Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory: Festschrift in Honor of Volker Mehrmann

635 Pages·2015·8.06 MB·English

by  Peter Benner

Peter Benner
Matthias Bollhöfer
Daniel Kressner
Christian Mehl
Tatjana Stykel Editors
Numerical Algebra,
Matrix Theory,
Diff erential-Algebraic
Equations and Control
Theory
Festschrift in Honor
of Volker Mehrmann
Numerical Algebra, Matrix Theory,
Differential-Algebraic Equations
and Control Theory

Peter Benner • Matthias BollhoRfer (cid:129)
Daniel Kressner (cid:129) Christian Mehl (cid:129)
Tatjana Stykel
Editors
Numerical Algebra,
Matrix Theory,
Differential-Algebraic
Equations and Control
Theory
Festschrift in Honor of Volker Mehrmann
123
Editors
PeterBenner MatthiasBollhoRfer
MaxPlanckInstituteforDynamics TUBraunschweig
ofComplexTechnicalSystems Carl-Friedrich-Gauß-Fakultät
Magdeburg,Germany InstituteComputationalMathematics
Braunschweig,Germany
DanielKressner ChristianMehl
ÉcolePolytechniqueFédéraledeLausanne TUBerlin
(EPFL) InstituteofMathematics
MATHICSE Berlin,Germany
Lausanne,Switzerland
TatjanaStykel
UniversityofAugsburg
InstituteofMathematics
Augsburg,Germany
ISBN978-3-319-15259-2 ISBN978-3-319-15260-8 (eBook)
DOI10.1007/978-3-319-15260-8
LibraryofCongressControlNumber:2015938103
Mathematics Subject Classification (2010): 15A18, 15A21, 15B48, 49M05, 65F08, 65F15, 65H10,
65L80,93B36,93B40,93D09,93D15
SpringerChamHeidelbergNewYorkDordrechtLondon
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hisresearchwork
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Preface
ThisbookisdedicatedtoVolkerMehrmann,ourmentor,co-author,colleague,and
friend. Each chapter in this book highlights one of the many topics Volker has
worked on. Inevitably, there are omissions. In this preface, we therefore make an
attemptto notonlyconnectthebookchapterstoVolker’sbibliographybutto also
provide additional details on topics that receive less attention in the subsequent
chapters.
M-Matrices andH-Matrices andTheirFriends
A complexor real square matrix A is called a Z-matrix if its off-diagonalentries
are nonpositive. Equivalently, A D ˛I (cid:2) B holds for some scalar ˛ 2 R and
n
a nonnegative matrix B. If, additionally, ˛ is not smaller than (cid:2).B/, the spectral
radius of B, then A is called an M-matrix. Arising frequently,for example, from
thediscretizationofpartialdifferentialequations,matriceswithsuchstructureshave
manydesirablepropertiesthatfacilitate,amongotherthings,thedesignandanalysis
ofiterativemethodsforsolvinglinearsystems.
OstrowskicoinedthetermM-matrixandproposedthefollowinggeneralization:
AmatrixAiscalledanH-matrixifitscomparisonmatrixC (definedviac Dja j
ii ii
andc D(cid:2)ja jfori D1;:::;n,j 6Di)isanM-matrix.Incompletefactorizations
ij ij
forsuchH-matricesarethetopicofthefirstscientificpaperbyVolkerMehrmann,
a1980publication[A1]jointlywithRichardS.VargaandEdwardB.Saff.
Apart from M- and H-matrices, there is a myriad of other matrix classes
with similarly desirable properties, but different targets in mind. In his 1982
dissertation [T2] and subsequent publications [A2, A3, A5], Volker contributed
to this scene by proposing and analyzing the concepts of R- and V-matrices.
One major motivation for these new concepts was to find a unified treatment for
M-matricesandHermitianpositivesemidefinitematrices.
Let the number l.B/ denote the smallest real eigenvalue of a matrix B, with
the convention l.B/ D 1 if B has no real eigenvalue. A matrix A is called an
v
vi Preface
Fig.1 YoungVolkerdoing
Math
!-matrixifthisnumberisfiniteandsatisfiesamonotonicitypropertyforallprinci-
palsubmatricesofA.If,moreover,l.A/(cid:3)0,thenAiscalleda(cid:3)-matrix.Sincetheir
introductionbyEngelandSchneider,therehasbeensignificantinterestinstudying
!- and (cid:3)-matrices because they represent generalizations of several important
matrix classes, such as Hermitian matrices, totally nonnegative matrices, and
M-matrices. In [A4], Volker showed for n D 4 that every n (cid:4) n (cid:3)-matrix is
stable by establishing an eigenvalue inequality conjectured by Varga. More than
adecadelater,in1998,OlgaHoltzdisprovedtheconjecturethatthispropertyholds
forgeneralnbyconstructingawholefamilyofunstable(cid:3)-matrices.
InsubsequentcollaborationswithDanielHershkowitzandHansSchneideronthe
matrixclassesdiscussedabove,Volkerinvestigatedlinearpreserverproblems[A9],
eigenvalue interlacing properties [A11], and a generalization of sign symmetric
matrices[A12].Jointwork[A23]withLudwigElsnerestablishestheconvergence
ofblockiterativemethodsforblockgeneralizationsofZ-matricesandM-matrices
thatarisefromthediscretizationoftheEulerequationsinfluidflowcomputations.
This1991paperalsoseemstomarkatransitionofVolker’sworkintootherareas.
HamiltonianandSymplectic Matrices
