for linear response eigenvalue problems A symmetric structure-preserving ΓQR algorithm Linear Algebra and its Applications

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Linear Algebra and its Applications

www.elsevier.com/locate/laa

A symmetric structure-preserving ΓQR algorithm for linear response eigenvalue problems

Tiexiang Li^a,1,Ren-Cang Li^b,∗,2, Wen-Wei Lin^c,3

aSchoolofMathematics,SoutheastUniversity,Nanjing,211189,People’s Republic ofChina

bDepartmentofMathematics,UniversityofTexasatArlington,P.O.Box19408, Arlington,TX76019,UnitedStates

cDepartmentofAppliedMathematics,NationalChiaoTungUniversity, Hsinchu 300,Taiwan

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received4January2016 Accepted2January2017 Availableonline10January2017 SubmittedbyH.Fassbender MSC:

15A18 15A23 65F15

Keywords:

Π^±-matrix Γ-orthogonality Structurepreserving ΓQR algorithm

Linearresponseeigenvalueproblem

Inthispaper,wepresentanefficientΓQR algorithmforsolv- ingthelinearresponseeigenvalueproblem_Hx=λx,where H is Π⁻-symmetric with respect to Γ0 = diag(In,−In).

BasedonnewlyintroducedΓ-orthogonaltransformations,the ΓQR algorithmpreservestheΠ⁻-symmetricstructureof_H throughoutthewholeprocess,andthusguaranteesthecom- putedeigenvaluestoappearpairwise(λ,−λ) astheyshould.

WiththehelpofanewlyestablishedimplicitΓ-orthogonality theorem,weincorporatetheimplicitmulti-shifttechniqueto acceleratetheconvergenceoftheΓQR algorithm.Numerical experimentsaregiven toshow theeffectivenessof thealgo- rithm.

©2017ElsevierInc.Allrightsreserved.

* Correspondingauthor.

E-mailaddresses:[email protected](T. Li),[email protected](R.-C. Li),[email protected] (W.-W. Lin).

1 SupportedinpartbytheNSFCgrants11471074and91330109.

2 SupportedinpartbyNSFgrantsDMS-1317330andCCF-1527104,andNSFCgrant11428104.

3 SupportedinpartbytheNationalCenterofTheoreticalSciences,andtheSTYauCentreattheNational ChiaoTungUniversity.

http://dx.doi.org/10.1016/j.laa.2017.01.005 0024-3795/©2017ElsevierInc.Allrightsreserved.

1. Introduction

Inthispaper, weconsiderthestandardeigenvalueproblemoftheform Hx≡

A B

−B −A x1

x2

=λx, (1.1)

whereAandB aren×nrealsymmetricmatrices.Werefertoitalinearresponseeigen- valueproblem(LREP).Anycomplexscalarλandnonzero2n-dimensionalcolumnvector xthatsatisfy(1.1)are calledaneigenvalueand itsassociatedeigenvector,respectively, and correspondingly,(λ,x) iscalled aneigenpair.

Our consideration of this problem is motivated by Casida’s eigenvalue equations in [1–4]. In computational quantum chemistry and physics, the excitation states and responsepropertiesofmoleculesandclustersarepredictedbythelinear-responsetime- dependent density functional theory. The excitation energies and transition vectors (oscillator strengths)of molecular systemscanbe calculatedby solvingCasida’s eigen- value equations [1–3]. There has been a great deal of recent work on and interest in developingefficient numericalalgorithmsandsimulationtechniquesforcomputingexci- tation responsesofmoleculesandformaterialdesignsinenergyscience[5–12].

Let

Γ₀=

In 0 0 −In

, Π ≡Π_2n=

0 In

In 0

. (1.2)

Thematrix_H in(1.1)satisfies Γ_0H =

A B B A

and_HΠ=−Π_H. (1.3)

Asaresultofthesecondequationin(1.3),if(λ,x) isaneigenpairof_H,i.e.,_Hx=λx, then (−λ, Πx) is alsoaneigenpair of_H, andifalsoλ∈/R, thenλ,¯ x¯

and

−λ, Π¯ x¯ areeigenpairsofH aswell,whereλ¯isthecomplexconjugateofλandx¯takesentrywise complexconjugation.

Previouslyin[5,6,13],LREP(1.1)waswell-studiedundertheconditionthatΓ0H is positivedefinite.Forthecase,alleigenvaluesof H arereal.Withoutthepositivedefinite condition,themethodsdevelopedin[5,6,13]arenotapplicable.

LetJn betheset ofalln×ndiagonalmatriceswith±1 onthediagonalandset Γ_2n ={diag(J,−J) : J ∈Jn}.

