Computing selected eigenvalues of sparse unsymmetric matrices using subspace iteration

Transcription

1 RAL Computing seected eigenvaues of spase unsymmetic matices using subspace iteation by I. S. Duff and J. A. Scott ABSRAC his pape discusses the design and deveopment of a code to cacuate the eigenvaues of a age spase ea unsymmetic matix that ae the ight-most, eft-most, o ae of agest moduus. A subspace iteation agoithm is used to compute a sequence of sets of vectos that convege to an othonoma basis fo the invaiant subspace coesponding to the equied eigenvaues. his agoithm is combined with Chebychev acceeation if the ight-most o eft-most eigenvaues ae sought, o if the eigenvaues of agest moduus ae nown to be the ight-most o eft-most eigenvaues. An option exists fo computing the coesponding eigenvectos. he code does not need the matix expicity since it ony equies the use to mutipy sets of vectos by the matix. Sophisticated and nove iteation contos, stopping citeia, and estat faciities ae povided. he code is shown to be efficient and competitive on a ange of test pobems. Centa Computing Depatment, Atas Cente, Ruthefod Appeton Laboatoy, Oxon OX11 0QX. August 1993.

2 1 Intoduction We ae concened with the pobem of computing seected eigenvaues and the coesponding eigenvectos of a age spase ea unsymmetic matix. In paticua, we ae inteested in computing eithe the eigenvaues of agest moduus o the ight-most (o eft-most) eigenvaues. his pobem aises in a significant numbe of appications, incuding mathematica modes in economics, Maov chain modeing of queueing netwos, and bifucation pobems (fo efeences, see Saad 1984). Athough agoithms fo computing eigenvaues of spase unsymmetic matices have eceived attention in the iteatue (fo exampe, Stewat 1976a, Stewat and Jennings 1981, Saad 1980, 1984, 1989), thee is a notabe ac of genea pupose obust softwae. he best-nown codes ae SRRI (Stewat 1978) and LOPSI (Stewat and Jennings 1981). Both SRRI and LOPSI use subspace iteation techniques and ae designed to compute the eigenvaues of agest moduus. Many ea pobems, howeve, equie a nowedge of the ight-most eigenvaues. Fo exampe, common bifucation pobems invove computing the eigenvaue λ of agest ea pat (the ight-most eigenvaue) of a stabiity matix and then detecting when Re(λ) becomes positive as the matix changes (see exampe 2 in Section 3). In the Hawe Suboutine Libay, outine EA12 uses a subspace iteation method combined with Chebychev acceeation (Rutishause 1969) to cacuate the eigenvaues of agest moduus and the coesponding eigenvectos of a age spase ea symmetic matix. hee is no anaogous outine in the Hawe Suboutine Libay fo the unsymmetic pobem. In the NAG Libay, outine F02BCF cacuates seected eigenvaues and eigenvectos of ea unsymmetic matices by eduction to Hessenbeg fom, foowed by the QR agoithm and invese iteation fo seected eigenvaues whose modui ie between two use-suppied vaues. his outine is intended fo dense matices since a the enties of the matix (incuding the zeo enties) must be passed by the use to the outine, which stoes the matix as a two-dimensiona aay, maing it unsuitabe fo age spase matices. Since it was the intention that EB12 shoud povide a code fo unsymmetic pobems that was anaogous to the Hawe code EA12 fo symmetic pobems, EB12 uses subspace iteation techniques and in the design of the code we have not consideed empoying any of the othe methods fo computing seected eigenvaues and eigenvectos of age unsymmetic matices such as Anodi s method and the unsymmetic Davidson s method which have been discussed ecenty in the iteatue (fo exampe, see Saad 1980, Ho 1990, Ho, Chatein, and Bennani 1990, and Sadane 1991). his pape descibes the agoithms empoyed by EB12 and iustates the use of EB12 on epesentative test pobems. In Section 2 we intoduce the agoithms and discuss some of the design featues of the code. he esuts of using EB12 to find seected eigenvaues and the coesponding eigenvectos of a set of test exampes taen fom pactica pobems ae pesented in Section 3. hese esuts iustate the effect of vaying the code s paamete vaues and demonstate the supeioity of Chebychev acceeated subspace iteation ove simpe subspace iteation fo those pobems whee the ight-most (o eft-most) eigenvaues coincide with those of agest moduus. Concuding emas ae made in Section 4. 1

3 2 he agoithm Let A be a ea n n matix with eigenvaues λ, λ,..., λ 1 2 n odeed so that λ λ... λ. (2.1) 1 2 n Let X be a subspace of dimension m with m < n (in genea, m < < n). If λ > λ then, unde mid m m+1 estictions on X, as inceases, the subspaces A X tend towad the invaiant subspace of A coesponding to λ, λ,..., λ (a poof is povided by Stewat 1975). he cass of methods based on 1 2 m using the sequence of subspaces A X, = 0,1,2,... incudes subspace (o simutaneous) iteation methods. In the specia case m = 1, the subspace iteation method educes to the powe method in which the dominant eigenvecto of A is appoximated by a sequence of vectos A x = 0,1,2,... Subspace iteation methods ae paticuay suitabe when the matix A is age and spase, since this cass of methods ony equies the mutipication of sets of vectos by A. Stating with an n m matix X 0 whose coumns fom a basis fo X, the subspace iteation method descibed by Stewat (1976a) fo a ea unsymmetic matix A geneates a sequence of matices X accoding to the fomua AX = X R, (2.2) whee R is an uppe tiangua matix chosen so that X has othonoma coumns. his coesponds to appying the Gam-Schmidt othogonaisation pocess to the coumns of AX. It is staightfowad to veify the eationship A X 0 = XRR 1...R 1, (2.3) so that the coumns of X fom a basis of A X. Convegence of the i th coumn of X to the i th basis vecto of the invaiant subspace of A coesponding to λ 1, λ 2,..., λ m is inea with convegence atio max{ λ i/λ i 1, λ i+1/λ i }, which may be intoeaby sow. A faste ate of convegence may be achieved by pefoming a Schu-Rayeigh-Ritz (SRR) step (Stewat 1976a) in which B = X AX is fomed and educed by a unitay matix Z to the ea Schu fom (2.4) = Z B Z, (2.5) whee is a boc tiangua matix, in which each diagona boc ( ) ii is eithe of ode 1 o is a 2 2 matix having compex conugate eigenvaues, with the eigenvaues odeed aong the diagona bocs in descending ode of thei modui. he matix X is then epaced by Xˆ = XZ. Fo an abitay matix X 0, if λ i 1 > λ i > λ i+1, the i th coumn of Xˆ wi in genea convege to the i th basis vecto of the invaiant subspace of A coesponding to λ 1, λ 2,..., λ m ineay with convegence atio λ m+1/λ i. hus convegence is faste and the fist coumns of Xˆ tend to convege moe quicy than the ate coumns. If the eigenvaues of A of agest modui ae equied, it is usua to iteate with m (m > ) tia vectos, the additiona vectos being caed guad vectos. he sowest ate of convegence wi be fo the th basis vecto, which has a convegence atio λ /λ. m+1 he pupose of the othogonaisation of AX (2.2) is to maintain inea independence among the coumns of AX. o educe oveheads, it shoud not be done unti thee is eason to beieve that some coumns have become ineay dependent. hus, in pactice, AX is epaced by A X fo some 1 ( = ()), and each iteation of the subspace iteation agoithm then consists of fou main steps: 2

