A Subspace Iteration for Calculating a Cluster of Exterior Eigenvalues

Transcription

1 Advances in Linear Algebra & Matrix heory Published Online Seteber 05 in SciRes htt://wwwscirorg/ournal/alat htt://dxdoiorg/0436/alat A Subsace Iteration for Calculating a Cluster of Exterior Eigenvalues Achiya Dax Hydrological Service Jerusale Israel Eail: dax0@watergovil Received 5 June 05; acceted 9 August 05; ublished Seteber 05 Coyright 05 by author and Scientific Research Publishing Inc his wor is licensed under the Creative Coons Attribution International License (CC BY) htt://creativecoonsorg/licenses/by/40/ Abstract In this aer we resent a new subsace iteration for calculating eigenvalues of syetric atrices he ethod is designed to coute a cluster of exterior eigenvalues For exale eigenvalues with the largest absolute values the algebraically largest eigenvalues or the algebraically sallest eigenvalues he new iteration alies a Restarted Krylov ethod to collect inforation on the desired cluster It is shown that the estiated eigenvalues roceed onotonically toward their liits Another innovation regards the choice of starting oints for the Krylov subsaces which leads to fast rate of convergence Nuerical exerients illustrate the viability of the roosed ideas Keywords Exterior Eigenvalues Syetric Matrices Subsace Iterations Interlacing Restarted Krylov Methods Introduction In this aer we resent a new subsace iteration for calculating a cluster of exterior eigenvalues of a given n syetric atrix G Other naes for such eigenvalues are eriheral eigenvalues and extree eigenvalues As with other subsace iterations the ethod is best suited for handling large sarse atrices in which a atrix-vector roduct needs only 0( n ) flos Another underlying assution is that is considerably saller than n Basically there are two tyes of subsace iterations for solving such robles he first category regards bloc versions of the Power ethod that use freuent orthogonalizations he eigenvalues are extracted with the Rayleigh-Ritz rocedure his ind of ethod is also called orthogonal iterations and siultaneous iterations he second category uses the Rayleigh-Ritz rocess to achieve aroxiation fro a Krylov subsace he Restarted Lanczos ethod turns this aroach into a owerful tool For detailed discus- How to cite this aer: Dax A (05) A Subsace Iteration for Calculating a Cluster of Exterior Eigenvalues Advances in Linear Algebra & Matrix heory htt://dxdoiorg/0436/alat

2 A Dax sions of these toics see for exale []-[9] he new iteration alies a Krylov subsace ethod to collect inforation on the desired cluster Yet it has an additional flavor: it uses an interlacing theore to irove the current estiates of the eigenvalues his enables the ethod to gain seed and accuracy If G haens to be a singular atrix then it has zero eigenvalues and any orthonoral basis of Null (G) gives the corresonding eigenvectors However in any ractical robles we are interested only in non-zero eigenvalues For this reason the coing definitions of the ter a cluster of exterior eigenvalues do not include zero eigenvalues Let r denote the ran of G and assue that < r hen G has r non-zero eigenvalues that can be ordered to satisfy or ˆ ˆ ˆ > 0 () λ λ λ r λ λ λ r () he new algorith is built to coute one of the following four tyes of target clusters that contain extree eigenvalues A doinant cluster A right-side cluster A left-side cluster { ˆ λ ˆ λ } (3) { λ λ } (4) { λ r λ + r λr} (5) and so forth Note that although the above definitions refer to clusters of eigenvalues the algorith is carried out by couting the corresonding eigenvectors of G he subsace that is sanned by these eigenvectors is called the target sace he restriction of the target cluster to include only non-zero eigenvalues eans that the target sace is contained in Range(G) For this reason the search for the target sace is restricted to Range(G) A two-side cluster is a union of a right-side cluster and a left-side cluster For exale { λ λr} { λ λ λr} Let us turn now to describe the basic iteration of the new ethod he th iteration = is coosed of the following five stes he first ste starts with a atrix V that contains old inforation on the l ( + l) target sace a atrix Y that contains new inforation and a atrix X = V Y that includes all the nown inforation he atrix X has = + l orthonoral coluns hat is X X = I (yical values for l are l = or l = 3 ) Ste : Eigenvalues extraction First coute the Rayleigh uotient atrix S = X GX hen coute eigenairs of S which corresond to the target cluster (For exale if it is desired to coute a right-side cluster of G then coute a right-side cluster of S ) he corresonding eigenvectors of S are assebled into a atrix U U U I = which is used to coute the related atrix of Ritz vectors V = + XU Ste : Collecting new inforation Coute a atrix sace he coluns of B are forced to stay in Range(G) B l that contains new inforation on the target 77

