cerfacs technical report no. tr/pa/01/84 franc¸oise ... · franc¸oise chaitin-chatelinand...

CERFACS TechnicalReport No. TR/PA/01/84

FrançoiseChaitin-ChatelinandElisabethTraviesas

Homotopic perturbationUnfolding the field of singularities of a matrix by a complexparameter:a global geometricapproach

Abstract This work discussesthe meaningwhich emergesfrom the linear relationbetweentwo matricesAandB, donevia the complex parametert suchthat A

�t �� A � t � B � A � . Sucha meaningis an explanation

drawn from anoutsideobservationof A andB. Onecouldpossiblywantto comparethis explanationwith themeaningwhich emergesfrom inside. Sucha line of researchis of epistemological importance.Thesecondideawhich emergesfrom the theoryof InexactComputingthat is presentedis the importanceofthedualityof viewpointswhich canbetakenin mathematicswith respectto singularities.Whencontinuity is theprevalentpoint of view, a finite setof singularitiesis nongenericandcanbe ignored:this is thebasisfor theeliminationof singularitiesfor continuoussubharmonicfunctions.This is aglobalpointof view. However, locally thesingularitiesdo haveaninfluencewhencomputationis inexact,suchasin finiteprecision.Theirneighborhoodis of positiveLebesguemeasure. Thehomotopicunfoldingof theresolventfieldz � � A � zI �� 1 is an extremely rich domain,wherewe areable to follow the subtleinterplaybetweenthecontinuousandthediscrete,throughtheactionof computation.

Key words. Homotopicunfolding,global, local, backwarderror, pseudospectrum,subharmonicity,synchronicity, knowledgeacquisition,deviation, InexactComputing,critical point

1. Intr oduction to homotopicperturbation

We briefly survey theuseof homotopicperturbationin thecontext of spectralapproximationoflinearoperatorsandmatrices.Thissurvey coversthework realizedor supervisedby oneof us(FCC)duringtheperiod1970to date.

In herbookSpectralapproximation of Linear Operators [15], whichappearedalmost20yearsago,FrançoiseChatelinusedtheredthreadof homotopicperturbationsto analyzethenumericalap-proximation,by Tn with finite rank,of aclosedlinearoperatorT in aBanachspace.Sheusedtheiden-tity T Tn �� T Tn � andlookedat thehomotopicfamily of linearoperatorsT � t � Tn t � T Tn � ,suchthatT � 0� Tn andT � 1� T, dependinglinearlyon thecomplexparametert � .The notion that � Tn � is a radial approximation of T as n � ∞ ([15], p 245) permits the useofpowerful analyticperturbationtheoryto getseriesexpansionof theexacteigenelementsin termsofthecomputedeigenelementsandin termsof theperturbationT Tn. This approachprovidedanim-pressively effective tool to presentin a unified mannermostof a posteriorierror boundsknown atthattimefor spectralcomputationsonmatricesaswell asonlinearoperators,boundedor closed.Thetheoryis quitegeneral,it requiresno assumptionof normalityor selfadjointness(symmetry).

Françoise Chaitin-Chatelin: Universit́e Toulouse 1 and CERFACS, 42 av. G. Coriolis, 31057 Toulouse cedex [email protected]

ElisabethTraviesas:CERFACS,42av. G. Coriolis,31057Toulousecedex - [email protected]

2 FrançoiseChaitin-ChatelinandElisabethTraviesas

Figure1.1.ThefamilyA�t � in � n � n, 0 � t � 1

Five yearslater, in 1988 was publishedthe book Valeurs propres de matrices [16] and anenlargedEnglishtranslationappearedin 1993[17]. Thesebookswerea sequelto thefirst oneandfocusedonly on matrices,that are linear operatorson finite Banachspaces.However in her useofhomotopicperturbationFrançoiseChatelinintroducedtwokey new ideas:

1. thenotionof block reducedresolventandpartialinverse([17], p. 76-79),2. theuseof Rayleigh-Schr̈odingerexpansions([17], p. 86-89)ratherthanRellich-Katoexpansions

([17], p.85-86)for iterative refinement.

Thefirst ideaallows to circumvent thediscontinuityof thespectralprojectionassociatedwith a sim-ple computedeigenvalue whenit approximatesa multiple defective one.The secondidealeadstomuchbetterbehavedalgorithmsto refinethecomputedeigenvalues.

Then,in 1996,FCCcoauthoredwith ValérieFraysśe Lectureson Finite PrecisionComputa-tions [6]. Thisthird bookconcentratesontheeffectof roundoff onthebehavior of numericalmethodswhichareprovedconvergentin exactarithmetic.It showshow thepowerful notionof backwarderroranalysiscanbeeasilyrecastin theframework of homotopicperturbation,with the importantdiffer-encethat theperturbationitself is not known in caseof inexact arithmetic[10]. Onehasaccessonlyto its normby meansof thefundamentalideaof thebackwarderror. This leadsto thekey notion,fornumericalmethodsrun in finite precision,of conditionalconvergencewith respectto thearithmeticof thecomputer.

This brief historicalaccountof threedecadesof researchon theapproximationof thesingular-ities of the resolvent field: z � � A zI �� 1, whereA is a linear operatoror a matrix andz is not aneigenvalueof A, putsin full light theubiquitousrole playedby homotopicapproximationin spectralapproximation.It allows to relateinformationonA andB A E by meansof thehomotopicfamilyA� t � A tE, t � . It hasbeenarguedthat,accordingto theclassicaldefinition of anhomotopicfamily, theparametert shouldbereal [22]. Theadjective “homotopic”of Greekorigin means“of thesamespatialform”, which refersto thefactthatthestructureof theperturbationtE stayshomotheticto the matrix E. In otherwords,the structureof the perturbationis fixed.This is true whethert isrealor complex. Thereasonto considert complex is that the treatmentof analyticitypropertiesfor� A� t � zI �� 1 � z � , is mucheasierin thatcontext. This reflectsthedifferencebetweenholomorphyin andanalyticity in . Goingcomplex simplifiestheanalysis,asis thecasefor the fundamentaltheoremof Algebra.NotethatwhenA andB arereal,thereis aneasygeometricinterpretationof thefamily A� t � , for t realand0 � t � 1, in n � n, illustratedin Figure1.1.BecauseA� t � A tE A t � B A� tB �� 1 t � A,it is clearthatA� t � represents,for any t in �0� 1� , apoint on thestraightinterval joining A to B.

