Accurate eigenvalue decomposition of arrowhead matrices and applications

Transcription

1 Accurate egenvalue decomposton of arrowhead matrces and applcatons N. Jakovčevć Stor a,1,, I. Slapnčar a,1, J. Barlow b,2 a Faculty of Electrcal Engneerng, Mechancal Engneerng and Naval Archtecture, Unversty of Splt, Rudjera Boškovća 32, Splt, Croata b Department of Computer Scence and Engneerng, The Pennsylvana State Unversty, Unversty Park, PA , USA Abstract We present a new algorthm for solvng an egenvalue problem for a real symmetrc arrowhead matrx. The algorthm computes all egenvalues and all components of the correspondng egenvectors wth hgh relatve accuracy n O(n 2 ) operatons. The algorthm s based on a shft-and-nvert approach. Double precson s eventually needed to compute only one element of the nverse of the shfted matrx. Each egenvalue and the correspondng egenvector can be computed separately, whch makes the algorthm adaptable for parallel computng. Our results extend to Hermtan arrowhead matrces, real symmetrc dagonal-plus-rank-one matrces and sngular value decomposton of real trangular arrowhead matrces. Keywords: egenvalue decomposton, arrowhead matrx, hgh relatve accuracy, sngular value decomposton 2000 MSC: 65F15 1. Introducton and Prelmnares In ths paper we consder egenvalue problem for a real symmetrc matrx A whch s zero except for ts man dagonal and one row and column. Snce egenvalues are nvarant under smlarty transformatons, we can symmetrcally permute the rows and the columns of the gven matrx. Therefore, we assume wthout loss of generalty that the matrx A s a n n real symmetrc arrowhead matrx of the form Correspondng author Emal addresses: nevena@fesb.hr (N. Jakovčevć Stor), van.slapncar@fesb.hr (I. Slapnčar), barlow@cse.psu.edu (J. Barlow) 1 The research of Ivan Slapnčar and Nevena Jakovčevć Stor was supported by the Mnstry of Scence, Educaton and Sports of the Republc of Croata under grant The research of Jesse L. Barlow was supported by the Natonal Scence Foundaton under grant CCF Preprnt submtted to Elsever March 1, 2013

2 where s dagonal matrx of order n 1, [ D z A = z T α D = dag(d 1, d 2,..., d n 1 ) ], (1) z = [ ζ 1 ζ 2 ζ n 1 ] T (2) s a vector and α s a scalar. Such matrces arse n the descrpton of radatonless transtons n solated molecules [3], oscllators vbratonally coupled wth a Ferm lqud [8], quantum optcs [15] (see also Example 4). Such matrces also arse n solvng symmetrc real trdagonal egenvalue problems wth the dvde-and-conquer method [11]. In ths paper we present an algorthm whch computes all egenvalues and all components of the correspondng egenvectors wth hgh relatve accuracy n O(n 2 ) operatons. Wthout loss of generalty we may assume that A s rreducble, that s, ζ 0, for all and d d j, for all j,, j = 1,..., n 1. If A has a zero n the last column, say ζ = 0, then the dagonal element d s an egenvalue whose correspondng egenvector s the -th unt vector, and we can reduce the sze of the problem by deletng the -th row and column of the matrx, eventually obtanng a matrx for whch all elements ζ j are nonzero. If d = d j, then d s egenvalue of matrx A (ths follows from the nterlacng property (7)), and we can reduce the sze of the problem by annhlatng ζ j wth a Gvens rotaton n the (, j)-plane and proceedng as n the prevous case. Further, by symmetrc row and column pvotng, we can order elements of D such that d 1 > d 2 > > d n 1. (3) Hence, we wll consder only ordered and rreducble arrowhead matrces. Wthout loss of generalty we can also assume that ζ > 0 for all, whch can be attaned by pre- and post-multplcaton of the matrx A wth D = dag(sgn(ζ ))). Let A = V ΛV T (4) be the egenvalue decomposton of A. Here Λ = dag(λ 1, λ 2,..., λ n ) s a dagonal matrx whose dagonal elements are the egenvalues of A, and V = [ v 1 v n ] 2