3 A Dax Ste 3: Discard redundant inforation Orthogonalize the coluns of B against the coluns of V + here are several ways to achieve this tas In exact arithetic the resulting atrix Z satisfies the Gra-Schidt forula ( ) Z B V V B Ste 4: Build an orthonoral basis Coute a atrix = + + Y Y Y = I n l l l whose coluns for an orthonoral basis of Range( Z ) ran ( Z ) is saller than l then l is redefined as ran ( ) Ste 5: Define which ensures that X + by the rule X+ = V+ Y + X X I = + + his can be done by a QR factorization of Z (if Z ) he above descrition is aied to clarify the urose of each ste Yet there ight be better ways to carry out the basic iteration he restriction of the search to Range(G) is iortant when handling low-ran atrices However if G is nown to be a non-singular atrix then there is no need to iose this restriction he lan of the aer is as follows he interlacing theores that suort the new ethod are given in the ( next section Let λ ) = and denote the Ritz values which are couted at Ste of the th iteration hen it is shown that each iteration gives a better aroxiation of the target cluster Moreover the seuence ( λ ) = roceeds onotonously toward the desired eigenvalue of G he rate of convergence deends on the inforation atrix B Roughly seaing the better inforation we get the faster the convergence is Indeed the heart of the algorith is the coutation of B It is well-nown that a Krylov subsace which is generated by G gives valuable inforation on eriheral eigenvalues of G eg [4] [5] [7] [3] [5] he basic schee of the new ethod uses this observation to define B see Section 3 Difficulties that arise in the coutation of non-eriheral clusters are discussed in Section 4 he fifth section considers the use of acceleration techniues; ost of the are borrowed fro orthogonal iterations Another related iteration is the Restarted Lanczos ethod he lins with these ethods are discussed in Sections 6 and 7 he aer ends with nuerical exerients that illustrate the behavior of the roosed ethod Interlacing heores In this section we establish a useful roerty of the roosed ethod We start with two well-nown interlacing theores eg [6] [7] [9] n heore (Cauchy interlace theore) Let G be a syetric atrix with eigenvalues Let the syetric atrix coluns Let H denote the eigenvalues of H hen λ λ λ n () be obtained fro G by deleting n rows and the corresonding n η η η () λ η for = (3) and η+ i λn+ i for i = (4) In articular for = n we have the interlacing relations λ η λ η λ λ η λ (5) 3 n n n Corollary (Poincaré searation theore) Let the atrix V have orthonoral coluns hat is 78

4 A Dax = V GV have the eigenvalues () hen the eigenvalues of H and G sa- V V H tisfy (3) and (4) he next theore sees to be new It sharens the above results by reoving zero eigenvalues heore 3 Assue that the non-zero eigenvalues of G satisfy () where r = ran ( G) Let the atrix V satisfy = I Let the atrix Let the atrix and H ( ) ( ) < r V V = I and Range V Range G (6) = V GV has the eigenvalues () hen the eigenvalues of G and H satisfy the ineualities λ η for = (7) η+ i λr+ i for i = (8) r Proof Let the atrix Y be obtained by coleting the coluns of V to be an orthonoral basis of Range( G ) hen Y GY is a full ran syetric atrix whose eigenvalues are the non-zero eigenvalues of G which are given in () Since the first coluns of Y are the coluns of V the atrix V GV is obtained by deleting fro Y GY the last r rows and the last r coluns Hence the ineualities (7) and (8) are direct corollary of Cauchy interlace theore Let us return now to consider the th iteration of the new ethod = 3 Assue first that the algorith is aied at couting a cluster of right-side eigenvalues of G and let the eigenvalues of the atrix be denoted as hen the Ritz values which are couted at Ste are and these values are the eigenvalues of the atrix Siilarly are the eigenvalues of the atrix herefore since the coluns of { λ λ λ } S = X GX = V Y G V Y ( ) ( ) ( ) ( ) λ λ λ λ ( ) ( ) ( ) λ λ λ V GV + + ( ) ( ) ( ) λ λ λ V GV V are the first coluns of ( ) ( ) λ On the other hand fro heore 3 we obtain that Hence by cobining these relations we see that X λ for = ( ) λ = λ for ( ) ( ) (9) λ λ λ for = and = 3 Assue now that the algorith is aied at couting a cluster of left-side eigenvalues of G 79