Homotopicperturbation 3

In the reportto follow we want to look at the notion of homotopicperturbationfrom a purelyinformation-theoretic point of view: whatsortof informationon A andB A E canberelatedbymeansof thehomotopicfamily

A� t � A tE � t � ?Thenovelty of thework is fourfold:

i) it interpretsthecomputingactivity of well known algorithmsto solve Ax b or Ax λx in termsof knowledgeacquisition,thus bridging a gapbetweennumericallinear algebraand cognitivesciences.

ii) it structuresthefield of singularitiesof � A� t � zI �� 1 by meansof two familiesof curvesa) thesingularrayswhichareobtainedfor t reiθ, with afixedθ in � 0� 2π � ,b) thesingularorbits whichcorrespondto t reiθ, with r fixed,r � 0.

iii) It givesa globalsyntheticview of theresolventfield z � � A� t � zI �� 1 for thefamily A� t � A

tE, when � t � variesin � 0;∞ � . Thereis no restriction onE, suchas E smallenough,asis usualin classicalperturbationtheory. Theview relieson theanalysisof themapz � ρ � E � A zI �� 1 � .

iv) It providesa theoreticalframework to analyzetheinterplaybetweenexactcomputationandfiniteprecisionresults.

2. Knowledgeacquisition via Inexact Computing

Wearegiventwo matricesA andE of ordern suchthat

B A E !We supposethat the resolventfield for A: z � � A zI �� 1 is known for any z not aneigenvalueof A.Thesetof sucheigenvaluesis denotedσ � A� , thespectrumof A. It is thefieldof singularitiesof A: thesetof complex pointszsuchthat � A zI �� 1 doesnotexist.Theresolventfield allows to solve thelinearsystem

� A zI � x y " x � A zI � � 1yfor any z #� σ � A� , andany vectory in n.

Now we considerthesimilar systemwith B:

� B zI � u y! (2.1)If weknew � B zI �� 1 (whenever it exists),wewouldwrite u � B zI �� 1y andtheproblemwouldbesolved.However, theruleof thelearninggameis thatweareallowedonly to useE andtheresolventfield z � � A zI �� 1 to solve (2.1).Suchagameis calledInexact Computing.2.1. Knowledge acquisition

How canwe play thelearninggame?We introducethehomotopicfamily A� t � A tE, t � , suchthatA� 0� A andA� 1� B. For any z #� σ � A� ,

A� t � zI A tE zI � A zI �$� I t � A zI �� 1E � � I tE � A zI �� 1 �$� A zI �


Figure2.1.Partition of % , where theshadedareais theinexactpseudo-spectrumΨσAs recalledin theAnnex, theNeumannseries

� A� t � zI � � 1 � I ∞∑k& 1� t � A zI � � 1E � k �$� A zI � � 1

convergesiff ρ �$� A zI � � 1E �(' 1. Notethatρ �$� A zI � � 1E � ρ � E � A zI � � 1 � . Indeed,E � A zI � � 1x λx� x ) 0 " � A zI � � 1Ey λy with y � A zI � � 1x ) 0!

Thereforethesolutionx� t � of � A� t � zI � x� t � y is obtainedasthelimit, whenk � ∞, of theseriesof vectors

xk � t � � I ∑kj & 1 � t � A zI �� 1E � j � x � A zI �� 1 � I ∑kj & 1 � tE � A zI �� 1 � j � y� (2.2)for any t suchthat � t � � 1. Therateof convergenceof xk � 1� towardu � B zI � � 1y dependson thevalueof ρ � E � A zI �� 1 � comparedto 1.Whenzvariesin outsideof σ � A� , themapz � ρ � E � A zI � � 1 � is anuppersemi-continuousfunctionof z (proposition2.2.7in [16,17]). Thereforetheset

Γ � A� E � *� z;ρ � E � A zI � � 1 � 1� (2.3)is a compactsetin consistingof a finite numberof closedcurves,certainof which encloseeigen-valuesof A. Seein Section5 aproofof thisassertion.It is theboundaryof thedomainof convergenceof the Neumannseries(2.2) for � t � ' 1. The setΓ Γ � A� E � divides the complex planeinto twocomplementarysets:

i) theinexactresolventset, which is anunboundedopensetdefinedby

Re *� z; u � B zI � � 1y l imk + ∞xk � 1�$� *� z;ρ � E � A zI � � 1 �,' 1�ii) the inexactpseudo-spectrumwhich is aboundedclosedsetdefinedby

Ψσ -� z; B zI is not invertiblewith theavailablecomputingmeans� Neumannseries�.� !-� z;ρ � E � A zI �� 1 ��/ 1�SeeFigure2.1.In general,Ψσ is a finite collectionof connectedregionswhich is not itself nec-

essarilysimplyconnected.Theregionsaresuchthatoneor severaleigenvaluesof A lie inside.Fromthepoint of view of Inexact Computation Reis thesetof complex pointswhere(2.1)canbe


solv0 edby meansof a convergentNeumannseries.On theotherhand,Ψσ is thesetof pointswhere(2.1) cannotbesolved,be it becausethis is impossibleor becausethesolutionexistsbut cannotbecomputedby a Neumannseries.The pseudo-spectrumΨσ canbe seenasa black hole becausenoinformationon theresolventfield z � � B zI �� 1 canbeobtainedinsideit.This point of view, the possibility of knowledgeacquisitionoutsideof Ψσ via the resolvent fieldz � � B zI �� 1, will becomplementedin thenext Sectionby thedualpoint of view of thesingulari-tiesof B, andmoregenerally, thesingularitiesof A tE.2.2. Thespeedof knowledge acquisition

This topic will beaddressedspecificallyin Section4, afterthepresentationof thedualpoint of viewin Section3.

3. Distanceto singularity in Inexact Computing

Thepoint of view in this sectionis, givenz in σ � A� , canthis point z beseenasaneigenvalueofA tE for oneor morevaluesof t?This canbeeasilystudiedby meansof thefactorization,valid for z )� σ � A� ,

A tE zI 1� I tE � A zI � � 1� � A zI � !Thereforez is aneigenvalueof A tE if f 1 tλi 0, for λi, i 1� !2!2! � n, any eigenvalueof E � A zI �� 1.We concludethat z canbe seenasan eigenvalue of at mostn matricesA tiE suchthat ti � 1λi ,i 1� !2!2! � n. This requiresthatbothλ andt arenonzero.Amongstthesen matricesin the family A� t � A tE, thereexistsonewhich is closest, this is theoneassociatedwith t3 of minimummoduluswhich realizes

� t34� mini� ti � 1maxi � λi � !

The matrix A� t3 � is the closestmatrix of the family A� t � for which z is an eigenvalue, when thedistanceis measuredby themodulusof t.