NotethatΓ₀= diag(I_n,−I_n)∈Γ_2n.Inthispaper,wewillstudyaneigenvalueproblem for which the condition that Γ_0H is positive definite is no longer assumed and it in fact includes LREP (1.1) as aspecial case. Specifically, we will consider the following eigenvalueproblem

Hx≡

A B

−B −A

x=λx (1.4a)

withthestructureproperty:

there isaΓ = diag(J,−J)∈Γ2n with J = diag(±1)∈ Jn such thatΓH =

J A J B J B J A

withJ A,J B∈R^n×n beingsymmetric. (1.4b) There are two reasonsfor consideringthis moregeneral eigenvalueproblem (1.4). The first reasonis thatthis includes (1.1), with/without Γ_0H being positive definite,as a specialcase,andthesecondoneisthattheintermediateeigenvalueproblemsinourlater iterativeΓQR algorithmforsolving(1.1)areofthis kind,i.e., withΓ =Γ0.

Itcanbeverifiedthatthesecond equationin(1.3),HΠ =−ΠH, stillholdsinthe caseof(1.4b).Thereforethesameresultsabouttheeigenvaluepatternwementionedfor (1.1)remainvalid.Namely,if(λ,x) isaneigenpair,then(−λ, Πx) isalsoaneigenpair, andifalsoλ∈/R,then ¯λ,x¯

and

−¯λ, Πx¯

areeigenpairsaswell.Anotherinteresting resultis about theΓ-orthogonalityamong the eigenvectorsof H. Specifically, fortwo eigenpairs (λ,x) and(μ,y) of H ifλ= ¯μ, then itholds thaty^HΓx= 0,where y^H is theconjugatetransposeofy. Thisisbecauseusing (1.4b),wehave

λy^HΓx=y^HΓHx=y^HH^HΓx= ¯μy^HΓx andthus(λ−μ)y¯ ^HΓx= 0 whichyields y^HΓx= 0 whenλ= ¯μ.

Thematrix H in(1.4)hassomeniceblockstructures.Infact,theeigenvalueproblem (1.4)canbe transformedintoaspecialHamiltonian eigenvalueproblem

0 J M J K 0

y₁ y₂

=λ y₁

y₂

withK=A−B, M=A+B, (1.5) y₁ = x₁−x₂, and y₂ = x₁+x₂. There are several existingstructure-preserving ap- proaches[14,15,8,16]thatcanbeappliedtosolvetheeigenvalueproblem(1.1),(1.4),or (1.5).

(a)AperiodicQR(PQR)algorithm [16]canbeusedto solvetheproducteigenvalue problem(J K)(J M)y₂=λ²y₂for(1.5).Notethatonlyy₂iscomputeddirectlythisway, buty₁canberecoveredthroughλy₁=J My₂ fornonzeroeigenvaluesλ.Forλ= 0, we maytakeanyy₁ suchthatKy₁= 0.Oncey₁andy₂areknown,x₁= (y₁+y₂)/2 and x₂= (y₂−y₁)/2 becomeavailable.Conceivably,thereisastabilityconcerninrecovering y₁foreigenvaluesλtinyinmagnitude.PQRusesorthogonaltransformations.Here,the n×n block structure in(1.4) is exploitedand the symmetry of the spectrum canbe preserved. However,the symmetric structures ofJ A and J B are destroyed duringthe iterations.

(b) The KQZ algorithm [8] can be appliedto solve H in (1.1). It uses orthogonal and Π-orthogonal transformations. The block structure HΠ = −ΠH is preserved

during the KQZ iteration. The reduced matrices J A and J B in (1.4b) are no longer symmetric (tridiagonal), but only Hessenberg. It is mathematically different from the PQR algorithm,buttheyhavethesimilar amountof computationalcosts.Laterat the end of Section5andinSection6,we will showthatthey aremoreexpensivethan the ΓQR algorithmto bepresented.

(c) AnHR process proposed by Brebner and Grad(BG) [14] is used to reduce the product eigenvalue problem (J K)(J M)y₂ = λ²y₂ to a pseudosymmetric form Cz = λ²Jz, where C is symmetric tridiagonal and J is the inertia sign-matrix of J K or J M. Thetridiagonalpseudosymmetry intheBGalgorithm is preservedduringtheHR iterations.TheBGalgorithmhasthesimilaramountofcomputationalcostsasourΓQR algorithm.However,ill-conditionedK andM maycausethenumericalinstabilityofthe BG algorithmduringconstructingJ.