5 A Dax hen siilar arguents show that { λr+ λ r λr} ( ) ( ) + i + i r+ i λ λ λ (0) for i = and = 3 Recall that a two-sides cluster is the union of a right-side cluster and a left-side one In this case the eigenvalues of S that corresond to the right-side satisfy (9) while eigenvalues of S that corresond to the left-side satisfy (0) A siilar situation occurs in the coutation of a doinant cluster since a doinant cluster is either a right-side cluster a left-side cluster or a two-sides cluster 3 he Krylov Inforation Matrix It is left to exlain how the inforation atrices are couted he first uestion to answer is how to define the starting atrix X For this urose we consider a Krylov subsace that is generated by the vectors 3 Gr G r G r G r (3) where r is a rando vector hat is a vector whose entries are uniforly distributed between and hen X is defined to be a atrix whose coluns rovide an orthonoral basis for that sace his definition ensures that range( X ) is contained in range(g) he actual coutation of X can be done in a nuber of ways he ain uestion is how to define the inforation atrix B [ b b b ] = l l (3) which is needed in Ste Following the Krylov subsace aroach the coluns of B are defined by the rule b = Gb Gb = l (33) where is soe vector nor In this way B is deterined by the starting vector b 0 he ability of a Krylov subsace to aroxiate a doinant subsace is characterized by the Kaniel-Paige- Saad (K-P-S) bounds See for exale ([5] ) ([7] 4-47) ([3] 7-74) and the references therein One conseuence of these bounds regards the angle between b and the doinant subsace: he saller the angle the better aroxiation we get his suggests that b 0 should be defined as the su of the current Ritz vectors hat is b0 = V + e (34) where vector of ones Another conseuence of the K-P-S bounds is that a larger Krylov subsace gives better aroxiations his suggests that using (3)-(33) with a larger l is exected to result in faster convergence since ( B) ( X + ) Range Range (35) A different arguent that suorts the last observation coes fro the interlacing theores: Consider the use of (3)-(33) with two values of l say l < l ˆ and let X and + X ˆ + denote the corresonding orthonoral atrices hen the first l coluns of X ˆ + can be obtained fro the coluns of X + herefore a larger value of l is liely to give better aroxiations However the use of the above arguents to assess the exected rate of convergence is not straightforward 4 reating a Non-Periheral Cluster A cluster of eigenvalues is called eriheral if there exists a real nuber α for which the corresonding eigenvalues of the shifted atrix G α I turn out to be a doinant cluster he theory develoed in Section suggests that the algorith can be used to coute certain tyes of non-eriheral clusters (see the coing exale) However in this case it faces soe difficulties One difficulty regards the rate of convergence as the K-P-S bounds tell us that aroxiations of eriheral eigenvalues are better than those of internal eigenvalues 80