4 () 1. Compute A X. () 2. Othonomaise A X. 3. Pefom an SRR step. 4. est fo convegence. Fom the computed eigenvaues of, the coesponding eigenvectos of can be detemined by a simpe bac-substitution pocess (see Petes and Wiinson 1970). he eigenvectos of can be used to obtain appoximate eigenvectos of A coesponding to the conveged eigenvaues. o see this, et w denote the eigenvecto of coesponding to λ. hen i i w = λ w. (2.6) i i i Fom (2.4) and (2.5), and hence = Xˆ AXˆ, (2.7) Xˆ (AXˆ w λ Xˆ w ) = 0. (2.8) i i i It foows that if y = Xˆ w, then as inceases, (y, λ ) conveges to the i th eigenpai of A. i i i i As aeady noted, in pactice the simpe subspace iteation agoithm uses A in pace of A. One possibe way of impoving the convegence ate achieved by the subspace iteation agoithm is to epace A by an iteation poynomia p (A). he use of Chebychev poynomias to acceeate the convegence of subspace iteation was suggested by Rutishause (1969) fo symmetic pobems. Fo the unsymmetic pobem, Saad (1984) discussed how the technique can be extended to find the ight-most (o eft-most) eigenvaues of a ea unsymmetic matix and to acceeate the convegence of the simpe subspace iteation agoithm when the eigenvaues of agest modui ae aso the ight-most (o eft-most) eigenvaues. We use many of the ideas poposed by Saad. Suppose we want to find the ight-most eigenvaues λ, λ,..., λ of A. Let E(d, c, a) denote an eipse with cente d, 1 2 foci d c and d + c, mao semi-axis a, and which is symmetic with espect to the ea axis (since A is ea the spectum of A is symmetic with espect to the ea axis) so that d is ea and a and c ae eithe ea o puey imaginay. Suppose E(d, c, a) contains the set S of unwanted eigenvaues λ +1, λ +2,..., λ m. he Chebychev acceeated iteation agoithm then chooses the iteation poynomia p (λ) to be the poynomia given by [(λ d)/c] p (λ) =, (2.9) [(λ d)/c] whee (λ) is the Chebychev poynomia of degee of the fist ind. his choice is made since the maximum moduus of p (λ) within the eipse is sma compaed to its moduus on the sought-afte eigenvaues. he denominato [(λ d)/c] in (2.9) is a scaing facto and λ is temed the efeence eigenvaue (Ho 1990 and Ho, Chatein, and Bennani 1990). In pactice, since λ is not nown, it is epaced by some appoximation γ, caed the efeence point; this is discussed in Section 2.7. Associated with each eigenvaue λ ε S is a convegence facto 2 2 (λ d) + ((λ d) c ) 2 2 (λ d) + ((λ d) c ) 1 2 R (d, c) =. (2.10) 1 2 3

5 he choice of d, c, a which give an eipse E(d, c, a) encosing a λ ε S and which minimises max R (d, c) defines the optima eipse. λ ε S Foming p (A) expicity may be avoided by computing the coumns of the matix p (A)X the thee-tem ecuence eation fo Chebychev poynomias using (λ) = 2λ (λ) (λ), q = 1,2,... (2.11) q+1 q q 1 with (λ) = 1, (λ) = λ. o see how (2.11) is used, fo an abitay vecto z, et z = p (A)z q q 0 Defining σ = ρ /ρ with ρ = [(λ d)/c], it foows fom (2.11) that the vecto z may be q+1 q q+1 q q q computed fo q = 1,2,... using the ecuence whee σ q+1 z = 2 (A d I)z σ σ z (2.12) q+1 q q+1 q q 1 c 1 σ q+1 =, (2.13) 2/σ σ 1 q σ with σ = c/(λ d) and z = (A d I)z. We note that if c is puey imaginay, povided the c efeence point γ used to appoximate λ is ea, the above ecusion can be caied out in ea aithmetic since in this case the scaas σ i ae puey imaginay and hence σ q+1/c and σ q+1σ q ae ea. It can be shown (Saad 1984) that if E(d, c, a) is the optima eipse and p (A) defined by (2.9) is used in the subspace iteation agoithm in pace of A, convegence is to the invaiant subspace coesponding to the eigenvaues of A outside the eipse and the convegence ate fo the i th basis vecto is η i whee 2 2 a + (a c ) 1 2 η i =, 1 i m, (2.14) a + (a c ) i i whee E(d, c, a i) is the eipse with cente d, foci d c and d + c, and mao semi-axis a i which passes though λ. If the m+1 ight-most (o eft-most) eigenvaues ae aso the m+1 eigenvaues of agest i moduus, the convegence ate η can be much bette than the vaue λ /λ achieved by the simpe i m+1 i subspace iteation agoithm. he effect of this faste ate of convegence is iustated in Section 3 and is expoited in EB12 by empoying Chebychev poynomias when the ight-most (o eft-most) eigenvaues ae sought and when the eigenvaues of agest moduus ae aso the ight-most (o eft-most) eigenvaues. he use of Chebychev poynomias is one way in which EB12 is a moe genea and fexibe code than the codes LOPSI and SRRI, which ae ony abe to compute the eigenvaues of agest moduus. Impicit in the above bief discussion of subspace iteation, thee ae many pactica questions such as how to choose the subspace dimension m fo a given vaue of, how to stat the iteation pocess, how to choose the degee of the iteation poynomia p (A), how to othonomaise a set of vectos, how to constuct the optima eipse E(d, c, a), and how to test fo convegence of an eigenvaue. We sha discuss these and othe questions and how we dea with them in EB12 in some detai in ate sections. Howeve, omitting these pobems fo the pesent, the agoithm empoyed by EB12 has the foowing genea stuctue. 4