Definition 3.1.The(homotopic) distanceto singularityof z is

min� � t � � z is an eigenvalueof A tE).It is alsocalled(homotopic)backwarderror, thatwedenoteBE � z� .It canbecomputedby theformula

BE � z� 1# ρ � E � A zI � � 1 � ! (3.1)The notionof distanceto singularityallows to reinterpretthecurve Γ definedby (2.3) asthecurveof all pointsz for which thedistanceto singularity(or equivalentlythebackwarderror)is equalto 1.This explainsthe:

Definition 3.2.Theborder Γ ∂Ψσ of thepseudo-spectrumis called thespectralcurve associatedwith A andE.


Weturn now to thetaskof characterizingthepointsz in σ � A� assingularitieswhich lie on Γ andinsideΨσ. For theeigenvalueson Γ, we shallalsospecifyat leastoneeigendirection,thanksto the

Lemma 3.1.y is aneigenvectorof E � A zI � � 1 associatedwith λ ) 0 iff s � A zI � � 1y is aneigen-vectorof A 1λ E associatedwith z.Proof

E � A zI � � 1y λy " λ � A zI � s Es " � A 1λE � s zs 5WhenE is of rank � 1, thereareseveral possiblematricesassociatedwith z aswe saw. HoweverwhenE is of rank 1, so is E � A zI �� 1. Thereforeit hasonly oneeigenvalue λ which is possiblynonzero.To thisuniqueλ ) 0 is associatedafinite t � 1λ andz is aneigenvalueof theuniquematrixA 1λE. If λ 0 thent is unbounded:� t � � ∞ asλ � 0.This discussionindicatesthat it is convenient to distinguishbetweenthe possiblevaluesof rankof E.

i) E is of rank1E � A zI �� 1 is of rank1 for z #� σ � A� andhasexactlyonepossiblynonzeroeigenvalue.ThereforezonΓ " ρ 1 " λ eiθ " t ei 6 π � θ 7 " Γ is thesetof all eigenvaluesof A tE, with � t � 1.Let uswrite E αβH , whereα � β � n aretwo nonzerovectors.Thereforeα definestheuniqueeigendirectionof E associatedwith the possiblynonzeroeigenvalue βHα dependingwhetherβHα ) 0 or not: Eα � βHα � α, α ) 0. Theothereigenvalue0 (of multiplicity n 1 if βHα ) 0)is associatedwith theeigenspaceKerE *� β �98 .Fromtheuniquenessof theeigendirectionα for E � A zI �� 1 associatedwith λ eiθ βH � A zI � � 1α, we deducetheuniquenessof theeigendirectionγ � A zI � � 1α associatedwith the(notnecessarilysimple)eigenvaluezof A:; A ei 6 π � θ 7 E. Theuniqueeigendirectionγ of A: associatedwith z, is deducedfrom theknowledgeof the resolvent field z � � A zI �� 1 andof the rangeofthedeviation matrixE: Im E *� α � .If we sett 1, we seein particularthat z on Γ is an eigenvalueof B A E if f λ < 1 is aneigenvalueof E � A zI � � 1, thatis βH � A zI � � 1α * 1.

ii) E is of rank � 1Thematrix E � A zI �� 1 hasat least2 possiblynonzeroeigenvalues.Thereforethereexist eigen-valuesof A tE, � t � 1, which lie insideΨσ andnoton its boundaryΓ.

In conclusion,if E is of rank1, any z in Ψσ is aneigenvalueof A tE for � t � � 1, for whichweknowtheuniqueeigendirection.WhenE is of rank � 1, onecanonly concludethat all eigenvaluesof A tE with � t � � 1 belongto Ψσ. The reciprocaldoesnot hold, which meansthat Ψσ may containeigenvaluesof A tE for� t � � 1 [5]. Thiswill beillustratedlaterin Section6.Theabove discussioncanbesummarizedby thefollowing proposition:

Proposition 3.1. i) WhenE is of rank1, thecurveΓ runs throughthen eigenvaluesof anymatrixA� t � A tE, � t � 1. For each distinct eigenvalue z is associatedat leastoneeigendirection


definedby γ � A zI � � 1α.ii) WhenrankE � 1, thecurveΓ runsthroughsome(but not necessarilyall) of then eigenvaluesof

A� t � , � t � 1. For such eigenvaluesonΓ, at leastoneeigenvectoris associatedanddefinedby thesameformula.

Proof:

i) If E is of rank1, thereis only onepossiblynonzeroeigenvalueλ for E � A zI �� 1. Therefore,then eigenvalueszi , i 1� !2!2! � n, of A tE with � t � 1 define(atmost)n matricesE � A ziI � � 1 whichhave thesamenonzeroeigenvalueλ, with λt = 1, hence� λ � 1.

ii) If rankE � 1, thenon-uniquenessof λ ) 0 impliesthatnot all of then eigenvaluesof A tE canalwaysrepresentedthatwaywith z � Γ, � t � 1. 5

WhenE is of rank1, all theeigenvaluesof A� t � reachthespectralcurve Γ in a unit lengthinterval� t � 1.WhenE is of rank � 1, a desynchronisationmayoccursincecertainpointsinsideΨσ requirea unitinterval � t � 1 to bereached,eventhoughthey arenot on Γ. At suchpoints,no knowledgecanbeacquired,becauseρ � 1. Thisphenomenonwill beillustratedin Section6.It is remarkablethatthespectralcurve Γ lendsitself to two complementaryinterpretations:

1. ontheonehand,it is theborderof theexteriordomainRewhereknowledgeacquisitionispossible,becausetheradiusof convergenceis 1,

2. ontheotherhand,it is theborderof theinteriordomainΨσ, acurveconsistingof pointsfor whichthedistanceto singularityequals1.

4. The rate of knowledgeacquisition

Weshow that,at afinite numberof pointsz, asuperfastconvergencecanbeachieved.

4.1. E is of rank1

We examinethe rate of convergenceof the seriesexpansion(2.2) for xk � t � , for z outsideΓ. Andwe supposethat E αβH , with α, β nonzerovectorsof n. In this particularcase,(2.2) can beconvenientlyrewrittenby introducingthetwo scalars:

ξ βH � A zI � � 1y and µ βH � A zI � � 1α µ� z� .µ is theuniquepossiblynonzeroeigenvalueof E � A zI �� 1, sothatρ � E � A zI �� 1 � >� µ � .Lemma 4.1.With E αβH of rank1, thelimit solutionx� t � is givenby x� t � � A zI � � 1 � y s� t � α� ,where thescalars� t � � ξt1? tµ is computedfor � µt � ' 1, asthelimit of a geometricsequencesk � t � .


Proof:From(2.2),follows that

xk � t � � A zI � � 1 � y @� k∑j & 1� t � jµj � 1ξ � α�A!