(d) The symplectic QR-like algorithm [15] can also be applied to solve the special Hamiltonian eigenvalue problem (1.5). It uses symplectic transformations. Here the Hamiltonian matrixisreducedto acondensedHamiltonian form

H₁ H₃ H₂ −H₁

withH1

and H2 being diagonal and H3 being symmetric tridiagonal, and then the H3-block will be made to converge to aquasi-diagonal matrix duringthe iterative process. The symplectic QR-like algorithm does not really exploit the symmetry properties of J A and J B. Instability can occur when the symplectic Gaussian elimination matrices at somestepshavelargeconditionnumbers.Thisphenomenoncannotbeavoidedbecause to maintain theHamiltonian structure only theGaussian elimination withoutpivoting isallowed.Furthermore,thesymplecticmatricesQinthesymplecticQR-likealgorithm satisfyQJ Q=J,whereJ =Γ0Π.TheΓ-orthogonalmatricesQinourΓQR algorithm (to bepresented)satisfyQΠ =ΠQandQΓ Q=Γ withΓ,Γ∈Γ2n.

(e) TheHamiltonian QR-algorithm [17] uses symplecticorthogonal transformations butitisonlysuitableforaveryspecialHamiltonianeigenvalueproblemsuchasin(1.1) or (1.5)withrank(B) beingone,orasin(1.5)withrank(K),or rank(M) being one.

Themain taskofthispaperisto developiterativeΓQR algorithmsforsolving(1.4), whileexploitingtheinherentstructuresinH andΓ forbetternumericalefficiency,based on (Γ,Γ)-orthogonaltransformations withΓ,Γ∈Γ2n to bedefinedinSection2. The transformations preserve the symmetry structures inΓH and the diagonal structure ofΓ.TheseΓQR algorithmsmaybeviewedasextensionsoftheHRalgorithmproposed in[18],apioneeringworkforsolvingtheeigenvalueproblemofann×nmatrix A having theproperty thatthere existsaso-calledpseudo-orthogonalmatrixH inthesense that HJ H =J forsomeJ = diag(±1) andJ= diag(±1) suchthatH⁻¹_AH=Risupper triangular. TheHRalgorithm,however,cannotbe appliedtoLREP(1.4)directlyasit doesnotrecognizethe2×2 blockstructure,among others.

Throughout thispaper, weassumethat_H in(1.1)isnonsingular, andthus0 isnot itseigenvalue.

The restof this paperis organizedasfollows. InSection2,we introducesomebasic definitions and state their immediately implied properties. In Section 3, we first give

two kinds of Γ-orthogonal transformations, and then prove existence and uniqueness of the ΓQR factorization and propose an algorithm to compute the factorization for agiven matrix G with GΠ =−ΠG. In Section4, we present the Π⁻-upper Hessen- berg reduction/tridiagonalization and provethe implicit Γ-orthogonality theorem of a Π⁻-matrix G. In Section 5, we develop ΓQR algorithms for computing all eigenpairs of H and analyze their convergence with the goal of an efficient implicit multi-shift ΓQR algorithm.NumericalresultsoftheΓQR algorithmcomparedtotheotherexisting algorithmareshowninSection6.Finally,someconclusionsare drawninSection7.

Notation.R^n×misthesetofalln×mcomplexmatriceswithrealentries,Rⁿ=R^n×1, andR=R¹.WedenotebyInand0m×n(0m) then×nidentitymatrixandm×n(m×m) zeromatrix,respectively,andtheirsubscriptsmaybedroppediftheirsizescanbe read fromthecontext.Γ₀ andΠ_2n arereservedas givenby(1.2),andoftenthesubscriptto Π2nisdropped,too,whennoconfusionispossible.Thejthcolumnoftheidentitymatrix Iisej whosesizewillbe determinedbythecontext.WeshallalsoadoptMATLAB-like conventionto access theentriesof vectorsand matrices.Let i:j be theset ofintegers fromitojinclusive.ForavectoruandamatrixX,u_(j)isthejthentryofuandX_(i,j) isthe(i,j)thentry ofX;X_(I1,I2)isthesubmatrixof X consisting ofintersections ofall rowsi∈I1and allcolumnsj ∈I2;X isthetranspose ofX.Wedenotebyeig(A) the spectrumofmatrixA,anddiag(X,Y) isthe2-by-2 block-diagonalmatrixwithdiagonal blocksX andY.

2. Definitionsandpreliminaries

Inthissection, weintroduceseveralkindsofmatrixclassesand theiressentialprop- erties.RecallΠ2n definedin(1.2).