6 1. Stat: Choose the subspace dimension m and an n m matix X with othonoma coumns. Set = 1, p (λ) = Iteation: Compute X p (A)X. 3. SRR step: Othonomaise the coumns of X. Compute B = X AX. Reduce B to ea Schu fom = Z BZ, whee each diagona boc ii is eithe of ode 1 o is a 2 2 matix having compex conugate eigenvaues, with the eigenvaues odeed aong the diagona bocs. Set X XZ. 4. Convegence test: If the fist coumns of X is a satisfactoy set of basis vectos spanning the invaiant subspace coesponding to the sought-afte eigenvaues of A then stop, ese detemine the degee of the iteation poynomia p (λ) fo the next iteation. If the eigenvaues of agest moduus ae equied and they ae aso the ight-most (o eft-most) eigenvaues, o if the ight-most (o eft-most) eigenvaues ae equied, find the eipse E(d,c,a) update the efeence point γ, and set p accoding to equation (2.9) with λ epaced by γ. Othewise, set p (λ) = λ. Go to 2. In the foowing subsections we discuss how this agoithm is impemented in EB12. In Section 2.1 we descibe the ovea design of EB12 and, in paticua, the use of evese communication. In Section 2.2 the dimension m of the iteation subspace, which is a paamete which must be set by the use, is consideed. he initia matix X chosen by EB12 is discussed in Section 2.3. In Section 2.4 the convegence citeion is given. he detemination of the degee of the iteation poynomia is discussed in Section 2.5. In Section 2.6 the ocing stategy empoyed by EB12 is outined. In Section 2.7 the cacuation of the eipse is consideed, and in Section 2.8 we discuss the computation of eigenvectos once the sought-afte eigenvaues have been detemined. 2.1 Ovea conto and design he code EB12 is witten in FORRAN 77 and has two enties: (a) EB12A (EB12AD in the doube pecision vesion) cacuates the seected eigenvaues of A. (b) EB12B (EB12BD in the doube pecision vesion) uses the basis vectos cacuated by EB12A to cacuate the eigenvectos coesponding to the computed eigenvaues. Optionay, the scaed eigenvecto esiduas (see equation (2.36)) ae computed. EB12 does not equie the use to suppy the matix A expicity. Each time EB12 needs a set of vectos to be mutipied by A, conto is etuned to the use. his aows fu advantage to be taen of the spasity and stuctue of A and of vectoisation o paaeism. It aso gives the use geate feedom in cases whee the matix A is not avaiabe and ony the poduct of A with vectos is nown. Within the code EB12, ony dense inea ageba opeations ae pefomed and fo efficiency these expoit the BLAS (Basic Linea Ageba Subpogams) enes. his incudes the use of both eve 2 BLAS outines (Dongaa, Du Coz, Hammaing, and Hanson 1988) and the eve 3 BLAS outine _GEMM (Dongaa, Du Coz, Duff, and Hammaing 1990) to pefom matix-matix mutipications of the fom B = X Y, whee Y = AX has been computed by the use. In addition to using BLAS outines, duing the Schu-Rayeigh-Ritz step (see (2.4), (2.5)), EB12A empoys the EISPACK outines ORHES and ORRAN to educe B = X AX to Hessenbeg fom H (Wiinson and Reinsch 5

7 1971), which is then educed to the ea Schu fom = V HV using a modified vesion of the outine HQR3 given by Stewat (1976b). We have modified the outine HQR3 so that the diagona bocs of ae odeed with the eigenvaues appeaing in descending ode of thei modui if the simpe subspace iteation agoithm is being used, and in descending (o ascending) ode of thei ea pats if Chebychev acceeation is being empoyed. his odeing is convenient so that we can pic-off the eigenvaues in tun as they convege (see Section 2.4). 2.2 he numbe of tia vectos he use must suppy EB12 with the numbe of equied eigenvaues and the dimension m of the iteation subspace to be used. he vaue of the paamete m is impotant. It infuences the effectiveness of EB12 since the amount of stoage equied by the code and the numbe of matix-vecto mutipications at each iteation depends upon m, which impies that, fo a specified, m shoud not be chosen unnecessaiy age. But if m is too sma, the numbe of iteations equied fo convegence may be high. he numbe of tia vectos m must theefoe be chosen with some cae. he vaue of m must exceed, the numbe of sought-afte eigenvaues, to povide some guad vectos. Fo the simpe subspace iteation agoithm, m must be at east +1. If Chebychev poynomias ae empoyed, in ode to be abe to constuct the eipse at each iteation and to aow fo compex conugate pais of eigenvaues, EB12 nomay equies m to be at east +2, but if the +1 ight-most (o eft-most) eigenvaues ae nown to be ea, m may equa +1 (see Section 2.7 fo moe detais). In pactice, it is advisabe to tae m age than this minimum vaue (see the discussion foowing equation (2.5)). In typica uns, we have taen m to be about 2, but the best vaue fo m fo a given is pobem-dependent. At any stage of the computation, the use is abe to incease (o decease) the vaue of m and estat EB12. he esuts of empoying diffeent vaues of m fo a given fo ou test pobems ae pesented in Section 3. Fo some pobems, if eigenvaues ae sought, it can be advantageous to un EB12 with epaced by 1, whee 1 exceeds. he paamete m must then be chosen to satisfy m 1+1 fo simpe subspace iteation and m 1 +2 fo subspace iteation with Chebychev acceeation. he computation may be teminated once eigenvaues have conveged. his stategy may be usefu if, fo exampe, some of the unwanted eigenvaues of A have ea pats which ae amost equa to the ea pat of one o moe of the wanted eigenvaues. his is discussed futhe in Section 2.7 and is iustated in Exampe 2 of Section he stating matix X EB12A aows the use to suppy an initia estimate of the basis vectos which span the invaiant subspace coesponding to the sought-afte eigenvaues of A. If the use wishes to suppy an estimate, on the fist enty to EB12A the estimated vaues shoud be stoed in the fist coumns of X. he emaining m guad vectos ae geneated using the Hawe Suboutine Libay andom numbe geneato, FA01AS, which geneates andom numbes in the ange [ 1,1]. he esuting set of m vectos is then othonomaised using the modified Gam-Schmidt agoithm (see, fo exampe, Goub and Van Loan 1989). he impementation of the modified Gam-Schmidt agoithm empoyed in EB12 uses eve 2 BLAS enes and was witten by Van Loan (1989). If the use does not wish to suppy an estimate of the initia basis, a nomaised andom vecto, x, is 1 6