3 s an orthonormal matrx whose columns are the correspondng egenvectors. The egenvalues of A are the zeros of the Pck functon (see [4, 16]) n 1 ζ 2 f(λ) = α λ d λ = α λ zt (D λi) 1 z, (5) =1 and the correspondng egenvectors are gven by v = x [ (D λ I) 1 z, x = x 2 1 ], = 1,..., n. (6) Dagonal elements of the matrx D, d, are called poles of the functon f. Notce that (3) and the Cauchy nterlacng theorem [10, Theorem 8.1.7] appled to matrces D and A mply the nterlacng property λ 1 > d 1 > λ 2 > d 2 > > d n 2 > λ n 1 > d n 1 > λ n. (7) Snce A s symmetrc, ts egenvalues may be computed by nvokng any of a number of standard programs (LAPACK [1]). However, these programs usually begn wth an ntal reducton of the matrx to trdagonal form [17], or as proposed n [16], wth an alternatve whch takes advantage of the structure of A by fndng the zeros of the Pck functon gven n (5), for the egenvalues of A. Ths results n an algorthm whch requres only O(n 2 ) computatons and O(n) storage. Although the dea s conceptually smple and n fact has been used to solve other egenvalue problems of specal structure [2, 5, 6, 7], the computaton s not always stable [11]. Namely, f the computed egenvalues λ are not accurate enough, then the computed egenvectors v may not be suffcently orthogonal (see Example 3). The exstng algorthms for arrowhead matrces [11, 16] obtan orthogonal egenvectors wth the followng procedure: - compute the egenvalues λ of A by solvng (5); - construct a new matrx [ D z à = z T α ] by solvng nverse problem wth the prescrbed egenvalues λ, and dagonal matrx D, that s, compute new z and α as ) ) ζ = ( d λ ) ) ( λj d n 1 ( λj d n ( λ1 d (d j 1 d ) (d j d ), - compute egenvectors of à by (6). j=2 α = λ n 1 ) n + ( λj d j. j=1 j=+1 3

4 Snce the formulas for ζ nvolve only multplcatons, dvson and subtractons of exact quanttes, each ζ s computed wth relatve error of O(ε M ), where ε M denotes the machne precson. 3 Therefore, à = A + δa, where δa 2 = O(ɛ M ). Here 2 denotes the spectral matrx norm. We conclude that the computed egenvalues λ satsfy standard perturbaton bounds lke those from [10, Corollary 8.1.6]. Further, snce λ are the egenvalues of the matrx à computed to hgher relatve accuracy, the egenvectors computed by (6) are orthogonal to machne precson. For detals see [11, 16]. Our algorthm uses a dfferent approach. Accuracy of the egenvectors and ther orthogonalty follows from hgh relatve accuracy of the computed egenvalues and there s no need for follow-up orthogonalzaton. The algorthm s based on shft-and-nvert technque. Bascally, the egenvalue λ s computed as the largest or the smallest egenvalue of the nverse of the matrx shfted to the pole d whch s nearest to λ, that s, λ = 1 ν + d, (8) where ν s ether smallest or largest egenvalue of the matrx A 1 (A d I) 1. Inverses of arrowhead matrces are structured n the followng manner (here stands for non-zero element): the nverse of an arrowhead matrx wth zero on the shaft s a permuted arrowhead matrx wth zero on the shaft, 0 1 =, 0 and the nverse of the full arrowhead matrx s a dagonal-plus-rank-one (DPR1) matrx, 1 = ± uut. 0 3 The machne precson ε M s defned as a smallest postve number such that n the floatng-pont arthmetc 1 + ε M 1. In Matlab or FORTRAN REAL(8) arthmetc ε M = , thus the floatng-pont numbers have approxmately 16 sgnfcant decmal dgts. The term double of the workng precson means that the computatons are performed wth numbers havng approxmately 32 sgnfcant decmal dgts, or wth the machne precson equal to ε 2 M. 4