Definition2.1.LetG∈R^2n×2mwithm≤n.GiscalledaΠ^±-matrix if GΠ_2m=±Π_2nG, i.e., G=

G1 G2

±G2 ±G1

with G₁, G₂∈R^n×m. (2.1) DenotebyΠ^±_2n×2mthesetofall2n×2mΠ^±-matrices,andΠ^±_2n :=Π^±_2n×2n forshort.

Inthisdefinitionandthosebelow,tosavespaceandavoidrepetitions,weoftenpack twostatementsintoone:oneforΠ⁺-matrixandtheotherforΠ⁻-matrix.Inthesame spirit, in definitions/statementsin the rest of this paper, any of them with phrasesin parenthesesisunderstoodthatthephrasescanbeused toreplace thephrasesimmedi- atelybeforeforanotherdefinition/statement.

WesayamatrixX,possiblynonsquare,isupperHessenbergifX_(i,j)= 0 fori> j+ 1, upper triangular if X_(i,j) = 0 for i > j, and diagonal if X_(i,j) = 0 for i = j. This is consistent with thestandard definitionsof upper Hessenberg, upper triangular, and diagonalmatriceswhichareusuallyforsquare matrices.

Aquasi-uppertriangularmatrixmeansthatitisablockuppertriangularmatrixwith diagonalblocksbeing1×1 or2×2.Similarly,aquasi-diagonalmatrix meansthatitis ablockdiagonalmatrixwith diagonalblocksbeing1×1 or2×2.

Definition 2.2.LetG∈Π^±_2n×2mas in(2.1).

1. GisΠ^±-upper Hessenberg ifG₁ isupperHessenbergandG₂ isuppertriangular.

GisunreducedΠ^±-upper Hessenberg ifG1 isunreducedupperHessenberg.

2. GisΠ^±-upper (Π^±-quasi-upper)triangular ifG₁isupper(quasi-upper)triangular andG₂ isstrictlyuppertriangular.

Denote byU^±_2n×2m (qU^±_2n×2m)theset ofall2n×2mΠ^±-upper(Π^±-quasi-upper) triangularmatrices,and write,forshort,U^±2n :=U^±2n×2n and qU^±2n:= qU^±2n×2n. 3. GisΠ^±-diagonal (Π^±-quasi-diagonal)ifG₁ isdiagonal(quasi-diagonal)andG₂is

diagonal.

Denote by D^±2n×2m (qD^±2n×2m) the set of all 2n×2n Π^±-diagonal (Π^±-quasi- diagonal)matrices,andwrite,forshort,D^±2n:=D^±_2n×2n andqD^±2n:= qD^±2n×2n. Definition 2.3.

1. LetG∈Π⁺_2n as in(2.1). Gis Π⁺-symmetric (Π⁺-sym-tridiagonal), ifG1 issym- metric(symmetrictridiagonal)andG2 issymmetric(diagonal).

2. LetG∈Π⁻_2n asin(2.1).GisΠ⁻-symmetric(Π⁻-sym-tridiagonal)withrespectto Γ ∈Γ2n ifΓ GisΠ⁺-symmetric(Π⁺-sym-tridiagonal).

3. G is unreduced Π^±-sym-tridiagonal if it is Π^±-sym-tridiagonal and G1 is unre- duced.

The following propositions are direct consequences of Definitions 2.1–2.3 and are rather straightforwardto verify.

Proposition 2.1.

(i) G∈Π^±_2n×2m if andonly if Γ G∈Π^∓_2n×2m andGΓ ∈Π^∓_2n×2m for any Γ ∈Γ_2n andΓ∈Γ2m.

(ii) The inverse of a Π^±-(upper triangular) matrix is still a Π^±-(upper triangular) matrix.

(iii) The product GG of two 2n×2n matrices G and G in their respective categories belongs totheoneaslisted inthefollowingtables:

PPP^G PPP^{G Π+ Π−}

Π⁺ Π⁺ Π⁻ Π⁻ Π⁻ Π⁺

PPP^G PPP^{G U+ U−}

U⁺ U⁺ U⁻ U⁻ U⁻ U⁺

Proposition 2.2. Let G∈Π⁻_2n.Then G has 2n eigenvalues,appearing in pairs(λ,−λ) forrealorpurelyimaginaryeigenvaluesλandinquadruples(±λ,±¯λ)forcomplexeigen- valuesλ.

Proof. ByDefinition 2.1,itholdsthat

det(G−λI) = det(ΠG−λΠ) = det(GΠ+λΠ) = det(G+λI) = 0.