8 geneated using FA01AS and conto is passed to the use fo the matix-vecto mutipication Ax 1. In the next ca to EB12A Ax1 is othonomaised with espect to x 1 using the modified Gam-Schmidt agoithm to give x. he pocess is epeated unti x, x,..., x have been computed. he esuting set m of othonoma vectos x 1, x 2,..., x m ae taen to be the coumns of the stating matix X. his choice of stating matix amounts to using one step of Anodi s method (see, fo exampe, Saad 1980). In genea we found that this stating matix yieded bette esuts than wee obtained using a stating matix with andom othonomaised coumns (that is, fewe matix-vecto mutipications and fewe iteations wee equied fo convegence). his was paticuay tue when Chebychev acceeation was empoyed since in this case the use of Anodi s method on the fist step povided bette initia eipse paametes than wee obtained fom a andom stating matix. If the use has some pio nowedge of the spectum of A and fees that one step of Anodi s method is uniey to povide a good stating matix, a andom stating matix may be empoyed by pacing andom vectos in the fist coumns of X. 2.4 he convegence citeion We test fo convegence afte an SRR step. he convegence citeion used in EB12A essentiay amounts to demanding that the eation AX = X (2.15) is amost satisfied. In paticua, the i th coumn of X is consideed to have conveged if the foowing inequaity is satisfied (AX X) EPS(2) (AX), (2.16) i 2 i 2 whee EPS(2) is a convegence paamete. he use is ased to assign to the paamete EPS(1) a vaue in the ange (u, 1.0), whee u is the machine pecision. If the use suppies a vaue which is out of ange, EB12A issues a waning and sets EPS(2) to the defaut vaue set equa to EPS(1). u; othewise, EPS(2) is initiay EB12A equies the coumns of X to be accepted in the ode i = 1, 2,..., so that coumn is ony tested fo convegence if the peceding coumns i = 1, 2,..., 1 have a conveged. If coumns and +1 of X coespond to a compex conugate pai of eigenvaues, then (2.16) must hod simutaneousy fo i = and i = +1. If 1 eigenvaues have conveged, then unti the th eigenvaue is accepted, the code monitos the scaed esidua R given by R = (AX X) / (AX). (2.17) 2 2 Let λ () and R () denote, espectivey, the computed appoximation to the th eigenvaue and the coesponding scaed esidua on the th iteation. Fo a sufficienty age, the scaed esiduas shoud satisfy If and R (+1) R (). (2.18) R (+1) > R () > EPS(2) (2.19) 7

9 2 λ (+1) λ () < EPS(2) 10 max ( λ (+1), λ () ), (2.20) then EB12A accepts λ (+1), issues a waning that the convegence toeance equested by the use was not achieved, and sets EPS(2) to the vaue fo which (2.16) is satisfied fo i =. EB12A aso checs fo sow convegence. Convegence of the th eigenvaue is consideed to be intoeaby sow if, fo some, and and 2 R (+1) EPS(2) 10, R (+1) max R ( ), = 0,1,2,3 (2.21a) (2.21b) R (+1) 0.9 max R ( ). = 0,1,2,3 (2.21c) In this case, EB12A again accepts λ (+1), issues a waning, and sets EPS(2) to the vaue fo which (2.16) is satisfied fo i =. If EB12A does etun with EPS(2) EPS(1) (o EPS(2) u if EPS(1) was suppied out of ange), the equested accuacy can often be achieved by inceasing m and ecaing EB12A he degee of p (λ) At each iteation, EB12A must detemine the degee of the iteation poynomia p (λ) to be used on the next iteation. We want to ensue that is chosen so that the coumns of p (A)X emain ineay independent. Böc (1967) showed that the modified Gam-Schmidt agoithm appied to a matix X = [x, x,..., x ] poduces a computed othonoma matix Xˆ which satisfies 1 2 m Xˆ Xˆ = I + E, E uκ (X), (2.22) 2 2 whee κ (X) = max Xy / min Xy y 2 = 1 y 2 = 1 is the condition numbe of X and u is the machine pecision. hus the modified Gam-Schmidt agoithm shoud ony be used to compute othonoma bases when the vectos to be othonomaised ae easonaby independent. Let () denote the degee of the iteation poynomia on the th iteation. We use (2.22) to obtain an estimate of κ 2(X) and then use this to ty and ensue that (+1) wi not be chosen so age that some of the coumns of p (A)X become ineay dependent. In paticua, we equie (+1) 1 whee (+1) q q () (1 + og 10(κ 2(X) 10 ) ), κ 2(X) 10 1 = q 1 q () (1 + og 10(κ 2(X) 10 )), κ 2(X) > 10, (2.23) with q = 3. his bound, which we have not seen in the pubished iteatue, was chosen to ensue 1 vaies fom () in a contoed manne. If κ (X) is not age, so that the coumns of X ae easonaby 2 ineay independent, wi be geate than (), but if κ (X) is age, is smae than (). he vaue of q = 3 was seected as a esut of ou numeica expeiments. Povided the othe estictions on discussed in the emainde of this section ae imposed, ou esuts wee not found to be vey sensitive to changing the vaue of q to 2 o 4, but age vaues of q sometimes ed to an unnecessay amount of wo being done befoe the eipse is updated, whie smae vaues sometimes caused an SRR step to be pefomed unnecessaiy eay. 8

10 When Chebychev acceeation is used, it is necessay to estict the degee of the iteation poynomia when the cuent eipse is not a good eipse, since othewise (2.23) may ead to a age numbe of matix-vecto mutipications being pefomed befoe thee is an oppotunity to update the eipse. he estiction EB12A imposes on the degee of the iteation poynomia fo both the simpe subspace and Chebychev acceeated agoithms is taen fom Stewat and Jennings (1981) and is (+1) 2 with 1 2 = 0.5 (1 + og 10(u )/og 10(atio)), (2.24) whee atio is the atio of the convegence ates of the sowest and fastest conveging eigenvaue. If the eipse is poo, atio is age and wi be sma but as the agoithm conveges, inceases and 2 2 ou numeica expeiments found that the degee of the iteation poynomia is then govened by (2.23) athe than (2.24). Nea convegence, if (2.23) and (2.24) ae used to detemine () and its vaue is age, the iteation poynomia may yied appoximate eigenvaues which ae moe accuate than equied by the convegence citeion. In this case, unnecessay matix-vecto mutipications may be pefomed. o avoid this we monito the scaed esidua R fo the th eigenvaue, which is given by equation (2.17) with =. Fo vaues of R cose to EPS(2) we want to estict. As a esut of ou numeica expeiments, in EB12A we set (+1) 3 whee = t (1 + og (R /EPS(2)) ). (2.25) 3 10 with t = 40. Ou numeica esuts wee not vey sensitive to the choice t = 40; simia esuts wee obtained fo t = 30 and t = 50, but age vaues of t did not pevent unnecessay mutipications and, if t was too sma, moe iteations wee needed fo convegence. Fo some of ou test pobems, if ony a sma numbe of eigenvaues wee equied, the savings esuting fom imiting nea convegence wee significant. Fo exampe, fo Exampe 2 of Section 3 with n = 400, if the two eft-most eigenvaues wee equied and m = 5 was chosen, the estiction (2.25) gave a saving of moe than 50 pe cent in the numbe of matix-vecto mutipications equied. When the simpe subspace agoithm is used, if λ 1 > 1 the enties of the matix A X wi gow as inceases. Let x ( x = 1) be the fist coumn of X. hen A x 1 = λ 1x 1 + z (2.26) fo some z. o pevent ovefow, fo the simpe subspace iteation agoithm we equie (+1) 4 whee 4 satisfies 4 2 λ 1 < M 10, (2.27) whee M is the ovefow imit. his gives a bound on og 10M 2 4 <. (2.28) og ( λ ) 10 1 If i eigenvaues have aeady conveged, the estiction (2.28) becomes og 10M 2 4 <. (2.29) og ( λ ) 10 i+1 Since λ is not nown, the cuent estimate of λ is used in (2.29). Fo the Chebychev acceeated i+1 i+1 4 of 9