Therefores� t � = ξ∑∞j & 0 � t � jµj

= ξt1? tµ !s� t � limk + ∞ sk � t � , wheresk is thesumof thefirst k termsof theseriesaslongas � tµ � ' 1, which istheconditionzoutsideΓ. 5Whens� t � is computedby meansof (2.2),therateof convergenceis boundedby � tµ � ' 1. Sinceµ µ� z� dependson z, it is conceivablethatfor certainz in Re,µ is zero.This canhappenwhentherank1 matrixFz E � A zI � � 1 is in additiondef ective, thatis whenF2z 0 " µ βH � A zI � � 1α 0.Indeedµ� z� is a rationalfraction,whosedenominatoris thecharacteristicpolynomialin z, of degreen, andthenumeratoris a polynomialin z of degreed, 0 � d � n 1. Soit hasd rootsin (d 0 ifn 1). At suchpointsz in , thecondition � tµ � ' 1 is satisfiedfor all t andthe learningprocessisthereforesuperfastfor any t: s� t � * tξ and

x� t � � A zI � � 1 � y tξα � x tξγif we set γ � A zI � � 1α. At suchpoints,x� t � is l inear in t for all t. We note that µ µ� z� � 0as � z � � ∞. We prove in [12] that when A JT is the companionmatrix of xn and E B aeTn ,a � a0 � !2!2! � an � 1 � T , sothatA E is thecompanionmatrixassociatedwith

p� x� xn n � 1∑i & 0 aixi xn q� x�

thenµ� z� q6 z7zn is zeroat therootsof q� z� . Thispolynomialof degreed � n 1 hasn 1 rootsin ,unlessan� 1 0. Therearetherefored � n 1 pointszof superfastconvergence,for which p� z� znandx� t � is linearin t for any t. Moreover, thechoicey parallelto α yieldsξ 0, s� t � 0 andx� t � xfor all t: thesolutionis stationary(independentof t).This meansthat the solution x � A zI � � 1y we get at the first iterationhappensto be the exactsolutionof � B zI � u y with B A E, which wasour goal for the learningprocess.And this istruefor any solutionx� t � of � A tE zI � x� t � y, t � .This happensfor thechoiceof a vectory parallel to thedirectionα which definesthe rangeof thedeviation matrixE αβH . Weset

� A zI � γ α " γ � A zI � � 1α !Pointsz whereµ� z� 0 arespecialin two ways. First, Lemma3.1 indicatesthata point z suchthatµ� z� 0 cannotbe interpretedasan eigenvalueof A tE becauset would be infinite. We have tointerpretit asa regularpoint for A tE zI : thematrix is regular.Secondthe inverse � A tE zI � � 1 canberepresentedmoreeffectively thanby the limit of an infi-nite Neumannseriesconvergentfor � t � ρ ' 1. Thebehaviour is now finite. Thereis no questionofconvergencewith respectto t anymore,becauseonehasthefinite expansion

� I tE � A zI � � 1 � � 1 I tE � A zI � � 1


valid0 for any t. This is in sharpcontrastwith theinfinite Neumannseriesexpansionwhich is requiredwhenµ� z� ) 0.4.2. Andif rankE � 1?WhenE is of rankk, n / k / 1,wemaywrite E UVH , whereU andV arematricesin n � k, of rankk suchthatVHU is invertible.ThereforeFz UVH � A zI �� 1 of rankk hasthesamek possiblynonzeroeigenvaluesasthek C k matrixMz VH � A zI � � 1U . Theelementsof Mz arerationalfractionsin z, that is they areratiosof a polynomialof degree � n 1 to thecharacteristicpolynomialof A ofdegreen.An analysis,similar to the previous onedonefor k 1, canbe carriedout to look for conditionsof superconvergence. The fact thatnow Mz is a matrix of orderk � 1, ratherthanthescalarµ� z� βH � A zI � � 1α, allows morefreedomandthereforericherconclusionscanbeachieved.Thereaderisreferredto [12] for acompleteanalysisin thecasek 2.5. The global geometryof the map ϕ : z � ρ � E � A zI �� 1 �5.1. A relatedproblem:themapΨ : z �1 � A zI �� 1 During thedecadeof 90s,a lot of attentionhasbeengivento themapΨ definedfor z � , z #� σ � A� ,into ? . SuchamapΨ is oftenencounteredin stability issuesrelatedto thedistributionandinfluenceof eigenvaluesof A. Theoreticalinsight canbe gainedby inspectingthe2D level curvesof Ψ. It iseasyto show ([6], p. 177-178)thatther-level curve = � z; � A zI � � 1 D 1r � , r � 0, is theborderofthesetof all theeigenvaluesof A ∆A, for all matrices∆A suchthat ∆A D� r. Sucha setis knownasthenormwiser-pseudo-spectrumof A, whereA canbea matrixor a linearoperator[19,25,26].

As for the Γ curve, the 1-level curve � z; � A zI � � 1 D 1� for Ψ canbe interpretedby meansofthe(normwise)backwarderroror (normwise)distanceto singularityof z [6].The1-level curvesfor Ψ � z� andϕ � z� canbeeasilyrelated[24] when E D 1 becauseof theinequal-ity

ρ � E � A zI � � 1 � �> E � A zI � � 1 D�> � A zI � � 1 E!An importantapplicationof normwisepseudo-spectrais theanalysisof finite precisioncomputationby meansof scalednorms[6,10,24].

5.2. Ψ andϕ assubharmonicfunctionsfor z in σ � A�For any z #� σ � A� , � A zI � � 1 is ananalyticfunctionwhich canbewritten undertheCauchyintegralformula:if thedisk � s � � � s z � � r � containsnoeigenvalueof A, then

� A zI � � 1 12iπFHG

s� zG& r � A sI � �

1

s z ds 12πF 2π

0�A � z reiθ � I � � 1dθ !

Thispowerful resultis whatis neededto prove thatϕ � z� ρ � E � A zI � � 1 � andΨ � z� I � A zI � � 1 aresubharmonicfunctionsin σ � A� . Werecallthatasubharmonicfunction f : Ω J � , whenΩ is anopendomain,is continuous(or at leastuppersemicontinuous)in Ω andsatisfies,for any z inΩ suchthat � s� � s z � ' α � J Ω, the inequality


f � z� � 12π K 2π0 f � z reiθ � dθ,for any r, 0 ' r � α.Thefollowing is standardin Complex Analysis[2].

Proposition 5.1.A subharmonicfunction f in Ω satisfiesa MaximumPrinciple:

supzL Ω f � z� maxsL ∂Ω f � s� !

Fromthis follows that f cannothavea localmaximum,unlessit is constantlocally. Morecanbesaidfor ϕ andψ becauseof theirbehaviour at infinity:

limGz

G+ ∞ ρ � E � A zI � � 1 � �M E lim

Gz

G+ ∞ � A zI � � 1 0

by Corollary2.2.5in [16,17].