Theassertionfollows immediately. 2

Definition2.4.Q∈Π⁺_2n×2m isΓ-orthogonal withrespectto Γ ∈Γ_2n ifΓ:=QΓ Q∈ Γ_2m.DenotebyO^Γ2n×2mthesetofall2n×2m Γ-orthogonalmatrices,andO^Γ2n:=O^Γ2n×2n

forshort.

Often,forshort,wemaysayQ∈Π⁺_2n×2m isΓ-orthogonal, bywhichwemeanthere isΓ ∈Γ_2n thathastherequirementofthedefinitionsatisfied.Similarly,wemaysimply sayQisΓ-orthogonal.Thesameunderstandingappliesto theexpression Q∈O^Γ_2n×2m. Proposition 2.3. Let Q_i ∈ O^Γ2n with respect to Γ_i ∈ Γ_2n for i = 1,2, and suppose Γ₂=Q₁Γ₁Q₁.

(i) Q1Q2∈O^Γ2n with respecttoΓ1.

(ii) Qi isnonsingular andQ⁻¹_i =Γi+1Q_i Γi,where Γ3:=Q₂Γ2Q2. (iii) Ifalso Q_i∈U⁺2n,thenQ_i= diag(J_i,J_i)forsome J_i∈Jn.

Proof. Items(i)and(ii)followfromDefinition 2.4directly.Foritem(iii),Q⁻_i¹∈U⁺2n by Proposition 2.1(ii).Ontheotherhand,byitem(ii),Q⁻_i¹=Γi+1Q_i ΓiwhichisΠ⁺-lower triangular.ThisimpliesthatQi = diag(Ji,Ji) forsomeJi∈Jn,completingtheproofof item (iii). 2

3. ΓQR factorization

Definition 3.1. G=QR iscalled aΓQR factorization ofG ∈Π⁻_2n×2m withrespect to Γ ∈Γ2n ifR∈U⁻2n×2m andQ∈O^Γ2n withrespecttoΓ or ifR∈U⁻2mandQ∈O^Γ_2n×2m withrespectto Γ.

The case when R ∈ U⁻2m and Q ∈ O^Γ_2n×2m with respect to Γ in this definition correspondsto theso-calledskinny QRfactorizationinnumericallinearalgebra.

Definition3.2.LetM =

M1 M2

−M2 −M1

∈Π⁻_2n,andset

M1i= (M1)_(1:i,1:i), M2i= (M2)_(1:i,1:i).

M_1i M_2i

−M2i −M1i

iscalledtheithΠ⁻-leadingprincipalsubmatrix ofM anditsdetermi- nantiscalled theithΠ⁻-leadingprincipalminor ofM.

The next theorem shows that almost every Π⁻-matrix G ∈ Π⁻_2n×2m has a ΓQR factorization with respect to a given Γ ∈ Γ2n and the factorization is unique if it is requiredthatthetop-leftquarteroftheR-factorhaspositivediagonalentries.

Theorem 3.1.Supposethat G∈Π⁻_2n×2m(m≤n) hasfull column rankandΓ ∈Γ2n. (i) If G=QR=QR (with Q,Q ∈O^Γ2n×2m andR,R∈U⁻2m)are twoΓQR factoriza-

tionsofG withrespecttoΓ,then

QΓQ=QΓ Q∈Γ2m

andthereisa Π⁺-diagonal matrixD= diag(J,J)with J ∈Jm suchthat Q=QD andR=DR.Inparticular,ifthetop-leftquartersofRandRhavepositivediagonal entries,thenD=I2m,Q=Q, andR=R.

(ii) Ghas aΓQRfactorizationwith respecttoΓ ifandonlyif noΠ⁻-leadingprincipal minorof GΓ Gvanishes.

Proof. Wefirstproveitem(i).LetΓ =QΓ QandΓ=QΓQ. Fromtheassumption we have

ΓR=QΓ QR=QΓQR ⇒ QΓQ=ΓRR⁻¹. Similarly,QΓ Q=ΓRR ⁻¹.Therefore

ΓRR ⁻¹=QΓ Q= (QΓQ) = (ΓRR⁻¹)=R⁻RΓ. (3.1) Because ΓRR ⁻¹ ∈U⁻2m and at the same time R⁻RΓ is Π⁻-lower triangular, we concludethatRR ⁻¹ andR⁻R mustbediagonal.Set

D=RR ⁻¹∈Π⁺_2m (3.2)

whichimpliesR⁻R= (RR ⁻¹)⁻=D⁻¹.Thus,from(3.1)and(3.2),wehaveΓD= QΓ Q andΓD⁻¹=QΓQ. ThisimpliesΓ=DΓD, andthus

D²=I2m, Γ=Γ ∈Γ2m.