5 Our algorthm s completely parallel, snce the computaton of one egenvalue and ts egenvector s completely ndependent of the computaton of other egenvalues and egenvectors. In Secton 2 we descrbe the basc dea of our algorthm named aheg (ArrowHead EIGenvalues). In Secton 3 we dscuss the accuracy of the algorthm. In Secton 4 we present the complete algorthm whch uses double of the workng precson, f necessary. In Secton 5 we llustrate algorthm wth few examples and n Secton 6 we apply our results to egenvalue decomposton of Hermtan arrowhead matrx, sngular value decomposton of real trangular arrowhead matrx and egenvalue decomposton of real symmetrc dagonal-plus-rank-one matrx. The proofs are gven n Appendx A. 2. Basc shft-and-nvert algorthm Let λ be an egenvalue of A, let v be ts egenvector, and let x be the unnormalzed verson of v from (6). Let d be the pole whch s closest to λ. Clearly, from (7) t follows that ether λ = λ or λ = λ +1. Let A be the shfted matrx where A = A d I = D z ζ 0 0 D 2 z 2 z1 T ζ z2 T a D 1 = dag(d 1 d,..., d 1 d ), D 2 = dag(d +1 d,..., d n 1 d ), z 1 = [ ζ 1 ζ 2 ζ 1 ] T, z 2 = [ ζ +1 ζ +2 ζ n 1 ] T, a = α d. Notce that D 1 (D 2 ) s postve (negatve) defnte. Obvously, f λ s an egenvalue of A, then A 1 = µ = λ d, (9) s an egenvalue of A, and vce versa, and they both have the same egenvector. The nverse of A s D1 1 w w1 T b w2 T 1/ζ 0 w 2 D /ζ 0 0, (10) 5

6 where where Notce that w 1 = D1 1 z 1 1, ζ w 2 = D2 1 z 1 2, ζ b = 1 ( a + z T ζ 2 1 D1 1 z 1 + z2 T D2 1 z 2). (11) b = f (d ) /ζ 2 f (d ) = α d z T ( D d I ) 1 z where D s the dagonal matrx D wthout d and z s z wthout ζ. The egenvector x from (6) s gven by (D 1 µi) 1 z 1 x = x 1. x n = ζ µ (D 2 µi) 1 z 2 1. (12) If λ s an egenvalue of A whch s closest to the pole d, then µ s the egenvalue of matrx A whch s closest to zero and ν = 1 µ = ± A 1 2. In ths case, f all entres of A 1 are computed wth hgh relatve accuracy, then, accordng to standard perturbaton theory, ν s computed to hgh relatve accuracy (by any reasonable algorthm). In Secton 3 we show that all entres of A 1 are ndeed computed to hgh relatve accuracy, except possbly b (see (11)). If b s not computed to hgh relatve accuracy and t nfluences A 1 2, t s suffcent to compute t n double of the workng precson (see Secton 4). Further, f µ s not the egenvalue of A whch s closest to zero, then ν < A 1 2, and the quantty A 1 K ν = 2 (13) ν tells us how far s ν from the absolutely largest egenvalue of A 1. If K ν 1, then the standard perturbaton theory does not guarantee that the egenvalue µ wll be computed wth hgh relatve accuracy. Remedes of ths stuaton are descrbed n Remark 2. Fnally, the exteror egenvalues λ 1 and λ n of A are excluded from the above reasonng. The absolutely larger of those s equal to A 2 and s, accordng to standard perturbaton theory, computed wth hgh relatve accuracy. The 6