Theorem 5.1. If ϕ andΨ are locally constant,they haveto bezero on σ � A� .Proof:

1. Webegin with themapz � Ψ � z� . Since � A zI � � 1 D max

xLON n � A zI �� 1x x �thereexistsaparticularx, x D 1,suchthat � A zI � � 1 D> � A zI � � 1x . Now thisvectornormcanberepresentedby meansof ascalarproduct:thereexistsy, y P3Q 1 (y linearfunctionalon

n where E!R 3 denotesthedualvectornorm)suchthat � A zI � � 1x D yH � A zI � � 1x!The vectorsx andy dependcontinuouslyon z, the function g� z� yH � A zI �� 1x is thereforeholomorphic for z in σ � A� . It cannotbeconstanton anopendomainunlessit is constantev-erywherein σ � A� . Thatthisconstantshouldbe0 follows from thefactthatlim

Gz

G+ ∞ Ψ � z� 0.

2. We turn to themapz � ϕ � z� . We first remarkthat,in theparticularcaseE αβH of rank1, theabove proof appliesverbatim,wherex (resp.y, g) becomesα (resp.β, ϕ).For ageneralE of rank � 1, wesupposethatthereexistsanopendomainΩ in σ � A� onwhichϕ � z� ρ � E � A zI �� 1 � hasthe constantvaluec ) 0. We choosea z in Ω, by definition it is aneigenvalueof a matrix A� t � A tE with � t � 1c ' ∞. We considera sequence� tn � in suchthat � tn � ' 1c for all n, andtn � t asn � ∞. For all n largeenough,n � N, A� tn � is closeenoughto A� t � that thereexistsat leastoneeigenvaluezn of A� tn � in Ω. But suchzn � Ω hasa distanceto singularityequalto � tn � ' 1c . This is impossiblefor all z in Ω. Hencethe contradiction.Theconstantvaluec hasto bezerofor all z in S σ � A� . 5

Corollary 5.1 For z in S σ � A� , onehasonlyoneof thetwopossibilitiesfor ϕ (resp.ψ)i) thefunctionϕ (resp.ψ) is nonconstantonopensubsetsof S σ � A� ,ii) ϕ (resp.ψ) is zero on S σ � A� .Proof: It follows from Theorem5.1. 5


5.3.T ThegeometricpictureSubharmonicityfor ϕ (resp.ψ) hasimportantconsequenceswith respectto maximality properties.First at all, therecannotexist a local maximumunlessthevalueis ∞ (which happensat theeigen-valuesin σ � A� ). Second,becauseof corollary 5.1, level setsfor ϕ (resp.ψ) have to be closedsets,andmoreover closedcurves.The r-level curvesencloseeigenvaluesof A (local minimumat ∞) for r largeenough.They canen-closea local minimum for r in a certaininterval � r1 � r2�,JU� 0� ∞ � . Suchexamplescanbe found in[18,24], andin theVeniceexamplebelow.

For r1 0, we have seenin Section4, that the 0-level curve at finite distance,when it exists, isreducedto a finite numberd, 0 � d � n 1, of isolatedpointsin , wherewe getsuperconvergencefor (2.2). This is a particularcaseof the theorem([2], théor̀eme6, p. 303)which statesthat thesetof singularities(i.e. zerovalues)for a positive subharmonicfunction is non generic(its interior isempty).To distinguishthesespoints from the eigenvalueswhich arealsosingularpoints,we shallreferto theformerascritical points.

6. Emergenceof meaningvia an increaseof complexity: the interpretation with spectral raysand orbits

Wewishto structurethepseudo-spectrumwhichis left unstructuredby thepointof view of solv-ing (2.1)via seriesexpansion(a“black hole”). Intuitively, werealizethatwehave to turn to thepointof view of thesingularitiesof theresolventfield in orderto getanunderstanding.

In termsof complexity, this meanschangingfrom a systemof n linearequationsto then rootsof acharacteristicpolynomial,thatis apolynomialof degreen. Sothecomplexity of thenew pointofview (aneigenproblem)is vastly increasedascomparedto theoriginalone(solvingalinearsystem).The parametert is the threadwhich will help us orderΨσ, that is make senseof it. It will help usunderstandhow it is to be interpreted,by meansof t, from thedataA andE in termsof knowledgeacquisitionasbefore,but of knowledgeof higher complexity.

The eigenvaluesλ � t � of A� t � A tE lie in the inexact spectrumΨσ for � t � � 1. We shall ana-lyze thisdomainof thecomplex planeby meansof

i) thespectral raysΛ � θ � , whicharedefinedbyΛ � θ � =� λ � t � eigenvalueof A� t � A tE for t reiθ, 0 � r � 1 andθ fixedin � 0� 2π � � ,

ii) thespectral orbits Σ � r � whicharedefinedbyΣ � r � =� λ � t � eigenvalueof A� t � A tE, for t reiθ, 0 � θ ' 2π andr fixed,r � 0� !

Theanalysisof thestructureof Ψσ is illustratedbelow onanexamplewhich is now described.

6.1. Thestructure of theinexactpseudo-spectrumwith a companionmatrix

A of order8 is thecompanionmatrixassociatedwith thepolynomialof degree8:

p� x� � x 1� 3 � x 3� 4 � x 7� x8 22x7 198x6 958x5 2728x4 4674x3 4698x2 2538x 567


Wecall Venicetheassociatedcompanionmatrix:

A

0 5671

... 2538

..... . 4698... .. . 4674

... .. . 2728... ... 958

... 0 1981 22

It has1 astriple eigenvalue,3 asaquadrupleeigenvalueand7 asa simpleone.Thedeviation matrixE will besuccessively taken:

– of rank1, asE1 eeT8 , with e � 1!2!2! 1� T ande8 � 0!2!2! 01� T ,– of rank2, asE2 E1 e1eT6 ,– of rank4, asE3 E2 e2eT4 e3eT2 ;

Thesameanalysiswill beconductedin paralleli) onA� E andii) onA E � E. In casei), themultipleeigenvaluesareinsideΓ � A� E � , whereasin caseii) they areon Γ � A E � E � .

6.2. E E1 eeT8 is of rank1Thespectralcurve Γ (black)and8 spectralraysΛ � θ � in blackCasei) A� E Caseii) A E � E

0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

θ 0


Casei) A� E Caseii) A E � E

0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

θ π4

Thespectralcurve Γ (black)and8 spectralraysΛ � θ � in blackCasei) A� E Caseii) A E � E

0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

θ 3π2

As expected,all raysendon Γ. Thecolor for 0 � r � 1 is parameterizeduniformly, seeSection7.