So D= diag(J,−J) forsomeJ ∈Jmand R =DR. Furthermore,sinceG=QR=QR hasfullcolumnrank,itfollowsthatQ=QD.

Now if also the top-left quarters of R and R have positive diagonal entries, then R=DRimpliesD=I2m,as expected.

Nextwe proveitem(ii).

Necessity.LetP bethepermutationmatrix

P = [e₁,e₃,· · ·,e_2m−1|e₂,e₄,· · ·,e_2m]∈R^2m×2m. (3.3) SupposethatG=QR isa ΓQR factorization with respectto Γ,and letΓ =QΓ Q.

Then

PGΓ GP = (PRP)(PΓP)(PRP) =:R_pΓ_pR_p, whereRp=PRP isuppertriangularandΓ_p =PΓP isdiagonal,asin

R_p=

⎡

⎣

R11 · · · R1m

... . .. ... 0 · · · R_mm

⎤

⎦, Γ_p =

⎡

⎣

Γ₁₁ 0 . ..

0 Γ_mm

⎤

⎦ (3.4)

with R_ij ∈Π⁻₂, R_ii =

di 0 0 −di

, and Γ_ii ∈Γ₂ for i,j = 1,· · ·,m. Since G hasfull columnrank,itfollowsthatdet(R_ii)= 0 fori= 1,· · ·,m.Therefore,thereisnoleading principal minor of PGΓ GP of even order vanishes, i.e., no Π⁻-leading principal minorofGΓ Gvanishes.

Sufficiency.SupposethatG∈Π⁻_2n×2m andnoΠ⁻-leadingprincipalminor ofM :=

GΓ Gvanishes.Thenthere isanLUfactorizationofMp:=PM P,i.e.,Mp=LpRp

withnonsingular

Lp=

⎡

⎢⎢

⎣

I2 0

L21 I2

... . .. . ..

L_m1 · · · L_m,m−1 I₂

⎤

⎥⎥

⎦, Rp=

⎡

⎢⎣

R11 · · · R1m

. .. ...

0 Rmm

⎤

⎥⎦, (3.5)

whereL_ij ∈Π⁺₂ andR_ij ∈Π⁻₂.DecomposeR_pas

Rp=

⎡

⎢⎣

R11 0 . ..

0 R_mm

⎤

⎥⎦

⎡

⎢⎢

⎢⎣

I2 R12 · · · R1m

. .. ... ... . .. R_m−1,m

I2

⎤

⎥⎥

⎥⎦=:DpRp (3.6)

withRij =R⁻_ii¹Rij ∈Π⁺₂.Thenwehave

Mp=LpRp=LpDpRp=R_pD_pL_p =M_p. (3.7)

The uniquenessoftheLUfactorization impliesthatL_p =R_p.Since Mp issymmetric, it follows thatD_p = diag({R_ii}^mi=1) is symmetric. Because R_ii ∈ Π⁻₂, R_ii must be of theform Rii =

d_i 0 0 −d_i

fori= 1,· · ·,m.Write R_ii=

|di| 0

0

|di|

sgn(di) 0

0 −sgn(di) |di| 0

0

|di|

and denote D^1/2_p = diag(

|d_i| 0

0

|d_i| m

i=1

), Γ_p = diag(

sgn(d_i) 0 0 −sgn(d_i)

m i=1

). (3.8) From (3.7)and(3.8)wehave

GΓ G=P M_pP = (P L_pD^1/2_p P)(P Γ_pP)(P D^1/2_p L_pP) =:RΓR, (3.9) where R=P D^1/2p L_pP∈U⁺2mand Γ=P Γ_pP.Let

Q₋ :=GR⁻¹Γ ∈Π⁺_2n×2m. (3.10) Withthehelpof(3.9), itcanbe verifiedthat

Q₋Γ Q₋= (ΓR⁻G)Γ(GR⁻¹Γ) =ΓR⁻(RΓR)R⁻¹Γ=Γ whichsaysQ₋∈O^Γ_2n×2m. Therefore

G= (GR⁻¹Γ)(ΓR) =:Q₋R₋ with R₋∈U⁻2m (3.11) to giveG=QR,aΓQR factorization. 2

Our goal in this paper is to develop a structure-preserving QR-like algorithm to compute all eigenvalues of H ∈ Π⁻_2n. The basic idea is to calculate a sequence of Γ-orthogonalmatrices{Qi},basedonaΓQR factorization,suchthat

Hi+1=Q⁻_i¹HiQi, Q_i ΓiQi=Γi+1 fori= 1,2, . . . ,

where initially H0 = H. For this purpose, at first, we introduce two elementary Γ-orthogonal transformations which will be used to zero out a specific entry or en- tries of a vector. Specifically, given Γ ∈ Γ2n and u ∈ R²ⁿ, we seek Q ∈ O^Γ2n to zero out someportion of u. Twodifferent kindsof matrices Qwill be used to deal with all possible scenariosthatwilloccurincomputingtheΓQR factorizationsinAlgorithm 3.1 later.