7 other exteror egenvalue can also be computed wth hgh relatve accuracy f t s of the same order of magntude as the larger one. If ths s not the case, the absolutely smaller egenvalue s computed as nteror egenvalues of A. Wth ths approach the componentwse hgh relatve accuracy of the egenvectors computed by (12) follows from hgh relatve accuracy of the computed egenvalues (see Theorem 3). Componentwse hgh relatve accuracy of the computed egenvectors mples, n turn, ther orthogonalty. The descrbed procedure s mplemented n algorthm aheg basc (Algorthm 1). The computaton of the nverse of the shfted matrx, A 1, accordng to formulas (10) and (11), s mplemented n Algorthm 2. Algorthm 3 computes the largest or the smallest zero of the Pck functon (5) by bsecton. Gven egenvalue λ, Algorthm 4 computes the correspondng egenvector by (6) or (12), respectvely. 3. Accuracy of the algorthm We now consder numercal propertes of Algorthms 1, 2, 3, and 4. We assume tha standard model of floatng pont arthmetc where subtracton s preformed wth guard dgt, such that [9, 18, 10, 19] fl(a b) = (a b)(1 + ε ), ε ε M, {+,,, /}, where ε M s machne precson. In the statements of the theorems and ther proofs we shall use the standard frst order approxmatons, that s, we neglect the terms of order O(ε 2 M ) or hgher. Moreover, we assume that nether overflow or underflow occurs durng the computaton. We shall use the followng notaton: Matrx Exact egenvalue Computed egenvalue A λ λ A µ à = fl(a ) µ µ = fl( µ) A 1 ν (A ) = fl(a 1 ) ν ν = fl( ν) (14) Here à = fl (A ) = D 1 (I + E 1 ) 0 0 z ζ 0 0 D 2 (I + E 2 ) z 2 z1 T ζ z2 T a (1 + ε a ), where E 1 and E 2 are dagonal matrces whose elements are bounded by ε M n absolute values and ε a ε M. 7

8 Algorthm 1 [λ, v] = aheg basc (D, z, α, k) % Computes the k-th egenpar of an rreducble arrowhead matrx % A = [dag (D) z; z α] n = max(sze(d)) + 1 % Set the shft ndex ntally to zero = 0 % Compute exteror egenvalue (k = 1 or k = n) f k == 1 λ = bsect(d, z, α, R ) λ n = bsect(d, z, α, L ) f λ n /λ < 10 % the egenvalue s accurate v = vect(d, z, λ) else % determne the shft σ, the shft ndex, and whether λ s on the left % or the rght sde of the nearest pole σ = d 1 = 1 sde = R end elsef k == n λ = bsect(d, z, α, L ) λ 1 = bsect(d, z, α, R ) f λ 1 /λ < 10 % the egenvalue s accurate v = vect(d, z, λ) else % determne the shft, shft ndex, and sde σ = d n 1 = n 1 sde = L end 8

9 Algorthm 1 (Contnued) else % Compute nteror egenvalue (k {2,..., n 1}) % Determne the shft (the nearest pole) and the shft ndex Dtemp = D d k atemp = α d k mddle = Dtemp k 1 /2 F mddle = atemp mddle (z 2./(Dtemp mddle)) f F mddle < 0 σ = d k = k sde = R else σ = d k 1 = k 1 sde = L end end f 0 % Compute the nverse of the shfted matrx, A 1 [nvd 1, nvd 2, w 1, w 2, w ζ, b] = nva(d, z, α, ) % Compute the leftmost or the rghtmost egenvalue of A 1 ν = bsect([nvd 1 ; 0; nvd 2 ], [w 1 ; w ζ ; w 2 ], b, sde) % Compute the correspondng egenvector µ = 1/ν v = vect(d σ, z, µ) % Shft the egenvalue back λ = µ + σ end Algorthm 2 [nvd 1, nvd 2, w 1, w 2, w ζ, b] = nva (D, z, α, ) % Computes the nverse of an arrowhead matrx A = [dag(d d ) z; z α d ] % accordng to (10) and (11). n = max(sze(d)) + 1 D = D d a = α d w 1 = z 1: 1./D 1: 1 /z w 2 = z +1:n 1./D +1:n 1 /z w ζ = 1/z nvd 1 = 1./D 1: 1 nvd 2 = 1./D +1:n 1 b = ( a + sum(z 1: 1.^2./D 1: 1 ) + sum(z +1:n 1.^2./D +1:n 1 ))/z ^2 9

10 Algorthm 3 λ = bsect (D, z, α, sde) % Computes the leftmost (for sde= L ) or the rghtmost (for sde= R ) egenvalue % of an arrowhead matrx A = [dag (D) z; z α] by bsecton. n = max(sze(d)) + 1 % Determne the startng nterval for bsecton, [lef t, rght] f sde == L left = mn{d z, α z 1 } rght = mn d else rght = max{d + z, α + z 1 } left = max d end % Bsecton mddle = (left + rght)/2 whle (rght left)/abs(mddle) > 2 eps F mddle = α mddle sum(z.^2./ (D mddle)) f F mddle > 0 left = mddle else rght = mddle end mddle = (left + rght)/2 end % Egenvalue λ = rght Algorthm 4 v = vect (D, z, λ) % Computes the egenvector of an arrowhead matrx A = [dag(d) z; z α] % whch corresponds to the egenvalue λ by usng (6). v = [z./(d λ); 1] v = v/ v 2 10