Thespectralorbit Σ � r � and8 spectralraysΛ � 0� in blackCasei) A� E Caseii) A E � E

0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

r 1


0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

r 12

As expected,the spectralorbit Σ � 1� is identical to the spectralcurve Γ. The 8 spectralrays Λ � 0�definein 8 sectors,in which θ variesfrom 0 to 2π. This decomposesthespectralorbit Σ � θ � in 8curvilinearsegments,eachof whichhasa full color span(blueto red)correspondingto 0 � θ � 2π.Thecolor for 0 � θ � 2π is parameterizeduniformly, seesection7.


6.3.T E E2 is of rank2


0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

θ 0


0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8

θ π4



0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

θ 3π2


0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8

r 1



0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8

r 12In thiscaseof arank2 deviation,all raysstill endonΓ which is identicalto Σ � 1� . Theonly importantdifferenceis that now, in casei), Σ � 1� is not connected: it consistsof threeclosedcurves,the twoinner curvesdo not encloseany eigenvalueof A. However sucha phenomenonis not relatedto therank of E: it canhappenwhenE is rank 1 [24]. As a consequence,Γ ) Σ � 1� andthereareonly 6curvilinearsegmentson Γ, eachcorrespondingto 0 � θ � 2π. The two othercurvesareinsideΨσ:they arenot seenon Γ.

6.4. E E3 is of rank4Thespectralcurve Γ (black)and8 spectralraysΛ � θ � in blackCasei) A� E Caseii) A E � E

−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10

−6

−4

−2

0

2

4

6

θ 0



−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10

−6

−4

−2

0

2

4

6

θ π4


−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

θ 3π2


Thespectralorbit Σ � r � and8 spectralraysΛ � 0�Casei) A� E Caseii) A E � E

−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10

−6

−4

−2

0

2

4

6

r 1Thespectralorbit Σ � r � and8 spectralraysΛ � 0�


−2 0 2 4 6 8 10 12

−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10

−6

−4

−2

0

2

4

6

r 12The moststriking phenomenonwith this rank 4 deviation is thedifferencebetweenΓ andΣ � 1� forA� E andevenmoresofor A E � E: in thelattercase,theorbit tendsto become“curly”. Whenacurl(or loop)of Σ � 1� happensto beinsideΓ withoutenclosingany eigenvalueof A, or A E, thespectralraysΛ � θ � , whichalwaysendon Σ � 1� , mayendata point insideΨσ ratherthanon theborderΓ. Thesignificanceof loopsinsideΓ versusoutsideΓ will bespecificallyaddressedin Section6.9.


−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.5 1 1.5 2 2.5 3 3.51.4

1.6

1.8

2

2.2

2.4

2.6

Casei) A� E, rank(E)=4 (r 1) Caseii) A E � E (r 1)Zoomon [-0.24;0.24] C [-0.24;0.24] Zoomon [0,4;3,5]C [1,4;2,7]

6.5. Themapϕ : z � ρ � E � A zI �� 1 �It is moreinformative to considera logarithmicscalefor ρ, becauseof its wide rangeof variationin � 0; ∞ � . Thegraphicaldisplayof z � logρ � E � A zI �� 1 � canbedonein 2D with level curves,orrenderedin 3D.


0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8

2D displayfor E of rank1



0 24 6

8 1012

−10−5

05

10−6

−4

−2

0

2

4

6

8

−5

0

5

10

15

−10

−5

0

5

10−5

−4

−3

−2

−1

0

1

2



0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8




0 24 6

8 1012

−10−5

05

10−2

−1

0

1

2

3

4

5

6

7

−4 −20 2

4 68 10

12 14

−10

−5

0

5

10−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5



−2 0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8

−2 0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8




−4 −20 2

4 68 10

12 14

−10

−5

0

5

10−2

−1

0

1

2

3

4

5

6

7

8

−4 −20 2

4 68 10

12 14

−10

−5

0

5

10−1.5

−1

−0.5

0

0.5

1

1.5


6.6. A meshof raysθ 0� !2!2! � 2πThesegment �0;2π� is evenly dividedby 25 pointswhich realizea discretizationof size π12. We plotthe24 raysΛ � θi � correspondingto thediscretevaluesθi i π12, i 0� 1� !2!2! � 23.

Casei) A� E Caseii) A E � EE of rank1

0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8



0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8


−2 0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

−2 0 2 4 6 8 10

−6

−4

−2

0

2

4

6

6.7. A meshof orbits r 0� !2!2! � 1� !2!2! � 2

Thesegment �0;2� is evenlydividedby 11pointssuchthatthemeshsizeis 0.2.Weplot the10orbitsΣ � r i � correspondingto thediscretevaluesr i i5, i 1� 2� !2!2! � 10.



0 5 10 15−8

−6

−4

−2

0

2

4

6

8

0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8


0 5 10 15−8

−6

−4

−2

0

2

4

6

8

−2 0 2 4 6 8 10 12 14−8

−6

−4

−2

0

2

4

6

8



−2 0 2 4 6 8 10 12 14 16

−8

−6

−4

−2

0

2

4

6

8

−2 0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8

6.8. Weavingraysandorbits

Weplot in 3D for r rangingin �0;1� and �1;2� 6 orbitsalongthe8 raysΛ � 0� , correspondingto θ 0.Thedisplayscorrespondto caseii) A E � E.

0 � r � 1 1 � r � 2E of rank1

0

2

4

6

8

10

−10−5

05

10

0

0.2

0.4

0.6

0.8

1

0246

81012

−10

−5

0

5

10

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2


0 � r � 1 1 � r � 2E of rank2

02

46

810

−10

−5

0

5

10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0246810

12 −10

−5

0

5

10

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 � r � 1 1 � r � 2E of rank4

−2

0

2

4

6

8

10

−10

−5

0

5

10

0

0.2

0.4

0.6

0.8

1

−5

0

5

10

15

−10−5

05

10

1

1.2

1.4

1.6

1.8

2

As θ variesin �0;2π� , theraysΛ � θ � createin 3D thesurfacewhoser-level curvesaretheorbitsΣ � r � .6.9. A closerlook at theloopsof Σ � 1�We wish to draw the attentionof the readeron the fact that, when Σ � 1� hasloops, thereis a bigdifferencebetweenan internal loop which is insideΨσ, but doesnot belongto Γ, andan externalloop which is partof Γ. Westudythe2 casessuccessively.