6 A Dax Hence when treating a non-eriheral cluster we exect a slower rate of convergence A second difficulty coes fro the following henoenon o silify the discussion we concentrate on a left-side cluster of a ositive sei-definite atrix that has several zero eigenvalues In this case the target cluster is coosed fro the sallest non-zero eigenvalues of G hen once the coluns of V + becoe close to eigenvectors of G the atrices B and Z turn out to be ill-conditioned his aes Y + vulnerable to rounding errors in the following anner he orthogonality of Y + is et but now Range( Y + ) ay fail to stay in Range(G) he sae rear alies to X + since Y + is art of X + In this case when Range( X + ) contains soe vectors fro Null(G) the sallest eigenvalues of the atrix X + GX + ay be saller than the sallest non-zero eigenvalue of G Conseuently the algorith ay converge toward zero eigenvalues of G Hence it is necessary to tae a recaution to revent this ossibility (Siilar ill-conditioning occurs when calculating a eriheral cluster but in this case it does no har) One way to overcoe the last difficulty is to force Range( Z ) to stay inside Range(G) his can be done by the following odification of Ste 3 Ste 3 * : As before the ste starts by orthogonalizing the coluns of B against the current Ritz vectors (the coluns of V + ) his gives us an l atrix Z hen the atrix GZ is orthogonalized against the Ritz vectors giving Z A second ossible reedy is to force Range( X + ) to stay inside Range(G) his can be done by correcting Ste 5 in the following way Ste 5 * : Coute the atrices X + = V+ Y + and X ˆ + = GX + hen X + is defined to be a atrix Range X ˆ + his can be done by a QR factorization of X ˆ + whose coluns for an orthonoral basis of ( ) In the exerients of Section 8 we have used Ste 3 * to coute a left-side cluster of Proble B and Ste 5 * was used to coute a left-side cluster of Proble C However the above odifications are not always helful and there ight be better ways to correct the algorith 5 Acceleration echniues In this section we outline soe ossible ways to accelerate the rate of convergence he acceleration is carried out in Ste of the basic iteration by roviding a better inforation atrix B All the other stes reain unchanged 5 Power Acceleration In this aroach the coluns of B are generated by relacing (33) with the rule b = G b G b = l (5) where is a sall integer Of course in ractice the atrix G is never couted Instead b is couted by a seuence of atrix-vector ultilications and noralizations he above acceleration is suitable for calculating a doinant cluster In other exterior clusters it can be used with a shift he ain benefit of (5) is in reducing the ortion of tie that is sent on orthogonalizations and the Rayleigh-Ritz rocedure (see able 4) 5 Using a Shift he shift oeration is carried out by relacing (5) with ( α ) ( α ) b = G I b G I b = l (5) where α is a real nuber It is well nown that Krylov subsaces are invariant under the shift oeration hat is for = the Krylov sace that is generated by (5) euals that of (33) eg ([7] 38) Hence the shift oeration does not accelerate the rate of convergence Yet it hels to reduce the deteriorating effects of rounding errors Assue first that G is a ositive definite atrix and that we want to coute a left-side cluster (a cluster of the sallest eigenvalues) hen (5) is relaced by (5) where α is an estiate for the largest eigenvalue of G Observe that the reuired estiate can be derived fro the eigenvalues of the atrices S 8

7 A Dax A siilar tactic is used for calculating a two-sides cluster In this case the shift is couted by the rule ( ) ( ) ( ) α = λ + λ l In other words the shift estiates the average value of the largest and the sallest (algebraically) eigenvalues of G A ore sohisticated way to ileent the above ideas is outlined below 53 Polynoial Acceleration Let the eigenvalues of G satisfy () and let the real nubers define the onic olynoial hen the eigenvalues of the atrix olynoial are deterined by the relations Moreover the atrices G and ( G) is an eigenvector of ( G) α α α ( ) ( )( ) ( ) Φ θ = θ α θ α θ α ( G) ( G α I)( G α I) ( G α I) Φ = ϕ =Φ ( λ) = n Φ share the sae eigenvectors: An eigenvector of G that corresonds to λ Φ that corresonds to ϕ and vice versa In olynoial acceleration the basic schee (33) is relaced with ( ) ( ) b =Φ G b Φ G b = l (53) he idea here is to choose the oints α α in a way that enlarges the size of ϕ when λ belongs to the target cluster and/or diinishes ϕ when λ is outside the target cluster As with orthogonal iterations the use of Chebyshev olynoials enables effective ileentation of this idea In this ethod there is no need in the nubers α α Instead we have to suly the end oints of the diinishing interval hen the atrix-vector roduct Φ ( G) b is carried out by the Chebyshev recursion forula See ([7] 94) for details In our iteration the end oints of the diinishing interval can be derived fro the current Ritz values (the eigenvalues of S ) 54 Inverse Iterations his aroach is ossible only in certain cases when G is invertible and the atrix-vector roduct 0 n flos In this case (33) is relaced with be couted in ( ) G b can b = G b G b = l (54) In ractice G is alost never couted Instead the linear syste Gx = b is solved for x For this urose we need an aroriate factorization of G he use of (54) is helful for calculating sall eigenvalues of a ositive definite atrix If other clusters are needed then G should be relaced with ( G α I) where α is a suitable shift his is ossible when the G α I x = b is easily solved as with band atrices shifted syste ( ) 6 Orthogonal Iterations In this section we briefly exaine the siilarity and the difference between the new ethod and the Orthogonal Iterations ethod he last ethod is also called Subsace Iterations and Siultaneous Iterations eg [] [4] [5] [7]-[0] [3]-[5] It is aied at couting a cluster of doinant eigenvalues as defined in (3) he silest way to achieve this goal is erhas to aly a bloc version of the Power ethod on an atrix In this way the Power ethod is siultaneously alied on each colun of the atrix he Orthogonal Iterations ethod odifies this idea in a nuber of ways First as its nae says it orthogonalizes the iteration atrix which 8

