Numerical Methods for Eigenvalue Problems

217 Pages·2012·0.95 MB·English

by  Steffen Borm

De Gruyter Graduate Lectures
Börm / Mehl · Numerical Methods for Eigenvalue Problems

Steffen Börm
Christian Mehl
Numerical Methods for
Eigenvalue Problems
De Gruyter
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Preface
Eigenvaluesandeigenvectorsofmatricesandlinearoperatorsplayanimportantrole
whensolvingproblemsfromstructuralmechanics,andelectrodynamics, e.g.,byde-
scribing the resonance frequencies of systems, when investigating the long-term be-
haviour of stochastic processes, e.g., by describing invariant probability measures,
andasatoolforsolvingmoregeneralmathematicalproblems,e.g.,bydiagonalizing
ordinarydifferentialequationsorsystemsfromcontroltheory.
Thisbookpresentsanumberofthemostimportantnumericalmethodsforfinding
eigenvaluesandeigenvectorsofmatrices. Itisbasedonlecturenotesofashortcourse
forthird-yearstudentsinmathematics,butitshouldalsobeaccessibletostudentsof
physicsorengineeringsciences.
Wediscussthecentral ideasunderlying thedifferent algorithms andintroduce the
theoretical concepts required to analyze their behaviour. Our goal is to present an
easily accessible introduction to the field, including rigorous proofs of all important
results,butnotacompleteoverviewofthevastbodyofresearch.
Foranin-depthcoverageofthetheoryofeigenvalueproblems,wecanrecommend
thefollowingmonographs:
(cid:2) J.H.Wilkinson,“TheAlgebraicEigenvalueProblem”[52]
(cid:2) B.N.Parlett,“TheSymmetricEigenvalueProblem”[33]
(cid:2) G.H.GolubandC.F.VanLoan,“MatrixComputations”[18]
(cid:2) G.W.Stewart,“MatrixAlgorithms”[43,44]
(cid:2) D.S.Watkins,“Thematrixeigenvalueproblem”[49]
We owe a great debt of gratitude to their authors, since this book would not exist
withouttheirwork.
Thebookisintendedasthebasisforashortcourse(onesemesterortrimester)for
third-orfourth-yearundergraduatestudents. Wehaveorganizedthematerialmostly
inshortsectionsthatshouldeachfitonesessionofacourse. Somechaptersandsec-
(cid:2)
tions are marked by an asterisk . These contain additional results that we consider
optional,e.g.,rathertechnicalproofsofgeneralresultsoralgorithmsforspecialprob-
lems. With one exception, the results of these optional sections are not required for
the remainder of the book. The one exception is the optional Section 2.7 on non-
unitary transformations, which lays the groundwork for the optional Section 4.8 on
theconvergenceofthepoweriterationforgeneralmatrices.
vi Preface
Inordertokeepthepresentationself-contained, anumberofimportantresultsare
provenonlyforspecialcases, e.g., forself-adjointmatricesoraspectrumconsisting
onlyofsimpleeigenvalues. Forthegeneralcase,wewouldliketoreferthereaderto
themonographsmentionedabove.
Deviatingfromthepracticeofcollectingfundamentalresultsinaseparatechapter,
weintroducesomeoftheseresultswhentheyarerequired. AnexampleistheBauer–
FiketheoremgivenasProposition3.11inSection3.4onerrorestimatesfortheJacobi
iterationinsteadofinaseparatechapteronperturbationtheory. Whilethisapproachis
certainlynotadequateforareferencework,wehopethatitimprovestheaccessibility
oflecturenoteslikethisbookthatareintendedtobetaughtinsequence.
WewouldliketothankDanielKressnerforhisvaluablecontributionstothisbook.
Kiel,December2011 Berlin,December2011
SteffenBörm ChristianMehl
Contents
Preface v
1 Introduction 1
1.1 Example: Structuralmechanics ............................... 1
1.2 Example: Stochasticprocesses ............................... 4
1.3 Example: Systemsoflineardifferentialequations ................ 5
2 Existenceandpropertiesofeigenvaluesandeigenvectors 8
2.1 Eigenvaluesandeigenvectors ................................ 8
2.2 Characteristicpolynomials .................................. 12
2.3 Similaritytransformations ................................... 15
2.4 SomepropertiesofHilbertspaces ............................. 19
2.5 Invariantsubspaces ........................................ 24
2.6 Schurdecomposition ....................................... 26
(cid:2)
2.7 Non-unitarytransformations ................................ 33
