systemsoflinearequations.
MathScriptarewellsuitedforLinearAlgebra.
6.1 Vectors
Givenavectorx
𝑥$
𝑥9
𝑥 = ⋮ ∈ 𝑅) 𝑥)
Example:Vectors
Giventhefollowingvector
1
𝑥= 2
3
>> x=[1; 2; 3]
x =
1
2
3
TheTransposeofvectorx:
𝑥 3 = 𝑥$
𝑥9
⋯
𝑥) ∈ 𝑅$-) >> x'
ans =
1
2
3
TheLengthofvectorx:
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31
𝑥 =
LinearAlgebraExamplesusingMathScript 𝑥$9 + 𝑥99 + ⋯ + 𝑥)9 𝑥3𝑥 =
Orthogonality:
𝑥 3 𝑦 = 0
6.2 Matrices
GivenamatrixA:
𝑎$$
𝐴= ⋮
𝑎)$
⋯
⋱
⋯
𝑎$&
⋮ ∈ 𝑅)-& 𝑎)&
Example:Matrices
Giventhefollowingmatrix:
𝐴=
0
1
−2 −3
>> A=[0 1;-2 -3]
A =
0
1
-2
-3
6.2.1
Transpose
TheTransposeofmatrixA:
𝑎$$
𝐴 = ⋮
𝑎$&
3
⋯
⋱
⋯
𝑎)$
⋮ ∈ 𝑅&-) 𝑎)&
Example:Transpose
Giventhematrix:
𝐴3 =
0
1
−2 −3
3
=
0 −2
1 −3
>> A'
ans =
0
1
-2
-3
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32
6.2.2
LinearAlgebraExamplesusingMathScript Diagonal
TheDiagonalelementsofmatrixAisthevector
𝑎$$
𝑎99
𝑑𝑖𝑎𝑔(𝐴) = ⋮ ∈ 𝑅:;<=>(-,&) 𝑎::
Example:Diagonal
FindthediagonalelementsofmatrixA:
>> diag(A)
ans =
0
-3
TheDiagonalmatrixΛisgivenby:
𝜆$
0
Λ=
⋮
0
0
𝜆9
⋮
0
⋯
⋯
⋱
⋯
0
0
∈ 𝑅)-) ⋮
𝜆)
GiventheIdentitymatrixI:
1 0 ⋯
0 1 ⋯
𝐼=
⋮ ⋮ ⋱
0 0 ⋯
0
0
∈ 𝑅)-& ⋮
1
Example:IdentityMatrix
Getthe3x3Identitymatrix:
>> eye(3)
ans =
1
0
0
6.2.3
0
1
0
0
0
1
Triangular
LowerTriangularmatrixL:
. 0
𝐿= ⋮ ⋱
. ⋯
0
0
.
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33
LinearAlgebraExamplesusingMathScript UpperTriangularmatrixU:
. ⋯
𝑈= 0 ⋱
0 0
6.2.4
.
⋮
.
MatrixMultiplication
Giventhematrices 𝐴 ∈ 𝑅)-& and 𝐵 ∈ 𝑅&-: ,then
𝐶 = 𝐴𝐵 ∈ 𝑅)-: where
)
𝑎UW 𝑏WV 𝑐UV =
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
6.2.5
MatrixAddition
Giventhematrices 𝐴 ∈ 𝑅)-& and 𝐵 ∈ 𝑅)-& ,then
𝐶 = 𝐴 + 𝐵 ∈ 𝑅)-& Tutorial:LinearAlgebrainLabVIEW
34
LinearAlgebraExamplesusingMathScript Example:MatrixAddition
Matrixaddition:
>> A=[0 1;-2 -3]
>> B=[1 0;3 -2]
>> A+B
ans =
1
1
1
-5
6.2.6
Determinant
Givenamatrix 𝐴 ∈ 𝑅)-) ,thentheDeterminantisgiven:
det 𝐴 = 𝐴 Givena2x2matrix
𝑎$$
𝐴= 𝑎
9$
𝑎$9
9-9
𝑎99 ∈ 𝑅 Then
det 𝐴 = 𝐴 = 𝑎$$ 𝑎99 − 𝑎9$ 𝑎$9 Example:Determinant
Findthedeterminant:
A =
0
1
-2
-3
>> det(A)
ans =
2
Noticethat
det 𝐴𝐵 = det 𝐴 det 𝐵 and
det 𝐴3 = det(𝐴)
Tutorial:LinearAlgebrainLabVIEW
35
LinearAlgebraExamplesusingMathScript Example:Determinant
Determinants:
>> det(A*B)
ans =
-4
>> det(A)*det(B)
ans =
-4
>> det(A')
ans =
2
>> det(A)
ans =
2
6.2.7
InverseMatrices
Theinverseofaquadraticmatrix 𝐴 ∈ 𝑅)-) isdefinedby:
𝐴I$ if
𝐴𝐴I$ = 𝐴I$ 𝐴 = 𝐼
