Block Krylov subspace methods for large-scale matrix computations in control

JournalofTaibahUniversityforScience9(2015)116–120

Availableonlineatwww.sciencedirect.com

ScienceDirect

Block

Krylov

subspace

methods

for

large-scale

matrix

computations

in

control

A.H.

Refahi

Sheikhani

,

S.

Kordrostami

DepartmentofMathematics,FacultyofMathematicalSciences,IslamicAzadUniversity,LahijanBranch,Lahijan,Iran

Availableonline4August2014

Abstract

InthispaperweshowhowtoimprovetheapproximatesolutionofthelargeSylvesterequationobtainedbyanarbitrarymethod. SuchproblemsappearinmanyareasofcontroltheorysuchasthecomputationofHankelsingularvalues,modelreductionalgorithms andothers.Moreover,weproposeanewmethodbasedonrefinementprocessandweightedblockArnoldialgorithmforsolving largeSylvestermatrixequation.Thenumericaltestsreporttheeffectivenessofthesemethods.

©2014TaibahUniversity.ProductionandhostingbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/3.0/).

MSC: 65F30;65F50

Keywords:Matrixequations;Sylvester;Arnoldi;Refinement

1. Introduction

AnimportantclassoflinearmatrixequationsistheSylvesterequation

XA+BX=C. (1.1)

SincethefundamentalworkofSylvesteronthestabilityofthemotion,thesematrixequationshavebeenwidelyused

instabilitytheoryofdifferentialequationsandplayanimportantroleincontrolandcommunicationstheory[1–5].For

smallproblems,directmethodsforsolvingtheSylvesterequationareattractive.Thestandardmethodsarebasedon

theHessenberg–Schurdecompositiontotransformtheoriginalequationintoaformthatcanbeeasilysolved.Iterative

methodsfor solvinglargeSylvester matrixequationshavebeen developedduringthepastyears[6–8].Aclass of

classicalmethodsknownastheKrylovsubspacemethods,thatincludetheblockArnoldiandweightedblockArnoldi,

etc.havebeenfoundtobe suitablefor sparsematrixcomputations.In thispaper,weextendtheideatoproposea

newprojectionmethodforsolving(1.1)basedonweightedblockKrylovsubspacemethod.Thepaperisorganizedas

∗_{Correspondingauthor.Tel.:+989113475547.}

E-mailaddresses:[email protected](A.H.RefahiSheikhani),[email protected](S.Kordrostami). PeerreviewunderresponsibilityofTaibahUniversity.

http://dx.doi.org/10.1016/j.jtusci.2014.07.006

1658-3655©2014TaibahUniversity.ProductionandhostingbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/3.0/).

follows.InSection2,weintroducethemainminimalresidualalgorithmforiterativelysolvingtheSylvestermatrix

equation(1.1).SeveralnumericalexamplespresentedinSection3.Finally,theconclusionisgiveninthelastsection.

2. BlockrefinementArnoldimethod

InthissectionweproposetoshowthattheobtainedapproximatesolutionoftheSylvesterequationbyanymethod

canbeimproved,inotherwordstheaccuracycanbeincreased.Ifthisideaisapplicablethenwehavefoundaniterative

method for solving of the Sylvester equation. Thereforelet the basis V_m=[v1,...,vm] andWm=[w1,...,wm]

constructedbytheblockArnoldiprocess,thuswehave

VT

mVm=Im WmTWm=Im

ThesquareblockHessenbergmatricesHmand ˆHm(m=r*lwhererandlarethedimensionsofblocks)whosenonzero

entriesarethescalarshijand ˆhij,constructedbytheblockArnoldiprocesscanbeexpressedas

Hm=VmTATVm Hˆm=WmTBWm

LetX(0)beaninitialapproximatesolutionoftheSylvesterequationXA+BX=CwhereA,B,C,X(0),∈Rn×n.

Alsointroducetheresidualmatrix

R(0)_=C−(X(0)_A+BX(0)₎

AndletYm∈Rm×mbethesolutionoftheSylvesterequation:

YmHmT + ˆHmYm=WmTR(0)Vm (2.1)

Ifset

X(1) _=X(0)_+W

mYmVmT (2.2)

thenthecorrespondingresidualR(1)=C−(X(1)A+BX(1))satisfies:

R(1)_=C−((X(0)_+W mYmVmT)A+B(X(0)+WmYmVmT)) =R(0)_−W mYmVmTA−BWmYmVmT =R(0)_−W mYmHmTVmT −WmHˆmYmVmT =R(0)_−W m(Y_mH_mT+ ˆH_mY_m)V_mT

SinceYmisthesolutionof(2.1)wehave:

R(1)_=R(0)_−W

mWmTR(0)VmVmT =0

AccordingtotheaboveresultswecandevelopaniterativemethodforthesolvingoftheSylvesterequationwhenthe

matricesA,BandCarelargeandsparse.Fordoingthisideaifwechoosem<n,theninsteadofsolvingXA+BX=C

wecansolve(2.1).Inotherwordsinthismethod,firstwetransformtheinitialSylvesterequationtoanotherSylvester

equationwithlessdimensions,thenineachiterationstepsolvethismatrixequationandextendtheobtainedsolution

tothesolutionofinitialequationby(2.2).Thealgorithmisasfollows:

Algorithm1(BlockrefinementArnoldimethod(BRA)). ChooseaninitialsolutionX(0),andatoleranceandselect

twonumbersrandlfordimensionsofblockandsetm=r*l(m<n).

Fork=0...untilConvergence.