3 Jacobiiteration 39
3.1 Iteratedsimilaritytransformations............................. 39
3.2 Two-dimensionalSchurdecomposition......................... 40
3.3 Onestepoftheiteration..................................... 43
3.4 Errorestimates ............................................ 47
(cid:2)
3.5 Quadraticconvergence .................................... 53
4 Powermethods 61
4.1 Poweriteration ............................................ 61
4.2 Rayleighquotient.......................................... 66
4.3 Residual-basederrorcontrol ................................. 70
4.4 Inverseiteration ........................................... 73
4.5 Rayleighiteration ......................................... 77
4.6 Convergencetoinvariantsubspace ............................ 79
viii Contents
4.7 Simultaneousiteration ...................................... 83
(cid:2)
4.8 Convergenceforgeneralmatrices ............................ 91
5 QRiteration 100
5.1 BasicQRstep ............................................ 100
5.2 Hessenbergform .......................................... 104
5.3 Shifting ................................................. 113
5.4 Deflation ................................................ 116
5.5 Implicititeration .......................................... 118
(cid:2)
5.6 Multiple-shiftstrategies ................................... 126
(cid:2)
6 Bisectionmethods 132
6.1 Sturmchains ............................................. 134
6.2 Gershgorindiscs .......................................... 141
7 Krylovsubspacemethodsforlargesparseeigenvalueproblems 145
7.1 Sparsematricesandprojectionmethods ........................ 145
7.2 Krylovsubspaces .......................................... 149
7.3 Gram–Schmidtprocess ..................................... 152
7.4 Arnoldiiteration .......................................... 159
7.5 SymmetricLanczosalgorithm ................................ 164
7.6 Chebyshevpolynomials ..................................... 165
7.7 ConvergenceofKrylovsubspacemethods ...................... 172
(cid:2)
8 Generalizedandpolynomialeigenvalueproblems 182
8.1 Polynomialeigenvalueproblemsandlinearization ................ 182
8.2 Matrixpencils ............................................ 185
8.3 DeflatingsubspacesandthegeneralizedSchurdecomposition....... 189
8.4 Hessenberg-triangularform .................................. 192
8.5 Deflation ................................................ 196
8.6 TheQZstep .............................................. 198
Bibliography 203
Index 206
Chapter 1
Introduction
Eigenvalueproblemsplayanimportantroleinanumberoffieldsofnumericalmath-
ematics: instructuralmechanicsandelectrodynamics,eigenvaluescorrespondtores-
onance frequencies of systems, i.e., to frequencies to which these systems respond
particularlywell(orbadly,dependingonthecontext). Whenstudyingstochasticpro-
cesses,invariantprobabilitymeasurescorrespondtoeigenvectorsfortheeigenvalue1,
andfindingthemyieldsadescriptionofthelong-termbehaviourofthecorresponding
process.
This book gives an introduction to the basic theory of eigenvalue problems and
focusesonimportantalgorithmsforfindingeigenvaluesandeigenvectors.
1.1 Example: Structural mechanics
Before we consider abstract eigenvalue problems, we turn our attention to some ap-
plicationsthatleadnaturallytoeigenvalueproblems.
Thefirstapplicationistheinvestigationofresonancefrequencies. Asanexample,
we consider the oscillations of a string of unit length. We represent the string as a
function
uWR(cid:3)Œ0;1(cid:2)!R; .t;x/7!u.t;x/;
where u.t;x/ denotes the deflection of the string at time t and position x (cf. Fig-
ure1.1).
Theoscillationsarethendescribedbythewaveequation
@2u @2u
.t;x/Dc .t;x/ forallt 2R; x 2.0;1/;
@t2 @x2
wherec > 0isaparameterdescribingthestring’sproperties(e.g.,itsthickness). We
assumethatthestringisfixedatbothends,i.e.,that
u.t;0/Du.t;1/D0 holdsforallt 2R:
Sincethedifferentialequationislinear, wecanseparatethevariables: wewrite uin
theform
u.t;x/Du .x/cos.!t/ forallt 2R; x 2Œ0;1(cid:2)
0