11 agoithm it is not necessay to impose the estiction (2.28) (o (2.29)) since the iteation poynomia (2.9) is scaed so that the matix enties do not ovefow. We ema that Stewat and Jennings (1981) impose a maximum vaue LMAX on the degee of the iteation poynomia p (λ) used in thei code LOPSI (see aso Saad 1984). In paticua, they suggest using LMAX = 20. We have consideed imposing a estiction LMAX in EB12A but have found that it usuay ed to pooe esuts. We epot on this futhe in Section he use of ocing Computation time may be educed when sevea eigenvaues ae desied by using a ocing technique. he idea behind ocing techniques, which ae sometimes aso temed impicit defation techniques (see, fo exampe, Stewat 1976a and Saad 1989), is to expoit the fact that the initia coumns of X tend to convege befoe the ate ones. Once the basis vectos x, x,..., x coesponding 1 2 i to λ, λ,..., λ (1 i < ) have satisfied the convegence citeion (2.16), they ae oced and no 1 2 i futhe computations ae caied out with these vectos. On the next and subsequent iteations, an iteation subspace of dimension m i is used to find the next eigenvaue. o conside in moe detai the ocing technique empoyed in EB12A, suppose i basis vectos have conveged and et X = (X 1, X 2), whee X1 is the n i matix containing the vectos which have conveged. On subsequent iteations, EB12A foms (X 1, p (A)X 2). his can ead to significant savings in the numbe of matix-vecto mutipications if sevea eigenvaues ae equied. hese savings ae iustated in Section 3. At the othogonaisation step, futhe savings ae made since the coumns of X the Schu-Rayeigh-Ritz step, the matix B = X AX has the fom whee 11 X1 AX2 2 1 X2 AX2 1 ae aeady othonoma. In B =, (2.30) X AX 11 = X1 AX1 is an i i boc uppe tiangua matix. Since the coumns of X1 have conveged AX = X + E (2.31) fo some matix E1 with nom dependent upon the convegence paamete EPS(2) (as in (2.16)). Assuming X X is sma (since the coumns of X have been othogonaised with espect to those of X ) and assuming X E is sma, we can wo with the paty tianguaized system X AX Bˆ =, B 2 = X2 AX 2. (2.32) 0 B 2 hus it is ony necessay to educe the (m i) (m i) matix B 2 to ea Schu fom. Locing may cause some of the computed eigenvaues to appea out of ode. o avoid this and to ovecome the eos which ae intoduced by teating X2 AX1 as zeo, once EB12A has tentativey accepted the fist coumns of X as basis vectos fo the invaiant subspace coesponding to the desied eigenvaues, the ocing device is switched off. EB12A then taes the computed n m matix X and estats the iteative pocess. he convegence citeion (2.16) must be satisfied simutaneousy fo = 1, 2,...,. In a of ou numeica expeiments ony one iteation with the unoced system was necessay. his ocing technique diffes fom that descibed by Stewat and Jennings (1981) since thei 10

12 technique finds a the eigenvaues of the m m matix B in equation (2.30) at each iteation and, if i basis vectos have aeady conveged, ony the vecto x (and x fo a compex conugate pai) is i+1 i+2 tested fo convegence. Since the matix B changes with each iteation, the eigenvaues λ, λ,..., λ of 1 2 i B and the coesponding basis vectos x, x,...,x which passed the convegence test on iteation 1 2 i coud fai the test on subsequent iteations. hee is theefoe a dange that eigenvaues and basis vectos which do not satisfy the convegence citeion wi be etuned. Moeove, when soting the eigenvaues, Stewat and Jennings have to pevent vectos that ae not oced fom changing positions with oced vectos. hey do this by computing the eigenvaues in an unodeed sequence (using the EISPACK outine HQR2) and then atificiay inceasing the moduus of each eigenvaue coesponding to a oced vecto immediatey pio to soting. Afte soting they estoe the vaues of the eigenvaue estimates. 2.7 Constucting the eipse When Chebychev acceeation is used, we want to find the optima eipse E(d, c, a) encosing the unwanted eigenvaues λ, +1. Manteuffe s technique (1975, 1977, 1978) fo doing this in the case of the soution of inea systems has been adapted by Saad (1984) to the unsymmetic eigenvaue pobem. Manteuffe s agoithm does not aow a compex efeence eigenvaue λ in equation (2.9) so Saad epaces λ by a ea efeence point γ and, in paticua, on the th iteation Saad taes γ to be the point on the ea ine which has the same convegence atio as λ with espect to the eipse found on the ( 1) st iteation. he best eipse fo the th iteation is then detemined to minimise the maximum convegence facto R (d, c) given by equation (2.10) with λ epaced by γ. When λ is ea, γ = λ. Since in genea γ λ, the eipse found using Saad s method is ony an appoximation to the optima eipse. Nevetheess, ou expeience is that this appoximation geneay wos we and so EB12A uses Saad s choice of γ. EB12A uses the eigenvaue estimates λ (), 1 m, computed on iteation to constuct a sequence of eipses E (d, c, a), = 1,2,... using the foowing pocedue. Hee it is assumed that the ight-most eigenvaues ae equied. fo := 1 step 1 unti s do begin end exit: If = 1, set p (λ) = 1; othewise et γ be the point on the ea ine with the same convegence ( 1) 1 () atio as λ with espect to E (d, c, a) and define p (λ) using (2.9) with λ epaced by γ. Compute the eigenvaue estimates λ agoithm of Section 2., 1 m, using steps 2 and 3 of the basic iteation if the convegence citeion (2.16) is satisfied fo i = 1, 2,..., go to exit () () () ese define the baie b = Re(λ ) and the set of unwanted eigenvaues S = {λ : Re(λ ) < b}. end Constuct the positive convex hu K containing S and the points (x, y) on the pevious 1 hu K fo which x < b. Find the best eipse E (d, c, a) using the agoithms of Manteuffe (1975, 1977) and Saad (1990). 11