8 A Dax ees it a full ran well-conditioned atrix Second although we are interested in eigenairs the iteration is alied on a larger atrix that has coluns where > his leads to faster convergence (see below) As before = + l where l is a sall ultile of he th iteration of the resulting ethod = 0 is coosed of the following three stes It starts with a atrix X that has orthonoral coluns Ste : Coute the roduct atrix Ste : Coute the Rayleigh uotient atrix and its doinant eigenvalues Y = GX S = X Y = X GX ( ) ( ) ( ) λ λ λ Ste 3: Coute a atrix Range Y his can be done by a QR factorization of Y ( ) Let the eigenvalues of G satisfy () hen for = the seuence λ ˆ λ = converges to zero at the sae asytotic rate as the seuence ˆ λ ˆ + λ = See for exale ([4] 57) or ([5] 368) herefore the larger is the faster is the convergence Moreover let the sectral decoosition of S have the for where X + whose coluns rovide an orthonoral basis of ( ) S = V DV ( ) ( ) ( ) ( ) ( ) = = { } V V I D diag λ λ and λ λ λ If at the end of Ste the atrix Y is relaced with YV then the coluns of X converge toward the corresonding eigenvectors of G However in exact arithetic both versions generate the sae seuence of Ritz values Hence the retrieval of eigenvectors can wait to the final stage he ractical ileentation of orthogonal iterations includes several further odifications such as siing Stes - 3 and locing For detailed discussions of these otions see [] [7]-[0] A coarison of the above orthogonal iteration with the new iteration shows that both ethods need about the sae aount of couter storage but the new ethod doubles the coutational effort er iteration he adatation of orthogonal iterations to handle other eriheral clusters reuires the shift oeration Another difference regards the rate of convergence In orthogonal iterations the rate is deterined by the ratio λ+ λ he theory behind the new ethod is not that decisive but our exerients suggest that it is uite faster 7 Restarted Lanczos Methods he current resentation of the new ethod is carried out by alying the Krylov inforation atrix (3)-(34) his version can be viewed as a Restarted Krylov ethod he Restarted Lanczos ethod is a sohisticated ileentation of this aroach that harnesses the Lanczos algorith to reduce the coutational effort er iteration As before the ethod is aied at couting a cluster of exterior eigenairs new inforation is gained fro an l -diensional Krylov subsace and = + l he th iteration = of the Ilicitly Restarted Lanczos ethod (IRLM) is coosed of the following four stes It starts with a tridiagonal atrix and the related atrix of Lanczos vectors n = L LL I which have been obtained by alying stes of Lanczos ethod on G 83

9 A Dax Ste : Coute the eigenvalues of Let ( ) ( ) ( ) ( ) + λ λ λ λ denote the couted eigenvalues where the first eigenvalues corresond to the target cluster Ste : Coute a atrix with orthonoral coluns such that Range( ) V V V I = λ λ he couta- V euals an invariant subsace of which corresonds to tion of V is carried out by conducting a QR factorization of the roduct atrix ( ) ( ) ( λ+ I) ( λ I) Ste 3: he above QR factorization is used to build a new tridiagonal atrix n L = LV R ( ) ( ) and the atrix his air of atrices has the roerty that it can be obtained by alying stes of Lanczos algorith on G starting fro soe (unnown) vector Ste 4: Continue l additional stes of Lanczos algorith to obtain + and L + he IRLM iterations are due to Sorensen [] he nae Ilicitly restarted refers to the fact that the starting vector (which initiates the restarted Lanczos rocess) is not couted For detailed descrition of this iteration see ([] 67-73) and [] See also [3] [5] A different ileentation of the Restarted Lanczos idea the hic-restarted Lanczos (RLan) ethod was roosed by Wu and Sion [7] he th iteration = of this ethod is coosed of the following five stes starting with and L as above Ste : Coute eigenairs of which corresond to the target cluster he corresonding eigenvectors of are assebled into a atrix U U U I = which is used to coute the related atrix of Ritz vectors ( ) ( ) ( ) ( ) X = LU Ste : Let λ λ and x x denote the couted Ritz airs he algorith uses these airs to n coute a vector r and scalars σ σ that satisfy the eualities and ( ) ( ) ( ) λ σ Gx x = r for = X r = o Ste 3: he vector y = r r is used to initiate a new seuence of Lanczos vectors he second vector ( ) y is obtained by orthogonalizing the vector Gy against y and the coluns of X Ste 4: Continue l additional stes of the Lanczos rocess that generate y y 3 l At the end of this rocess we obtain an atrix that has utually orthonoral coluns and satisfy is nearly tridiagonal: he ( ) ( ) ( ) ( ) ( ) ( ) ( ) L = x x y y y l L GL = where + + rincial subatrix of has an arrow-head shae ( ) ( ) with λ λ on the diagonal and σ σ on the sides Ste 5: Use a seuence of Givens rotations to colete the reduction of into a tridiagonal atrix + Siilarly L is udated to give L + For detailed descrition of the above iteration see [7] [8] Both IRLM and RLan are carried out with reor- 84