Leta∈R^k(1≤k≤n),J ≡diag(j1,...,jk)∈Jk.AssumethataJa= 0.LetPabea permutationwhichinterchangesrow1 andarowr(2≤r≤k)ofJ suchthatˆj1aJa= ˆ

j1aˆJaˆ > 0, where aˆ = Paa and J= PaJ Pa = diag(ˆj1,...,ˆjk). A Householder-like transformationisproposedby[18]to zeroouttheelementsofaˆ_(2:k) asfollows. Let

H(a)⁻¹=I−ˆj₁

β(ˆa−αe₁)(ˆa−αe₁)J , H(a)⁻¹=H(a)⁻¹P_a, (3.12a) whereα=−sign(ˆa₍₁₎)

ˆ

j₁aˆJaˆandβ =α[α−aˆ₍₁₎].Thenitcanbeverifiedthat H(a)⁻¹a= [H(a)⁻¹P_a]a=H(a)⁻¹aˆ=αe₁, H(a)JH(a) =J . (3.12b) Hyp_Householder (hyperbolic Householder) transformation: Suppose 1 ≤ < m ≤ n, u∈R²ⁿ and Γ = diag(γ₁,. . . ,γ_n,−γ₁,. . . ,−γ_n)∈Γ_2n. Therearetwocases:

Case 1. a←u₍:m)with =n+ andm=n+m,J = diag(γ,. . . ,γm);

Case 2. a←u_(:m),J = diag(γ,. . . ,γ_m).

Using(3.12)weconstructahyperbolicHouseholderΓ-orthogonaltransformationQwith respectto Γ throughitsinverseby

Q⁻¹ =

Qh( :m;u), case 1;

Qh(:m;u), case 2,

:= diag(I₋₁,Hˆ(a)⁻¹, I_n−m, I₋₁,Hˆ(a)⁻¹, I_n−m). (3.13) Thenitholdsthat

Q⁻¹u= ˆuwith

uˆ₍+1:m)= 0, case 1;

uˆ(+1:m)= 0, case 2, (3.14)

andQΓ Q=Γ,whereΓ= (γ₁,· · ·,γ_n,−γ₁,· · ·,−γ_n) isgivenby γ_s =γs , s= 1,· · ·, −1 andm+ 1,· · ·, n,

γ_{s+ −1}= ˆjs, s= 1,· · ·, m−+ 1. (3.15) Hyp_Givens (hyperbolic Givens) transformation: Suppose 1 ≤ ≤ n, u ∈ R²ⁿ and Γ = diag(γ1,· · ·,γn,−γ1,· · ·,−γn)∈Γ2n.Letα←u₍₎,β ←u_(n+).Define

⎧⎨

⎩

c= ^√₁¹_−r2, s= ^√₁^r_−r2 with r= ^β_α, if |α|>|β|,

c= ^√₁^r_−r2, s= ^√₁¹_−r2 with r= ^α_β, if |α|<|β|. (3.16)

Algorithm3.1ΓQR factorization.

Input: G∈Π⁻_2n×2m,Γ = diag(J,−J)∈Γ2nwithJ= diag(γ1,· · ·,γn),n←2n;

Output: Q∈O^Γ2nwithrespecttoΓ,Γ=QΓ Q∈Jn,andR∈U⁻2msuchthatG=QR;

1:Q←I2n,Γsav←Γ; 2:for= 1:mdo

3: ←n+,u←G_(:,);

4: computeHyp_HouseholderΓ-orthogonaltransformation: Q⁻¹= Qh( : n;u) withrespecttoΓ (by(3.13));

5: Q←QQ, G←Q⁻¹G,Γ ← −Γ(by(3.15));

6: α←G(,),β←G(,);

7: computeHyp_GivensΓ-orthogonaltransformation:Q⁻¹=Qg(;α,β) withrespecttoΓ(by(3.17));

8: Q←QQ, G←Q⁻¹G,Γ ←Γ(by(3.18));

9: u←G_(:,);

10: computeHyp_HouseholderΓ-orthogonaltransformation:Q⁻¹=Qh(:n;u) withrespecttoΓ (by (3.13));

11: Q←QQ, G←Q⁻¹G,Γ ←Γ(by(3.18));

12: end for

13: return Q←Q(:,[1:m,n+1:n+m]),Γ←Γ,Γ ←Γsav,andR=

G_(1:m,:) G(n+1:n+m,:)

.