1. Γ andΣ � 1� havean external loopSuchacaseis exemplifiedby thematrixVenicein caseii) A E � E, with E of all threeranks1, 2


θ1

θ2

θ2 π0

0

1

µ

Figure6.1.Mapsθ ��V µ1 � θ �WV (blue), V µ2 � θ �WV (red)

and4.This is aconsequenceof themultiplicity of two externaleigenvalues:theeigenvalues1 and3 of � A E �X E A areof respective multiplicity 3 and4. Eachindividual loopcontainscertaineigenvaluesof A E. It takesamultiple of 2π to describesucha loop.

2. Σ � 1� hasan internal loopwhich enclosesno eigenvalueof A andis nota part of ΓThis canhappenonly if rankE � 1. Suchanexampleis providedby Venice,caseii) A E � E, for Eof rank4, wheresevensuchloopsarepresent.Seealsothezoomaround0 for casei) A� E. Suchanoccurrenceis not anymorethe resultof themultiplicity of aneigenvalueof A� θ � � A E �Y eiθE,but is ratherdueto thecoincidenceatzof aneigenvalueof amatrixA� θ1 � with thatof amatrixA� θ2 � ,for θ1 ) θ2.This is easilyexplainedwith E of rank2.Let µ1 � θ � andµ2 � θ � bethetwo eigenvaluesof Fθ eiθE � A zI � � 1 which arenot necessarilyzero.We supposethat � µ1 � θ � � / � µ2 � θ � � . The representationθ �B�µ1 � θ � � , � µ2 � θ � is sketchedin Figure6.1. The equality � µ1 � θ � � >� µ2 � θ � � for θ1 ) θ2 (mod.2π) ispossiblebecauseE is of rank2. This impliesthatA� θ1 � andA� θ2 � sharethesameeigenvaluez. Thesameanalysiscanbecarriedon any orbit Σ � r � , r � 0. Sowe introducetheDefinition 6.1.We call synchronicitythecoincidenceat z of two eigenvalueson thesameorbit Σ � r �correspondingto two differentphasesθ1 ) θ2 (mod.2π).Because� θ1 θ2 � ' 2π, it takesθ lessthan2π to goaroundtheinternalloop.Thefamily Λ � θ � of n rayshastheglobalperiod2π in θ andit dividesany orbit Σ � r � into n curvilinearsegments.Eachindividual componentof anorbit Σ � r � , r � 0, is a curve which is dividedin asmanycurvilinearsegmentsasthereareeigenvaluesof A (countingtheir multiplicity) presentinsidethein-dividual component.


7. Computation in finite precision

7.1. InexactComputingin finiteprecision

Weexpectfinite precisiontohaveagreatinfluenceonthesolutionof any systemof matrixA tE zI,whenz is in a regionof wherelie thecritical points.This is becausein this region, � A tE zI �� 1hasafinite representation,ratherthananasymptoticone.

Although the (at most)n 1 pointsof superconvergence form a non genericsetof singularitiesinexactarithmetic,their influencein finite precisioncanbeimportant . Indeed,finite precisioncompu-tationexpressesthepoint of view of Lebesguemeasure ratherthanBairecategory (genericity).Weknow thattheneighborhoodof any exactroot of a polynomialis asetof positive Lebesguemeasure,thegreaterthemeasure(with respectto machineprecision),themoreill conditionedtheroot is.

We examinein greaterdetail two striking examplesof finite precisioncomputationin the regionaroundthecritical points,with adeviation matrixE of rank1.

7.2. Venicein finiteprecision

We studythematrix Venicewith E eeT8 of rank1. Thepointsof superconvergence arethe7 rootsof p7 � z� ∑7i & 0 zi � thatis the7 rootsof z8 1 differentfrom 1.

Casei) A� E Caseii) A E � EMapz � log r � r 1# ρ

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

−4 −2 0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8

Unit circle in black


z eZ i π4 [ i eZ i 3π4 -1ρ̃ 1\ 3 ] 10� 10 1\ 8 ] 10� 11 3\ 8 ] 10� 12 6\ 7 ] 10� 13

Table 7.1.ρ�E�A � zI � � 1 � computedatcritical points

Casei) A� E

−2 −1.5 −1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5

−2 −1.5 −1 −0.5 0 0.5 1

−1

0

1

2

2.5

3

3.5

4

4.5

5

2D, r ^� 100� 200� 2000� 4000� 8000� 16000� 3D, log r *� 2� 2! 5� 3� 3! 5� 4� 4! 5� 5�Thecomputedvaluesρ̃ � E � A zI � � 1 � arelistedin Table7.1 for the7 critical valuesof z.The 7 critical pointsareon theunit circle: they aresingularpointsfor themapϕ : z � log r withr 1# ρ, sothatϕ is equivalentto themap ϕ : z �_ log ρ. The2D displaysof ϕ clearlyshow theinfluenceof pointson theunit circle,differentfrom 1. Themapϕ is furtherdisplayedfor casei) A� Efor increasingvaluesof r, bothin 2D and3D.

7.3. ThefamilyOne(n)in finiteprecision

We considerthefamily of polynomialspn � x� xn andqn � x� ∑ni & 0 xi pn � x�X qn� 1 � x� , of degreen 1� 2� !2!2! , with q0 � x� 1.Thecompanionmatrixassociatedwith pn � x� is JTn , whereJn is theJordanblockof ordern:

Jn 0 1 0

... .. .. . . 1

0 0

!

Let En eeTn , with e � 1� !2!2! � 1� T . The family One� n� is the family An � t � JTn tEn definedby� JTn � En � . ClearlyAn � 1� JTn En is thecompanionmatrixassociatedwith qn � x� . Thecritical pointsof theunfoldingarethen 1 rootsof qn� 1 � x� .1. n 8

We give for casei) JT8� E8 the 2D displayof ϕ togetherwith the 7 orbits Σ � r � correspondingto


thevaluesr `� 0;0! 2;0! 5;0! 7;1! 1;1! 4;1! 7� . The local influenceof thecritical pointsis striking:aroundzero(r a 1) theorbitsarealmostperfectcircleswhichgetmodifiedwhenr increases.For comparison,thereverseunfoldingcorrespondingto thecaseii) JT8 E8 � E8 is alsogiven.The7 valuescorrespondto r ^� 0;0! 2;0! 5;0! 7;1! 1;1! 4;1! 7� .

Casei) JT8� E8 Caseii) JT8 E8 � E8

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

r *� 0;0! 2;0! 5;0! 7;1! 1;1! 4;1! 7�2. n 29

We considercasei) JT29� E29. On the left is a display of ϕ with 4 orbits Σ � r � correspondingto

r 0! 1� 0! 5� 1 and2.On the right is a displayof 4 raysΛ � θ � for θ 0� π2 � π � 3π2 . The local influenceof the criticalpointsis clearlyvisible:outsidetheunit circle theraysarejuststraight lines. NotethatΛ � π � doesnotextendhorizontallymuchbeyond � 1� 0� .