Fora2x2matrixwehave:
𝑎$$
𝐴= 𝑎
9$
𝑎$9
9-9
𝑎99 ∈ 𝑅 Theinverse 𝐴I$ isgivenby
𝐴I$ =
1
𝑎99
−𝑎
9$
det(𝐴)
−𝑎$9
9-9
𝑎$$ ∈ 𝑅 Example:InverseMatrices
Inversematrix:
A =
0
1
-2
-3
>> inv(A)
ans =
-1.5000
-0.5000
1.0000
0
Noticethat:
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36
LinearAlgebraExamplesusingMathScript 𝐴𝐴I$ = 𝐴I$ 𝐴 = 𝐼
→ProvethisinMathScript
6.3 Eigenvalues
Given 𝐴 ∈ 𝑅)-) ,thentheEigenvaluesisdefinedas:
det 𝜆𝐼 − 𝐴 = 0
Example:Eigenvalues
FindtheEigenvalues:
A =
0
1
-2
-3
>> eig(A)
ans =
-1
-2
6.4 SolvingLinearEquations
Giventhelinearequation
𝐴𝑥 = 𝑏
withthesolution:
𝑥 = 𝐴I$ 𝑏
(AssumingthattheinverseofAexists)
Example:SolvingLinearEquations
Solvingthefollowingequation:
Theequations
𝑥$ + 2𝑥9 = 5
3𝑥$ + 4𝑥9 = 6
maybewritten
𝐴𝑥 = 𝑏
Tutorial:LinearAlgebrainLabVIEW
37
LinearAlgebraExamplesusingMathScript 1 2 𝑥$
5
=
𝑥
3 4 9
6
where
𝐴=
1 2
3 4
𝑥$
𝑥= 𝑥 9
𝑏=
5
6
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.Thissyntax
canalsobeusedwhentheinverseofAdon’texists.
Example:SolvingLinearEquations
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
6.5 LUfactorization
LUfactorizationof 𝐴 ∈ 𝑅)-& isgivenby
𝐴 = 𝐿𝑈
where
Tutorial:LinearAlgebrainLabVIEW
38
LinearAlgebraExamplesusingMathScript Lisalowertriangularmatrix
Uisauppertriangularmatrix
TheMathScriptsyntaxis[L,U]=lu(A)
Example:LUFactorization
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
OrsometimesLUfactorizationof 𝐴 ∈ 𝑅)-& isgivenby
𝐴 = 𝐿𝑈 = 𝐿𝐷𝑈
where
Disadiagonalmatrix
TheMathScriptsyntaxis[L,U,P]=lu(A)
Example:LUFactorization
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)
Tutorial:LinearAlgebrainLabVIEW
39
LinearAlgebraExamplesusingMathScript TheSingularvalueDecomposition(SVD)ofthematrix 𝐴 ∈ 𝑅)-& isgivenby
𝐴 = 𝑈𝑆𝑉 3 where
Uisaorthogonalmatrix
Visaorthogonalmatrix
Sisadiagonalsingularmatrix
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
[L,U]=lu(A)
Description
LUFactorization
[L,U,P]=lu(A)
[U,S,V] = svd(A)
SingularValueDecomposition(SVD)
Tutorial:LinearAlgebrainLabVIEW
7 MathScriptNode
The“MathScriptNode”offersanintuitivemeansofcombininggraphicalandtextualcode
withinLabVIEW.Thefigurebelowshowsthe“MathScriptNode”ontheblockdiagram,
representedbythebluerectangle.Using“MathScriptNodes”,youcanenter.mfilescript
textdirectlyorimportitfromatextfile.
YoucandefinenamedinputsandoutputsontheMathScriptNodebordertospecifythedata
totransferbetweenthegraphicalLabVIEWenvironmentandthetextualMathScriptcode.