11 Further we defne the quanttes κ λ, κ µ and κ b as follows: λ = fl (λ) = λ (1 + κ λ ε M ), (15) µ = fl (µ) = µ (1 + κ µ ε M ), (16) b = fl (b) = b (1 + κb ε M ). (17) We also defne the quantty K b = a + z1 T D1 1 z 1 + z2 T D 1 a + z T 1 D1 1 z 1 + z2 T D Connecton between accuracy of λ and µ Let λ = µ + d 2 z 2 2 z 2. (18) be an egenvalue of the matrx A, where µ s the correspondng egenvalue of the shfted matrx A = A d from whch λ s computed. Let λ = fl( µ + d ) be the computed egenvalue. Theorem 1 gves us dependency of accuracy of λ n (15) upon accuracy of µ n (16). Theorem 1. For λ and λ from (15) and µ and µ from (16) we have κ λ d + µ λ ( κ µ + 1). (19) Proofs of ths theorem and subsequent theorems are gven n Appendx A. From Theorem 1 we see that the accuracy of λ depends on κ µ and the sze of the quotent d + µ. (20) λ Theorem 2 analyzes the quotent (20) wth respect to the poston of λ and sgns of µ and the neghborng poles. Theorem 2. Let the assumptons of Theorem 1 hold. () If (see Fgure 1 ()) then sgn (d ) = sgn (µ), d + µ λ = 1. () If λ s between two poles of the same sgn and sgn (d ) sgn (µ) (see Fgure 1 ()), then d + µ 3. λ 11

12 µ {}}{ µ {}}{ d +1 () λ d 0 d λ () d 1 0 Fgure 1: Typcal stuatons from Theorem 2 Theorem 2 does not cover the followng cases: (a) If d 1 < 0, then µ > 0. If, further, d 1 µ, then λ 1 s near zero, and ( d 1 + µ )/ λ 1 1 (see Fgure 2 (a)). 4 (b) If d n > 0, then µ < 0. If, further, d n µ, then λ n s near zero, and agan ( d n + µ )/ λ n 1. 5 (c) If λ s between two poles of the dfferent sgns and sgn (d ) sgn (µ), then ether d +1 < 0 < d and µ < 0, or d < 0 < d 1 and µ > 0. In both cases, f, addtonally, d µ, then λ s near zero, and ( d + µ )/ λ 1 (see Fgure 2 (c)). µ {}}{ µ {}}{ d 1 λ 1 (a) 0 d +1 0 (c) λ d Fgure 2: Typcal stuatons for specal cases Snce only one of these three cases can occur, Theorems 1 and 2 mply that for all egenvalues λ σ (A), but eventually one, t holds d + µ λ 3. If one of the above cases does occur, remedes are gven n the followng remark. Remark 1. If one of the cases (a), (b) or (c) occurs, then λ s an egenvalue of A nearest to zero, and we can accurately compute t from the nverse of A. 4 In ths case λ 1 s computed as a dfference of two close quanttes and cancellaton can occur. 5 In ths case λ n s computed as a dfference of two close quanttes and cancellaton can occur. 12