13 () We obseve that if λ is ea (espectivey, compex), S wi usuay be nonempty povided m +1 (espectivey, m +2). hus in genea the numbe of tia vectos m must satisfy m +2. Manteuffe (1975, 1977) descibes and impements an agoithm fo finding the eipse E (d, c, a) using the positive convex hu K descibed by Manteuffe but we have modified his code. We ema that it is necessay to use the pevious hu K (see aso Ashby 1985). In EB12A, we foow the pocedue 1 when constucting K. Suppose the ight-most eigenvaues ae sought. ypicay, duing the fist few iteations, the eipse E (d, c, a) wi not contain the actua eft-most eigenvaues of A. Since p (λ) is sma fo eigenvaues inside the eipse compaed with those ying outside the eipse, convegence wi be towads the eft-most (unwanted) eigenvaues as we as to the ight-most eigenvaues. At some stage, the computed eft-most eigenvaues wi ie within a hu which contains the actua eft-most eigenvaues of A and, povided a subsequent hus contain these computed eft-most eigenvaues, convegence wi be to the sought-afte ight-most eigenvaues. Some of the eipses E (d, c, a) may contain wanted eigenvaues, which wi sow convegence down. his is iey to be a pobem if thee is a custe of eigenvaues nea λ. In this case, it can be advantageous to set to be age than the actua numbe of equied eigenvaues (see Section 2.2). he effect of choosing a age vaue of is to move the baie b to the eft (o ight). If moe than one eigenvaue (o moe than one pai of compex conugate eigenvaues) is equied, once they have a conveged, we eca EB12A with set to the actua numbe of equied eigenvaues to ovecome the eos intoduced by ocing. his is iustated in Exampe 2 of Section Computing the eigenvectos Once EB12A has successfuy computed the equied eigenvaues of A, the use may ca EB12B to compute the coesponding eigenvectos. EB12B computes the eigenvectos wi of the boc tiangua matix using bac-substitution and then taes the appoximate eigenvectos of A to be y i = Xw i (see (2.8)). he computed eigenvectos y i ae nomaised. If the i th eigenvaue is compex with positive imaginay pat, on exit fom EB12B the i th and (i+1) th coumns of a matix Y wi hod the ea and imaginay pats of the i th eigenvecto, espectivey. Since the (i+1) th eigenvecto is the compex conugate of the i th eigenvecto, woing with compex aays is avoided. When computing the eigenvectos of using bac-substitution, we found it was necessay to set a the enties in the owe tiangua pat of (except those in the 2 2 diagona bocs coesponding to compex eigenvaues) to zeo. If we did not do this, the sma off-diagona enties in the matix computed by EB12A coud cause age eos in the computed eigenvectos of A. Suppose X = (X 1, X 2) whee the coumns of X1 have conveged and 11 = X1 AX 1. he esiduas fo the computed eigenvectos of A wi be sma if the esiduas fo the coesponding eigenvectos of ae sma. o demonstate this, et w ( w = 1) denote the computed eigenvecto of coesponding to the computed eigenvaue λ and et 1 be the esidua vecto = w λw. (2.33) 1 11 he coesponding appoximate eigenvecto of A is given by y = X w, and fom (2.31) and (2.33) we 1 have 12

14 E w = AX w X w = Ay λy X. (2.34) Hence, since X 1 has othonoma coumns, Ay λy E +. (2.35) he inequaity (2.35) shows that povided E and ae sma, the esidua fo the computed eigenvecto of A wi be sma. If the scaed eigenvecto esiduas (Ay λ y ) i i i 2, 1 i, (2.36) (Ay ) i 2 ae equied, the use must compute AY, whee Y has coumns y, y,..., y 1 2 m and eca EB12B. 2.9 Use of EB12 to obtain othe pats of the spectum he code EB12 can be used to find pats of the spectum othe than that coesponding to the eigenvaues of agest moduus o the ight-most (o eft-most) eigenvaues of A. If, fo exampe, we wish to compute a goup of inteio eigenvaues, say those cosest to p (that is, those fo which λ p is smaest), we can use the simpe subspace iteation agoithm option in EB12 and epace A by 1 1 (A p I). In this case, on each etun to the use, a matix-matix mutipication U = (A p I) W, with W of ode n (m i) (i is the numbe of oced vectos), must be pefomed. his is equivaent to soving the system of equations (A p I) U = W. (2.37) If A is age the soution of the system (2.37) may itsef be quite time-consuming but note that, if a diect method of soution is used, the decomposition of A p I into tiangua factos needs ony to be done once fo a vaue of the shift p. Having pefomed the decomposition, on each etun to the use it is ony necessay to pefom eativey cheap fowad and bacwad substitutions. In pactice, p may be an appoximation to a equied eigenvaue of A, in which case it may be advantageous to update the vaue of the shift as the computation poceeds and consequenty sevea factoizations may be 1 equied. Howeve, if the shift p is suitaby chosen, the matix B = (A p I) wi have a spectum with much bette sepaation popeties than the oigina matix A and the subspace iteation agoithm appied to B shoud equie fa fewe iteations fo convegence than when it is appied to A. hus, the ationae behind using a so-caed shift-and-invet stategy is that the additiona cost of the factoizations is ampy epaid by the eduction in the numbe of iteations equied by using B in pace of A. Note that if the shift p is compex, the use of compex aithmetic in the subspace iteation 1 agoithm may be avoided by epacing the compex opeato (A p I) by the ea opeato 1 1 Re [(A p I) ] (see Saad 1989). he use must fom the LU decomposition of the matix (A p I) 1 and evey time U = Re [(A p I) ]W is equied, must pefom fowad and bacwad soves in the usua way and tae the ea pat of the esuting matix to yied the matix U which is etuned to EB12. he code EB12 can aso be used fo the geneaised eigenvaue pobem Kx = λmx. In this case, on each etun to the use it is necessay to sove a system of the fom MU = KW. (2.38) If a diect method of soution is used, the factoization of M needs ony be done once fo the entie 1 cacuation. o gain faste convegence, the shifted and inveted opeato (K p M) M may be used. 13