We construct a hyperbolic Givens Γ-orthogonal transformation with respect to Γ throughitsinverseby

Q⁻¹=Qg(;α;β) = C S

S C

∈Π_2n⁺, (3.17)

where C is obtained from In by resetting C_(,) = c and S from On×n by resetting S_(,)=−s.Thenwe have

Q⁻¹u= ˆu with uˆ_(n+)= 0, and QΓ Q=Γ,where

γ=δγ, δ=c²−s²=±1,

γ_j=γj, j =. (3.18)

Remark3.1.

(i) UtilizingthespecialstructureofΠ⁻-matrixG∈Π⁻_2n×2m,Q⁻¹atlines4,7and10 ofAlgorithm 3.1eliminatesthe(n++ 1: 2n,) and(+ 1:n,) orthe(n+,)th and(,n+)thorthe(+ 1:n,) and(n++ 1: 2n,) entriesofGsimultaneously, for= 1,. . . ,m.

(ii) Upon exit, Algorithm 3.1 computes G = QR, where Q ∈ O^Γ2n is a Γ-orthogonal matrixwith respect to Γ and R∈U⁻_2n×2m. It is worthnotingthat Γ isunknown beforeG=QRiscomputedbutitisunique,accordingtoTheorem 3.1(i).

In the following, we use a small example with n = 3 and m = 2 by Wilkinson’s diagramto illustratetheeliminationprocessincomputingaΓQR factorizationofG.

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

× × × ×

× × × ×

× × × ×

× × × ×

× × × ×

× × × ×

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

Qh(4:6;u)

−−−→=Q⁻¹₁

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

× × × ×

× × 0 ×

× × 0 ×

× × × × 0 × × × 0 × × ×

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

Qg(1;α,β)

−−−→=Q⁻¹₂

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

× × 0 ×

× × 0 ×

× × 0 × 0 × × × 0 × × × 0 × × ×

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

Qh(1:3;u)

−−−→

=Q⁻¹3

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

× × 0 ×

0 × 0 ×

0 × 0 ×

0 × × ×

0 × 0 ×

0 × 0 ×

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

Qh(5:6;u)

−−−→

=Q⁻¹4

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

× × 0 ×

0 × 0 ×

0 × 0 0

0 × × ×

0 × 0 ×

0 0 0 ×

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

Qg(2;α,β)

−−−→

=Q⁻¹₅

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

× × 0 ×

0 × 0 0

0 × 0 0

0 × × ×

0 0 0 ×

0 0 0 ×

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

Qh(2:3;u)

−−−→

=Q⁻¹₆

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

× × 0 ×

0 × 0 0

0 0 0 0

0 × × ×

0 0 0 ×

0 0 0 0

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

In general, after m steps we have computed 3m Γ-orthogonal matrices Q⁻₁¹,. . . ,Q⁻_3m¹ suchthat(Q⁻¹_3m× · · · ×Q⁻¹₁ )G=Ris aΠ⁻-uppertriangular.

Remark3.2. Theorem 3.1(ii)reveals thatalmost allmatrices inΠ⁻_2n×2m(m≤n) have ΓQR factorizations. In practice, for a given 2n×2m Π⁻-matrix G, one way to con- struct itsΓQR factorization withrespect to given Γ ∈Γ2n is throughreducingGto a 2n×2m Π⁻-upper triangular matrix by asequence of Γ-orthogonal transformations:

Hyp_Householder Γ-orthogonal transformationsand Hyp_Givens Γ-orthogonal trans- formations.TheHyp_Householdertransformationin(3.12a)maynotexistifaˆJaˆ= 0.

Similarly,theHyp_Givenstransformationin(3.17)maynotexistif|α|=|β|.In[19],it issaidthatthesecasescanoccurwhenthematrixisartificiallydesigned.Thereisclearly anumericalstabilityissueifaˆJaˆ≈0 or|α| ≈ |β|.Thedangerofseverecancellationcan occur andisdiscussedin[20,21].Ifadangerouscancellationoccursat somethstepof Algorithm 3.1,itisrecommendedtopre-multiply thecurrentGbyarandomlygenerated Γ-orthogonal Q⁻¹ with QΓQ =Γ. Then we set G←Q⁻¹G, Γ ← Γ, and continue performingAlgorithm 3.1from thethstep. Itusually cansuccessfully circumventthe cancellationbythis[20,21].