Casei) JT29� E29 Caseii) JT29 E29� E29

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

r *� 0! 1;0! 5;1;2� θ *� 0� π2;π; 3π2 �


8. Relatedwork realizedin the Qualitati ve Computing Group

As wasshown in the introductorysurvey, the notion of homotopicperturbationis a very powerfultool to investigatetheoreticalquestionsrelatedto the spectralapproximationof matricesor linearoperatorsin Banachspaces.It hasthereforebeena key notion, explicitly or implicitly, for all theworksdevelopedby theQualitativeComputingGroupatCERFACS,duringmorethanadecade.Thelist of topics,givenbelow in chronologicalorder, reflectsthisunderlyinginfluence.

1. The notion of uniform radial convergence([15], p. 245) involves the spectralradius rσ � � T Tn �$� T z�� 1� which is nothingbut, in a differentnotation,ρ � E � A zI �� 1 � for z in a compactsubsetof theresolventsetof T (resp.A).

2. Computationin finite precisioninducesanunavoidableround-off errorwhich hasbeenanalyzedby abackwarderroranalysis([6], chapter5 and6 in particular).Finiteprecisioncomputationcanbeseenasuncertaincomputation, a particularcaseof inexact computation,wherethedeviationmatrix E is not known, only its size E is availableasa measureof thebackwarderror [6,10].It is importantto keepin mind that in finite precision,thenormshave to be scaledbecausethenotionof backwarderroris relativeto machineprecision[6,10].

3. The toolbox PRECISE(PRecisionEstimationandControl In Scientific andEngineeringcom-puting)offersa varietyof softwaretoolsdevelopedunderMatlab[6] andFortran[21] to conductcomputerexperimentsaboutstability. A detailedpresentationcanbefoundin [6], chapters8 and9, and in the user’s guideof the Fortranversion[21]. The key idea is to allow the sizeof theperturbationto vary in � 0� 1� for A� t � A tE.

4. Finite precisioncomputationof the Jordanform of a matrix is known to be unreliablebecausethe exact Jordanstructureis unstable,dueto the underlyingalgebraicsingularities.Homotopicperturbationshasbeensuccessfullyput at usein [20] to assessthemostinfluentJordanstructureof amatrix [8].

5. Whendealingwith linearoperatorsT in infinite dimensionalspacesthequestionarisessimilarlywhetherthemapsz � ρ � E � T z� � 1 � andz �1 � T z� � 1 canbeconstanton anopendomainof σ � T � . Thequestionhasbeenonly partially answeredsofar [7,19] : thesubharmonicityistruein a Hilbert space,andit is not yet known if it is truein generalin aBanachspace.

6. The incompleteArnoldi decompositioncanalsobe interpretedin thecontext of homotopicper-turbation[10,24]. Seealso[23].This leadsto anovel wayof interpretingthestoppingcriterioninKrylov methods[4,10,14,11].

7. Whenthe outeriterationis a Krylov-type method,inner-outer iterationsexhibit a surprisingro-bustnessto perturbationsdueto aninexactresolutionin theinnerloop [1]. In thiscontext too, thenotionof homotopicperturbationis abig helpto analyzethisseemingly“magical” behavior [9].

8. In thehomotopicviewpoint thatwepresented,theinexactpseudo-spectrumis structuredby meansof theeigenvaluesof A� t � of ordern. Sucheigenvaluesarethe rootsof polynomialsof degreenand,assuch,havebeenconsideredasn complex numbers(i.e.2D realvectors).Howeverpolyno-mialscanhave moreroots,indeedinfinitely many, if consideredasquaternions(4D realvectors)


or octonions(8Drealvectors)[3]. Theimportanceof consideringhypercomplex rootsis discussedin [13].

9. Conclusion

ThisworkhasdiscussedthemeaningwhichemergesfromthelinearrelationbetweentwomatricesA andB, donevia thecomplex parametert suchthatA� t � A t � B A� . Differentmeaningscanbe associatedwith differentpropertiesof the resolvent field z � � A� t � zI � � 1. The propertieshave to do with existenceor analyticityof theresolvent.Any z � which is not aneigenvalueofA� t � canbe interpretedeitherasan eigenvalueof A� t � , or equallyasa z suchthat � A� t � zI � � 1exists.Theonly instancewherezcannotbeinterpretedasaneigenvalueof A� t � is if it is acriticalpoint of � A� E � . In this case� A� t � zI �� 1 hasa finite representationin termsof t, ratherthananasymptoticone.The secondideawhich emergesfrom the theoryof Inexact Computingthat is presentedis theimportanceof thedualityof viewpointswhichcanbetakenin mathematicswith respectto singu-larities.Whencontinuityis theprevalentpointof view, afinite setof singularitiesis nongenericandcanbeignored:this is thebasisfor theeliminationof singularitiesfor continuoussubharmonicfunctions([2], p. 303).This is a global point of view. However, locally thesingularitiesdo have an influ-encewhencomputationis inexact,suchasin finite precision.Their neighborhoodis of positiveLebesguemeasure. Thehomotopicunfoldingof theresolventfield z � � A zI �� 1 is anextremelyrich domain,wherewe areable to follow the subtleinterplaybetweenthe continuousand thediscrete,throughtheactionof computation.


10. Annex

10.1. Convergenceof theNeumannseriesof a matrix

WeconsiderthematrixM I N andtheconditionof its invertibility. Let ν beany eigenvalueof N.It is clearthatM is invertibleiff ν )* 1 for all eigenvaluesν of N.Theconditionρ � N �b' 1 impliesthat � ν � ' 1. ThereforeM is invertibleandtheNeumannseries

� I N � � 1 I ∞∑k& 1� N � k

converges.Whenρ � N �,/ 1, M is still invertible iff ν )* 1. But theNeumannseriesdoesnot convergebecauseNk 0 ask � ∞. ρ � N � 1 is the radiusof convergenceof the Neumannseries(See[15], theo-rem2.21,p.101-102).TheNeumannseriesexpansionis thematrixequivalentto thegeometricseriesexpansionfor ascalarof moduluslessthan1: let m n 1, thenm� 1 11? n ∑∞k& 0 � n� k if f � n � ' 1.Thereis a significantdifferencebetweenthescalarandmatrix case.For a scalar, � n � 0 " n 0,but for amatrix,ρ � N � 0 doesnot impliesN 0: any nonzeronilpotent matrix satisfies:

ρ � N � 0 "_c l � 1 suchthatNl 0.ThereforetheNeumannseriesexpansionbecomesfinite:

� I N � � 1 I l � 1∑1� N � k !

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