10 A Dax thogonalization of the Lanczos vectors he RLan ethod coutes the Ritz vectors while IRLM avoids this coutation Yet the two ethods are nown to be atheatically euivalent One difference between the new ethod and the Restarted Lanczos aroach lies in the coutation of the Rayleigh uotient atrix In our ethod this coutation reuires additional atrix-vector roducts which doubles the coutational effort er iteration A second difference lies in the starting vector of the l diensional Krylov subsace that is newly generated at each iteration In our ethod this vector is defined by (34) In RLan this vector is r Yet it is difficult to recon how this difference effects the rate of convergence A third difference arises when using acceleration techniues Let us consider for exale the use of ower acceleration with G relacing G In this case the Restarted Lanczos ethods coute a tridiagonal reduction of allows us to use saller l and reduces the ortion of tie that is sent on the Rayleigh-Ritz rocedure See the next section) A siilar rear alies to the use of Polynoial acceleration he new ethod can be viewed as generalization of the Restarted Lanczos aroach he generalization is carried out by relacing the Lanczos rocess with standard orthogonalization his silifies the algorith and clarifies the ain reasons that lead to fast rate of convergence One reason is that each iteration builds a new Krylov subsace using an iroved starting vector A second reason coes fro the orthogonality reuireent: he new Krylov subsace is orthogonalized against the current Ritz vectors It is this orthogonalization that ensures successive iroveent (he Restarted Lanczos algoriths achieve these tass in ilicit ways) G and Ritz values of 8 Nuerical Exerients G while our ethod coutes Ritz eigenairs of G (Power acceleration In this section we describe soe exerients that illustrate the behavior of the roosed ethod he test atrices have the for where V n is a rando orthonoral atrix G n = VDV (8) V V = I and D is a diagonal atrix { } n D = diag d d n (8) he ter rando orthonoral eans that V is obtained by orthonoralizing the coluns of a rando atrix R whose entries are rando nubers fro the interval [ ] he rando nubers generator is n of unifor distribution All the exerients were carried out with n = 00 and = 6 he values of l are secified in the coing tables (he diagonal entries of D are the eigenvalues of G Hence the structure of D is the aor factor that effects convergence Other factors lie the size of the atrix or the sarsity attern have inor effects) he coutations were carried out with MALAB We have used the following four tyes of test atrices: ye A atrices where ye B atrices where ye C atrices where ye D atrices where and d = 0 for = 00 (83) d = 0 for = 00 and d = 0 for = 0 00 (84) d = 0 for = 50 and d = 0 for = 5 00 (85) ( ) d = 5 d = 5 for = 50 (86a) d = 0 for = 0 00 (86b) he difference between the couted Ritz values and the desired eigenvalues of G is couted as follows 85