Youcanassociate.mfilescriptvariableswithLabVIEWgraphicalprogramming,bywiring
Nodeinputsandoutputs.Thenyoucantransferdatabetween.mfilescriptswithyour
graphicalLabVIEWprogramming.Thetextual.mfilescriptscannowaccessfeaturesfrom
traditionalLabVIEWgraphicalprogramming.
TheMathScriptNodeisavailablefromLabVIEWfromtheFunctionsPalette:Mathematics→
Scripts&Formulas
40
41
MathScriptNode IfyouclickCtrl+HyougethelpabouttheMathScriptNode:
Click“Detailedhelp”inordertogetmoreinformationabouttheMathScriptNode.
UsetheNIExampleFinderinordertofindexamples:
Tutorial:LinearAlgebrainLabVIEW
42
MathScriptNode 7.1 TransferringMathScriptNodesbetween
Computers
IfascriptinaMathScriptNodecallsauser-definedfunction,LabVIEWusesthedefault
searchpathlisttolinkthefunctioncalltothespecified.mfile.Afteryouconfigurethe
defaultsearchpathlistandsavetheVIthatcontainstheMathScriptNode,youdonotneed
toreconfiguretheMathScriptsearchpathlistwhenyouopentheVIonadifferentcomputer
becauseLabVIEWlooksforthe.mfileinthedirectorywherethe.mfilewaslocatedwhen
youlastsavedtheVI.However,youmustmaintainthesamerelativepathbetweentheVI
andthe.mfile.
7.2 Examples
Example:UsingtheMathScriptNode
HereisanexampleofhowyouusetheMathScriptNode.Ontheleftborderyouconnect
inputvariablestothescript,ontherightborderyouhaveoutputvariables.Right-clickonthe
borderandselect“AddInput”or“AddOutput”.
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43
MathScriptNode Example:CallingaWindowsDLL:
Example:Usingm-filesintheMathScriptNode:
UsetheLabVIEWMathScripttocreateam-filescript(oryoumayuseMATLABtocreatethe
samescript):
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44
MathScriptNode Right-clickontheborderoftheMathScriptNodeandselect“Import”,andthenselectthemfileyouwanttoimportintotheNode.
Tutorial:LinearAlgebrainLabVIEW
45
MathScriptNode Right-clickontherightborderandselect“AddOutput”.Thenright-clickontheoutput
variableandselect“CreateIndicator”.
BlockDiagram:
Theresultisasfollows(clicktheRunbutton):
Ifyou,e.g.,addthefollowingcommandintheMathScriptNode:plot(x),thefollowingwindow
appears:
Tutorial:LinearAlgebrainLabVIEW
46
MathScriptNode 7.3 Exercises
UsetheMathScriptNodeandtestthesameexamplesyoudidinthepreviouschapter
(Chapter6-“LinearAlgebraExamplesusingMathScript”)
Tutorial:LinearAlgebrainLabVIEW
8 WhatsNext?
8.1 MyBlog
FormoreinformationaboutLabVIEW,visitmyBlog:http://home.hit.no/~hansha/
8.2 Training
ThisTrainingisapartofaserieswithotherTrainingKitsIhavemade,suchas:
•
•
•
•
•
•
•
•
•
IntroductiontoLabVIEW
DataAcquisitioninLabVIEW
ControlandSimulationinLabVIEW
LabVIEWMathScript
LinearAlgebrainLabVIEW
DataloggingandSupervisoryControlinLabVIEW
WirelessDataAcquisitioninLabVIEW
IntermediateTopicsinLabVIEW
AdvancedTopicsinLabVIEW
TheseTrainingKitsareavailablefordownloadfrommyblog:http://home.hit.no/~hansha
8.3 MathScriptFunctions
IntheHelpsystemthereisdetailedinformationaboutalltheMathScriptfunctionsavailable.
InadditiontotheMathScriptRTModulefunctions,differentadd-onmodulesandtoolkits
installsadditionalfunctions.TheLabVIEWControlDesignandSimulationModuleand
LabVIEWDigitalFilterDesignToolkitinstallsalotsofadditionalfunctions. 47
48
MathScriptNode Tutorial:LinearAlgebrainLabVIEW
QuickReference
49
50
Tutorial:LinearAlgebrainLabVIEW
QuickReference Hans-PetterHalvorsen,M.Sc.
E-mail:[email protected]
Blog:http://home.hit.no/~hansha/
UniversityCollegeofSoutheastNorway
www.usn.no