15 3 Numeica expeiments he code EB12 has been tested on a numbe of pobems. In this section, we descibe the esuts of using EB12 to cacuate seected eigenpais fo thee epesentative test exampes. In each exampe the 5 convegence paamete EPS(1) (see Section 2.4) was set to 10. he numeica expeiments wee 16 pefomed on a SUN SPARCstation using doube pecision (i.e. u ). he numbe of iteations equied is defined to be the numbe of times the iteation poynomia p (A)X is computed. houghout this section,, m, and max denote, espectivey, the numbe of eigenpais sought, the numbe of tia vectos used, and the highest degee of the iteation poynomia used by EB12A. A CPU timings ae in seconds. Exampe 1. he fist pobem is taen fom Stewat and Jennings (1981), who use this pobem to iustate the effectiveness of thei subspace iteation code LOPSI. he matix is a stochastic matix obtained duing the appication of Maov modeing techniques to the anaysis and evauation of compute systems. he matix is of ode 163 and has 1207 nonzeo enties. abe 3.1 compaes the convegence chaacteistics fo diffeent numbes of tia vectos fo the simpe subspace iteation agoithm and fo the Chebychev acceeated agoithm. Fo this exampe, the eigenvaues of agest moduus ae aso the ight-most eigenvaues, so the two agoithms may be compaed diecty. abe 3.1. A compaison of the simpe subspace iteation agoithm and the Chebychev acceeated agoithm fo Exampe 1 (ight-most eigenvaues). m Matix-vecto Iteations poducts max CPU time Simpe Chebychev Simpe Chebychev Simpe Chebychev Simpe Chebychev he eigenvecto coesponding to the dominant eigenvaue λ = 1 is nown to have a its eements 1 of equa vaue. In a the tests using this pobem, we used the option offeed by EB12A fo suppying an initia estimate of the basis vectos coesponding to the sought-afte eigenvaues to specify the fist basis vecto with ength 1 and a its eements of equa vaue; the initia estimate of each of the othe basis vectos fo this pobem was taen to be a andom vecto. he fist basis vecto passed the convegence test on the initia iteation and was then oced. Fo this pobem, the ocing faciity fo the emaining basis vectos did not come into effect unti ate in the computation so that most of the savings due to the ocing techniques empoyed by EB12A come fom the fist vecto. If ocing is switched off, with = 10 and m = 20, the simpe subspace and Chebychev acceeated agoithms equied 3540 and 2820 matix-vecto poducts, espectivey. Fo this exampe, we obseve that, in each case, the vaue of max fo the simpe subspace iteation agoithm exceeded that fo the Chebychev acceeated agoithm and that, with = 5, the simpe 14

16 subspace iteation agoithm too fewe iteations to convege than the Chebychev acceeated agoithm. Howeve, the numbe of matix-vecto mutipications and the computation times fo the Chebychev acceeated agoithm wee consideaby ess than fo the simpe subspace iteation agoithm. Because of the way Stewat and Jennings (1981) pesent thei esuts fo this pobem, it is difficut to mae a diect compaison between the esuts obtained by LOPSI and EB12. Howeve, compaed with LOPSI, EB12 appeas to equie significanty fewe iteations and, if Chebychev acceeation is used, the numbe of matix-vecto mutipications used by EB12 is aso consideaby smae. Exampe 2. he second test exampe is taen fom Gaatt, Mooe, and Spence (1991) and is concened with the detection of Hopf bifucation points in the paamete dependent noninea system dx n n n = f(x, ν), f : R R R, x R, ν R. (3.1) dt n+1 he set Γ := {(x, ν) R : f(x, ν) = 0} epesents the steady-state soutions of (3.1) and it is often impotant to detemine the (ineaised) stabiity of a banch of Γ. If λ 1 denotes the ight-most eigenvaue of the Jacobian matix A = f x(x, ν), then a steady-state soution is stabe (unstabe) if Re(λ 1) is negative (positive). It is aso desiabe to be abe to detect a Hopf bifucation point, that is a point of Γ whee Re(λ 1) changes sign as ν vaies. Exampe 2 aises fom a pai of equations of the fom (3.1) which mode a tubua eacto (see equations (1) (5) of Heinemann and Pooe 1981), and which ae discetised using simpe centa diffeences. We have taen n to be 200 and 400 and the paamete ν (the Damohe numbe) to be With this vaue fo ν the steady-state soution is stabe (see Gaatt et a.1989). he matix A is banded and has 796 and 1596 nonzeo enties when n = 200 and 400, espectivey. Fo n = 200, the ight-most eigenvaue is (appoximatey) ± i and the eft-most eigenvaue is ± i. Fo n = 400, the ight-most eigenvaue is ± i and the eft-most eigenvaue is ± i. Athough it is the ight-most eigenvaue that is of pactica impotance, we sha aso use EB12 to cacuate the eft-most eigenvaue since the eft-most eigenvaue is aso the eigenvaue of agest moduus, which aows us to compae the simpe subspace agoithm with the Chebychev acceeated agoithm. he esuts ae given in abe 3.2. In abe 3.2 we see that, when n = 200 and the simpe subspace iteation agoithm is used, fo each vaue of m, max = 78. In this exampe λ and the degee of the iteation poynomia is imited by (2.28), which pevents ovefow. Simiay, when n = 400, (2.28) imits the degee of the iteation poynomia used by the simpe subspace iteation agoithm to 68. hee is no coesponding imit fo the Chebychev acceeated agoithm, which again gives much bette esuts than the simpe subspace iteation agoithm. In paticua, when n = 400, = 2, and m = 5, the simpe subspace iteation agoithm equies moe than 21 times as many matix-vecto mutipications, 15 times as many iteations, and 18 times as much CPU time as the Chebychev acceeated agoithm. In abe 3.3 we pesent some esuts fo Exampe 2 fo the Chebychev acceeated agoithm used to find the ight-most eigenvaues. We obseve that fo n = 200 and m = 8, fewe matix-vecto poducts and ess CPU time ae equied fo convegence when = 4 than when = 2. In fact, with m = 8, the ight-most pai of compex conugate eigenvaues is found afte 2541 matix-vecto poducts and 76.6 seconds CPU time. his is an exampe which iustates that it can be advantageous to set to be age than the numbe of eigenvaues actuay equied. he esuts we obtained fo Exampe 2 using EB12 compae favouaby with those obtained by Gaatt (1991) using Chebychev acceeation techniques. 15