(cid:2) (cid:3)
0 I
LetJ D n .AmatrixH 2 R2n(cid:2)2n iscalledHamiltonianif.JH/T D JH
(cid:2)I 0
n
and a matrix S 2 R2n(cid:2)2n is called symplectic if STJS D J. Hamiltonian and
symplecticmatricesseemtohaveacertainfascinationforVolkerthatissharedby
severalof his co-authors.On the one hand, there are importantapplications,most
notably in optimal and robust control. On the other hand, there is rich algebraic
structure: the set of Hamiltonian matrices forms a Lie algebra, and the set of
symplectic matrices forms the corresponding Lie group. Additionally to being
Preface vii
Fig.2 CoverandtableofcontentsofVolker’sDiplomarbeit(equivalenttoaM.Sc.thesis),written
inGermanandpre-TEX
symmetricwithrespecttotherealline,thespectraofH andS aresymmetricwith
respecttotheimaginaryaxisandtheunitcircle,respectively.
The analysis and numerical solution of eigenvalue problems for matrices and
matrixpencilswithstructureisamajorthreadinVolkerMehrmann’sbibliography,
from the first works, his 1979 “Diplomarbeit” [T1] at Bielefeld (see Fig.2 for
acopyofthecoverandtableofcontents),untiltoday.Onemajorchallengeinthis
field had beenthe developmentof algorithmsthatwouldcomputethe eigenvalues
(and invariant subspaces) of Hamiltonian/symplectic matrices in a numerically
stable,efficient,andstructure-preservingmanner.Chapter1byBunse-Gerstnerand
FaßbenderaswellasChap.4byBennerexplainthesubstantialcontributionsVolker
madeinaddressingthischallengeforsmall-tomedium-sizeddensematrices.There
have been numerous extensions of the numerical algorithms resulting from this
work, most notably to large-scale eigenvalue problems, see Chap.2 by Watkins,
andmatrixpencilswithsimilarstructures,seeChap.4aswellasChap.5byPoloni.
Their extension to structured matrix polynomialswill be discussed in more detail
below.
viii Preface
Fig.3 VolkerandAngelika
Bunse-GerstnerinSan
Franciscoduringwhatseems
tobeanenjoyable(ofcourse
vegetarian!)dinneratGreens
RestaurantintheSan
Franciscoharborarea,1990
Thereisafascinating,moretheoreticalsidetothestoryonstructuredeigenvalue
problems.Still motivatedby algorithmicdevelopments,the 1991paperwith Greg
Ammar [A21] discussed the existence (or rather the lack thereof) of structured
Hessenberg forms for Hamiltonian and symplectic matrices. This paper is rather
unique in not only thanking Angelika Bunse-Gerstner (see Fig. 3 for a picture
of her and Volker around this time) for helpful discussions but also “the German
police fora speedingticket duringonediscussion”. As summarizedin Chap.6 by
MehlandXu, theworkofVolkerandhisco-authorsthenevolvedtowardsa more
general picture of structured canonical forms of structured matrices. Perturbation
theoryplays a major role in understandingthe potentialbenefits fromstructure in
thepresenceofuncertaintyinthematrixentries,forexample,duetoroundofferror.
Ofparticularinterestistheperturbationbehaviorofeigenvaluesthatarecriticalin
acertainsense,suchaspurelyimaginaryeigenvaluesofHamiltonianmatrices;see
Chap.8byBoraandKarowfora summaryofresultsin thisdirection.Chapter13
byRanandRodmangivesamoregeneralsurveyofthestabilityofmatrixanalysis
problemswithrespecttoperturbations.
Matrix Equations
Continuous-time linear-quadratic optimal control problems give rise to algebraic
Riccatiequationsoftheform
F CATX CXA(cid:2)XGX D0; (0.1)
whereA2Rn(cid:2)n,andF;G 2Rn(cid:2)naresymmetricand,often,positivesemidefinite.
Undercertainconditions,solutionsX of(0.1)canbeobtainedfromn-dimensional
Preface ix
Fig.4 Volker organized numerous workshops: one of the first was the 5-day Short Course on
Large Scale Scientific Computing in Bielefeld, 1992, jointly organized with Angelika Bunse-
Gerstner.Firstencountersofmanyinthispicture
invariantsubspacesofthe2n(cid:4)2nHamiltonianmatrix
(cid:2) (cid:3)
A G
:
F (cid:2)AT
As explained in Chaps.4 and 5, such quadratic matrix equations and their links
to structured matrices/matrix pencils have been a major theme in Volker’s work,
ever since his 1986 joint work [A6] with Angelika Bunse-Gerstner and his 1987
habilitation thesis [T3, B1]. Among others, this work has led to robust numerical
algorithmsforsolvingoptimalandrobustcontrolproblems.
In a collaboration with Mihail M. Konstantinov, Petko H. Petkov, and others,
theoretical properties of matrix equations have been investigated, establishing a
generalframeworkforderivinglocalandnonlocalperturbationbounds.Thiswork,
whichhasalsobeenextendedto theperturbationanalysisofeigenvalueproblems,
isdescribedinChap.7.
Differential-Algebraic Equations
A differential-algebraic equation (DAE) arises when one imposes algebraic con-
straints on the states of a physical system that is modelled with differential
equations. DAEs in all flavors are probably the most central theme of Volker