11 A Dax ( ) ( ) Let λ λ denote the eigenvalues of G which constitute the desired cluster Let λ λ denote the corresonding Ritz values which are couted at the th iteration = 0 hen the average difference between the corresonding eigenvalues is ( ) η = λ λ = 0 (87) = he figures in ables -4 rovide the values of η hey illustrate the way η converges to zero as increases he new ethod is ileented as described in Section 3 It starts by orthonoralizing a rando Krylov atrix of the for (3) he inforation atrix B is defined by (3)-(34) he Orthogonal iterations ethod is ileented as in Section 6 It starts fro a rando orthonoral atrix which is a coon default otion eg ([] 55) and ([3] 60) able Couting a doinant cluster with the new iteration Iter No ye A ye B ye C ye D l = l = 8 l = l = 8 l = l = 8 l = l = E 99E0 68E0 39E0 458E0 399E0 53E0 4E 48E0 96E0 E0 435E 985E 6E 409E 30E 4 5E0 93E 666E 695E 4 4E E 68E 4 345E E 8E 836E 4 0E 6 38E 3 804E 6 78E 5 0E E 47E 3 76E 5 33E 8 36E 5 46E 7 47E 7 49E 5 507E 7E 4 47E 6 386E 0 74E 7 59E 9 9E 8 5E 3 6 6E 3 493E 6 350E 8 45E 60E 9 770E 39E E 4 53E 7 368E 9 753E 575E E 394E 8 48E 5 0E 8 93E 5E 67E 8E 3 47E 0 5E 6 3E 0 6E 3 398E 3 900E 4 49E 7 3E 4 8E 9 3E able Couting a doinant cluster with orthogonal iterations Iter No ye A ye B ye C ye D l = 8 l = 30 l = 8 l = 30 l = 8 l = 30 l = 8 l = E 65E 579E 53E 663E 580E 399E 35E 355E 95E 43E 930E0 80E0 388E0 84E 6E 8E 74E 80E0 48E0 596E0 66E0 39E 96E 4 6E 807E0 36E0 00E0 33E0 07E0 649E0 77E E0 4E0 744E 89E 884E 39E 59E0 6E 54E0 80E 69E 05E 33E 44E 480E 9E 6 55E0 56E 357E 38E 3 938E 34E 3 30E 48E E 775E 76E 3 50E 4 70E 8E 4 373E E E 33E 69E 3 4E 5 79E 3 9E 5 E 875E 6 86

12 A Dax able 3 Couting a left-side cluster with the new iteration Iter No ye A ye B ye C ye D l = l = 8 l = l = 8 l = l = 8 l = l = E 94E E 997E0 499E0 470E0 69E0 55E0 30E 768E0 493E0 83E0 745E 36E 7E0 57E 77E0 300E0 64E0 689E 449E 3 45E 5 738E 8E E0 879E 45E0 80E 45E 5 3E 7 97E 4 3E E0 8E 4E0 563E 403E 6 806E 9 69E 5 3E 5 658E 67E 480E 87E 3 80E 8 360E 0 368E 7 67E E 36E 4 656E 87E 4 935E 0 566E 83E 9 533E E 573E 6 69E 3 909E 5 365E 74E 9E E 4 595E 7 973E 4 06E 5 34E 76E 3 454E 0 367E 5 474E 0 4E 4 98E 6 75E 3 84E 4 694E 7 970E 566E 5 584E E 8 47E 3 8E 5 7E 7 able 4 he use of Power acceleration to coute a doinant cluster of ye A atrix Iter No l = 6 l = l = 8 = = 4 = = 3 = 4 = = E 78E0 64E0 380E0 43E0 434E0 39E0 85E0 46E0 09E0 505E 84E 337E 6E 384E0 878E 94E 6E 38E 3 64E 30E E0 E 34E 348E 3 67E 7 773E 5 E E 55E 78E 4 504E 5 86E 8 746E 7 46E E 348E 874E 6 48E 7 49E 0 66E 8 96E 6 79E 83E 3 4E 6 34E 8 66E 7E 9 336E E 46E 4 33E 8 78E 0 3E 59E 8 09E 34E 5 35E 9 34E 33E 3 6E 0 33E 83E 7 934E 464E 3 393E 3 8E 3 793E 8 4E 4 76E 4 47E 9 6E E 6 3E able describes the coutation of doinant clusters with the new ethod he reading of this table is sile: We see for exale that after erforing 6 iterations on a ye B atrix η 6 = 350E 8 for l = and η 6 = 45E for l = 8 (he corresonding values of are = 8 and = 4 resectively) able describes the coutation of the sae doinant clusters with orthogonal iterations A coarison of the two tables suggests that the new ethod is considerably faster than Orthogonal iterations Another observation stes fro the first rows of these tables: We see that a rando Krylov atrix gives a better start than a rando starting atrix 87