17 abe 3.2. A compaison of the simpe subspace iteation agoithm and the Chebychev acceeated agoithm fo Exampe 2 (eft-most eigenvaues). n m Matix-vecto poducts Iteations max CPU time Simpe Chebychev Simpe Chebychev Simpe Chebychev Simpe Chebychev abe 3.3. Convegence esuts fo the Chebychev acceeated subspace iteation agoithm fo Exampe 2 (ight-most eigenvaues). Matix-vecto n m Iteations max poducts CPU time Exampe 3. he thid test pobem is anothe exampe of Maov chain modeing and is used by Saad (1984). his exampe modes a andom wa on a (+1) (+1) tiangua gid. Foowing Saad, we tae = 30, so that the ode of the matix is n = 496. Fo this pobem, matix-vecto poducts Ax can be pefomed by a simpe suboutine without expicity foming the matix A. Since EB12 etuns conto to the use fo a matix-vecto poducts, it is paticuay convenient fo soving pobems of this ind. In this pobem, the eigenvaues of agest moduus ae 1 and 1 and the ight-most eigenvaue is 1. In abe 3.4 we give convegence esuts fo obtaining the dominant pai of eigenvaues λ = ±1 using the simpe subspace iteation agoithm fo vaious vaues of the paamete m. In abe 3.5 we pesent esuts fo the Chebychev acceeated subspace iteation agoithm, but in this case ony the ight-most eigenvaue λ = 1 is obtained. he numbes in paentheses in the second coumn of abe 3.5 ae the figues epoted by Saad (1984). he diffeence in the numbe of matix-vecto poducts used by EB12 and by Saad is mainy attibutabe to the fact that Saad uses diffeent citeia fo choosing the degee of the iteation poynomia and, foowing Stewat and Jennings (1981), Saad imposes a maximum degee on the iteation poynomia. Fo this exampe, EB12 pefoms consistenty bette than the code used by Saad. 16

18 abe 3.4. Convegence esuts fo the simpe subspace iteation agoithm fo Exampe 3 ( = 2). Matix-vecto m Iteations max poducts CPU time abe 3.5. Convegence esuts fo the Chebychev acceeated subspace iteation agoithm fo Exampe 3 (ight-most eigenvaue). Matix-vecto m Iteations max poducts CPU time ( ) ( ) (645) (903) (909) Fom abes we see that, in genea, the best esuts ae obtained by choosing a vaue of m which is age than the minimum aowed vaue. We aso obseve that, in most of the exampes, inceasing the numbe of tia vectos educes (o eeps constant) the numbe of iteations equied fo convegence but the tota numbe of matix-vecto mutipications needed may incease. Even if the numbe of matix-vecto mutipications needed deceases as m inceases, the CPU time may incease since the ode of the matix which must be educed to ea Schu fom is m. Howeve, when the ode of the matix n is age, most of the CPU time is in the matix-vecto mutipication stage, and the tota CPU time taen is dependent upon the efficiency with which these mutipications can be caied out. In paticua, the time taen depends upon whethe the use is abe to expoit the stuctue of the matix and vectoisation o paaeism. We emaed in Section 2.5 that Stewat and Jennings (1981) impose a maximum vaue LMAX on the degee of the iteation poynomia p (λ) used in thei code LOPSI. Fo a ou test exampes we found that the maximum degee used by EB12 exceeded the vaue LMAX = 20 suggested by Stewat and Jennings. With the estiction LMAX = 20, the Chebychev acceeated agoithm appied to Exampe 2 with n = 200, = 2, and m = 4 equied 7043 matix-vecto poducts and 86 iteations fo convegence. his compaes unfavouaby with the coesponding esut of 3575 matix-vecto poducts and 10 iteations given in abe 3.3. Futhe numeica expeiments fo this and othe test exampes using a ange of vaues fo LMAX ed us to concude that, in genea, the esuts ae wose when a estiction LMAX is paced on. In Section 2.9 we discussed the use of EB12 to compute eigenvaues of the matix A othe than 17

19 those which ae ight-most, eft-most, o ae of agest moduus. We may, fo exampe, have an appoximation p to an eigenvaue of A and want to obtain a moe accuate appoximation. In this case 1 we woud epace A by (A p I) and empoy the simpe subspace iteation agoithm, soving equation (2.37) on each etun to the use. We have pefomed some numeica expeiments to do this fo the matix in Exampe 1. In these expeiments we used the Hawe Suboutine Libay outine MA28AD to facto the matix (A p I) once, and on each etun the factos ceated by MA28AD wee used by MA28CD to sove (2.37). Fu detais of the MA28 pacage may be found in Duff (1977). We obseved that fo this exampe EB12 conveged vey quicy. ypicay if an appoximation to a (ea) eigenvaue inside the spectum was nown to two decima paces, setting = 1 and m = 6, convegence 5 with EPS(2) equa to 10 (see equation (2.16)) was achieved in ony one o two iteations. he computed eigenvaues ageed with those given in abe I of Stewat and Jennings (1981). 4 Concuding emas he pupose of this pape was to discuss the design and deveopment of the code EB12 fo computing seected eigenvaues and the coesponding eigenvectos of a ea unsymmetic matix A. Existing codes LOPSI (Stewat and Jennings 1981) and SRRI (Stewat 1978) use subspace iteation techniques to compute the eigenvaues of agest modui. EB12 uses a subspace iteation agoithm, combined with Chebychev acceeation if eithe the ight-most (o eft-most) eigenvaues ae equied o it is nown that the eigenvaues of agest modui ae aso the ight-most (o eft-most) eigenvaues. EB12A wos in tems of the Schu vectos of A and a second optiona enty, EB12B, is used to obtain the eigenvectos once the Schu vectos have conveged. An impotant design featue of the code EB12 is that conto is etuned to the use each time a matix-vecto poduct Ax needs to be fomed. his use of evese communication maes the code suitabe fo age spase pobems and gives fexibiity ove the way in which the matix is stoed and matix-vecto poducts ae pefomed. Anothe featue of the code is that it empoys a new ocing technique which is designed to educe the numbe of matix-vecto mutipications equied fo convegence when moe than one eigenvaue is equied. he use of ocing has been found to be efficient in pactice. he use of EB12 must choose the dimension m of the iteation subspace. his is an impotant paamete which effects the efficiency of the agoithm. We have povided the use with some guidance egading the choice of m and, in addition, if a poo choice is made, we have designed EB12 so that the computation can be estated at any stage with a diffeent vaue of m whie taing advantage of the basis vectos which have aeady been computed. he usefuness of the code EB12 has been iustated on a numbe of epesentative pactica pobems. he numeica esuts show that subspace iteation combined with Chebychev acceeation is significanty supeio to simpe subspace iteation when the ight-most (o eft-most) eigenvaues ae aso those of agest moduus. he esuts aso show that the efficiency of ou subspace iteation agoithm (with o without Chebychev acceeation) is vey dependent upon how the agoithm chooses, the degee of the iteation poynomia. In paticua, we found that imposing a maximum vaue on, as suggested by othe authos (fo exampe, Stewat and Jennings 1981 and Saad 1984), coud ead to a consideabe degadation of the esuts. We have intoduced new citeia fo choosing 18