R(k)=C−(X(k)A+BX(k)).

ConstructtheorthonormalbasisVmandWm∈Rn×mbytheblockArnoldiprocess

Hm=VmTATVm Hˆm=WmTBWm.

SolvethereducedSylvesterequation

YmHmT+ ˆHmYm=WmTR(k)Vm.

X(k+1)_=X(k)_+WmYmVT m.

ifX(k+1)_X(−Xk)_(k)≤Stop

Table1

ImplementationofiterativeBRAmethodtosolvetheSylvesterequationwithdifferentvaluesofm.

m r l V_mTATV_m−H_m W_mTBW_m− ˆH_m Iteration Time 4 2 2 8.63E−014 4.05E−014 278 5.83 8 2 4 1.57E−013 5.49E−014 156 4.68 10 2 5 2.81E−013 1.98E−014 95 3.56 20 2 10 2.84E−013 1.77E−013 58 2.68 30 2 15 3.05E−013 4.46E−014 36 1.74 40 2 20 9.77E−013 8.36E−014 21 0.7811 50 2 25 3.17E−012 4.68E−013 2 0.0918 Table2

ImplementationofnewIterativemethodsandHessenberg–SchurmethodforsolvingtheSylvesterequation.

n Hessenberg–Schurmethod WeightedKrylovmethod BRAmethod cond(B)

Error Time Error Time Error Time

200 1.16E−010 0.6881 2.50E−012 0.4753 2.38E−014 0.2612 8.53E+003

400 3.89E−007 6.312 4.22E−008 4.642 6.15E−014 3.134 3.17E+004

600 0.0011 28.89 0.0033 65.76 6.95E−014 21.39 7.63E+005

800 8.931 85.74 13.01 174.32 8.37E−014 58.26 2.13E+007

1000 27.35 201.14 48.19 322.11 1.84E−013 121.53 1.50E+008

3. Numericalexamples

Inthissection,wepresentsomenumericalexamplestoillustratetheeffectivenessofalgorithmsdescribedinthis

paperforlargeandsparseSylvesterequations.AllnumericaltestsareperformedinMATLABsoftwareonaPCwith

2.20GHzwithmainmemory2GB.

Example3.1. ConsidertheSylvesterequation XA+BX=Cwithn=100.WeapplytheIterativeblockrefinement

Arnoldimethodforsolvingthismatrixequationandtake=10−6.InTable1,wereporttheresultsfordifferentvalues

ofm. A= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 10 1.2 .42 .8 2.3 .8 0 ... ... 0 1.8 10 1.2 .42 .8 2.3 .8 0 . .. 0 1.6 1.8 . .. . .. . .. . .. . .. . .. 0 ... 1.64 1.6 . .. . .. . .. . .. . .. . .. .8 0 1.3 1.64 . .. . .. . .. . .. . .. . .. 2.3 .8 1.61 1.3 . .. . .. . .. . .. . .. . .. .8 2.3 0 1.61 . .. . .. . .. . .. . .. . .. .42 .8 .. . 0 . .. . .. . .. . .. . .. . .. 1.2 .42 .. . ... 0 1.61 1.3 1.64 1.6 1.8 10 1.2 0 0 ... 0 1.61 1.3 1.64 1.6 1.8 10 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ n×n

B= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 10 2.1 .38 .7 1.5 .4 0 ... ... 0 1.21 10 2.1 .38 .7 1.5 .4 0 . .. 0 1.9 1.21 . .. ... ... ... ... . .. 0 ... .64 1.9 . .. ... ... ... ... . .. .4 0 1.9 .64 . .. ... ... ... ... . .. 1.5 .4 .87 1.9 . .. ... ... ... ... . .. .7 1.5 0 .87 . .. ... ... ... ... . .. .38 .7 .. . 0 . .. ... ... ... ... . .. 2.1 .38 .. . ... 0 .87 1.9 .64 1.9 1.21 10 2.1 0 0 ... 0 .87 1.9 .64 1.9 1.21 10 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ n×n C= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ .1 2.21 1.4 1.5 .13 2.62 0 ... ... 0 1.3 .1 2.21 1.4 1.5 .13 2.62 0 . .. 0 2.6 1.3 . .. . .. ... . .. . .. . .. 0 ... 1.7 2.6 . .. . .. ... . .. . .. . .. 2.62 0 2.3 1.7 . .. . .. ... . .. . .. . .. .13 2.62 2.6 2.3 . .. . .. ... . .. . .. . .. 1.5 .13 0 2.6 . .. . .. ... . .. . .. . .. 1.4 1.5 .. . 0 . .. . .. ... . .. . .. . .. 2.21 1.4 .. . ... 0 2.6 2.3 1.7 2.6 1.3 .1 2.21 0 0 ... 0 2.6 2.3 1.7 2.6 1.3 .1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ n×n

Example3.2. NowconsiderAandBandCarethesamematricesthatusedinExample3.1.WeapplytwoIterative

methodsandHessenberg–SchurmethodtosolvetheSylvesterequationwhenthedimensionofthematricesarelarge.

ResultsareshowninTable2.

4. Commentsandconclusion

Inthispaper,weintroducedanewmethodforcomputinglow-rankapproximatesolutionstolargeSylvestermatrix

equations.MoreoverRefinementprocesspresentedinSection2hasthecapabilityofimprovingtheresultsobtained

byanarbitrarymethod.ForexampleinthispaperweapplytherefinementprocesswithHessenberg–Schurmethod.

Theexperimentsillustratetheadvantagesofthemethodforlargesparsematrices.

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