13 A Dax he ability of the new ethod to coute a left-side cluster is illustrated in able 3 Note that the ye B atrix and the ye C atrix are ositive seidefinite atrices in which the left-side cluster is coosed fro the saller non-zero eigenvalues of G Both atrices have several zero eigenvalues In the ye B atrix the eigenvalues in the left-side cluster are uch saller than the doinant eigenvalues but the new ethod is able to distinguish these eigenvalues fro zero eigenvalues he erits of Power acceleration are deonstrated in able 4 On one hand it enables us to use a saller l which saves storage On the other hand it reduces the ortion of tie that is sent on orthogonalizations and the Rayleigh-Ritz rocess 9 Concluding Rears he new ethod is based on a odified interlacing theore which forces the Rayleigh-Ritz aroxiations to ove onotonically toward their liits he current resentation concentrates on the Krylov inforation atrix (3)-(34) but the ethod can use other inforation atrices he exerients that we have done are uite encouraging esecially when calculating eriheral clusters he theory suggests that the ethod can be extended to calculate certain non-eriheral clusters but in this case we face soe difficulties due to rounding errors Further odifications of the new ethod are considered in [3] References [] Bai Z Deel J Dongarra J Ruhe A and van der Vorst H (999) elates for the Solution of Algebraic Eigenvalue Probles: A Practical Guide SIAM Philadelhia [] Bauer FL (957) Das Verfahren der reeniteration und verwandte Verfahren zur Lösung algebraischers Eigenwertroblee Zeitschrift für angewandte Matheati und Physi ZAMP htt://dxdoiorg/0007/bf [3] Dax A Restarted Krylov Methods for Calculating Exterior Eigenvalues of Large Matrices ech Re Hydrological Service of Israel in Prearation [4] Deel JW (997) Alied Nuerical Linear Algebra SIAM Philadelhia htt://dxdoiorg/037/ [5] Golub GH and Van Loan CF (983) Matrix Coutations Johns Hoins University Press Baltiore [6] Horn RA and Johnson CR (985) Matrix Analysis Cabridge University Press Cabridge htt://dxdoiorg/007/cbo [7] Parlett BN (980) he Syetric Eigenvalue Proble Prentice-Hall Englewood Cliffs [8] Reinsch CH (97) Siultaneous Iteration Method for Syetric Matrices In: Wilinson JH and Reinsch CH Eds Handboo for Autoatic Coutation (Linear Algebra) Sringer-Verlag New Yor [9] Rutishauer H (969) Coutational Asects of F L Bauer s Siultaneous Iteration Method Nuerische Matheati htt://dxdoiorg/0007/bf06569 [0] Rutishauser H (970) Siultaneous Iteration Method for Syetric Matrices Nuerische Matheati htt://dxdoiorg/0007/bf09773 [] Sorensen DC (99) Ilicit Alication of Polynoial Filters in a -Ste Arnoldi Method SIAM Journal on Matrix Analysis and Alications htt://dxdoiorg/037/06305 [] Stewart GW (969) Accelerating the Orthogonal Iteration for the Eigenvalues of a Heritian Matrix Nuerische Matheati htt://dxdoiorg/0007/bf06543 [3] Stewart GW (00) Matrix Algoriths Volue II: Eigensystes SIAM Philadelhia [4] refethen LN and Bau III D (997) Nuerical Linear Algebra SIAM Philadelhia htt://dxdoiorg/037/ [5] Watins DS (007) he Matrix Eigenvalue Proble: GR and Krylov Subsace Methods SIAM Philadelhia htt://dxdoiorg/037/ [6] Wilinson JH (965) he Algebraic Eigenvalue Proble Clarendon Press Oxford [7] Wu K and Sion H (000) hic-restarted Lanczos Method for Large Syetric Eigenvalue Probles SIAM Journal on Matrix Analysis and Alications htt://dxdoiorg/037/s [8] Yaazai I Bai Z Sion H Wang L and Wu K (00) Adative Proection Subsace Diension for the 88

14 A Dax hic-restart Lanczos Method ACM ransactions on Matheatical Software 37-8 htt://dxdoiorg/045/ [9] Zhang F (999) Matrix heory: Basic Results and echniues Sringer-Verlag New Yor htt://dxdoiorg/0007/