13 Notce that the nverse s of an unreduced arrowhead matrx wth non-zero shaft s a dagonal-plus-rank-one (DPR1) matrx of the form [ ] A 1 D 1 = + ρuu T, 0 where u = [ z T D 1 1 ] T, ρ = 1 a z T D 1 z. Egenvalues of A 1 are zeros of (see [2, 14]) ϕ (λ) = 1 + ρ n j=1 u 2 j d j λ. Snce the absolutely largest egenvalue of A 1 s computed accurately accordng to standard perturbaton theory, and 1/ λ = A 1 2, λ s also computed wth hgh relatve accuracy. In computng matrx A 1, eventually ρ needs to be computed n hgher precson. For more detals see Remark 2. We stll need to bound the quantty κ µ from (19). Ths quantty essentally depends on the accuracy of fl(b). The bound for κ µ s gven n Theorem Accuracy of the egenvectors Snce the egenvector s computed by (12), ts accuracy depends on the accuracy of µ as descrbed by the followng theorem: Theorem 3. Let (16) hold and let x 1 (D 1 (I + E 1 ) µi) 1 z 1 x =. = fl( ζ µ (D 2 (I + E 2 ) µi) 1 z 2 ) (21) x n 1 be the computed un-normalzed egenvector correspondng to µ and λ. Then x j = x j ( 1 + εxj ), ε xj 3 ( κ µ + 3) ε M, j = 1,..., n. In other words, f κ µ s small, then all components of the egenvector are computed to hgh relatve accuracy. Snce the accuracy of λ and x depends on the accuracy of µ (on the sze of κ µ ) n the next three subsectons tells we dscuss the accuracy of µ. Snce µ s computed as an nverse of the egenvalue of the matrx fl(a 1 ), we frst dscuss the accuracy of that matrx. 13

14 3.3. Accuracy of the matrx A 1 We have the followng theorem: Theorem 4. For the computed elements of the matrx A 1 from (10) and (11) for all (j, k) (, ) we have ( A 1 )jk = fl ( A 1 )jk = ( A 1 ) For the computed element b ( A 1 where K b s defned by (18). ) jk (1 + ε jk), ε jk 3ε M. from (17) we have κ b 3K b, The above theorem states that all elements of the matrx A 1 are computed wth hgh relatve accuracy except possbly b. Therefore, we have to montor whether b s computed accurately, and, f not, t needs to be computed n double of the workng precson (see Secton 4 for detals) Accuracy of bsecton Let λ max be the absolutely largest egenvalue of a symmetrc arrowhead matrx A, an let λ max be the egenvalue computed by bsecton as mplemented n Algorthm 3. The error bound from [16, Secton 3.1] mmedately mples that λmax λ max = κ bs ε M, κ bs 1.06n ( n + 1 ). (22) λ max Notce that the smlar error bound holds for all egenvalues whch are of the same order of magntude as λ max Accuracy of exteror egenvalues of A 1 The desred nteror egenvalue and, n some cases, also absolutely smaller exteror egenvalue λ of A s n Algorthm 1 computed by (8), where ν s one of the exteror egenvalues of the matrx A 1. The followng theorem covers the case when ν s the absolutely largest egenvalue of A 1 2, and gves two dfferent bounds. Theorem 5. Let A 1 be defned by (10) and let ν be ts egenvalue such that ν = A 1 2. (23) Let ν be the exact egenvalue of the computed matrx Then (A 1 ) ( ) = fl A 1. Let ν = ν (1 + κ ν ε M ). (24) { κ ν mn 3 n max{1, K b }, 6 ( n + 1 n 1 ζ where K b s defned by (18). k=1 k ζ k )}, (25) 14

15 3.6. Fnal error bounds All prevous error bounds are summarzed as follows. Theorem 6. Let λ be the computed egenvalue of an unreduced arrowhead matrx A, let µ be computed egenvalue of the matrx à from (9), and let ν be the correspondng computed egenvalue of the matrx (A ) 1 from (10). If µ s the egenvalue of A closest to zero (or, equvalently, f (23) holds), then the error n the computed egenvalue λ s gven by (15) wth κ λ 3( κ ν + κ bs ) + 4, (26) and the error n the computed un-normalzed egenvector x s gven by Theorem 3 wth κ µ κ ν + κ bs + 1, (27) where κ ν s bounded by (25) and κ bs s defned by (22). Snce we are essentally usng the shft-and-nvert technque, we can guarantee hgh relatve accuracy of the computed egenvalue and hgh componentwse relatve accuracy of the computed egenvector f ν s such that ν = O( A 1 2 ) and t s computed accurately. Ths s certanly fulflled f the followng condtons are met: C1. The quantty K ν from (13) s moderate, and C2. () ether the quantty K b from (18) s small, or () the quantty 1 n 1 ζ ζ k from (25) s of order O(n). k=1 k The condton C1 mples that ν wll be computed accurately accordng to the standard perturbaton theory. The condtons C2 () or C2 () mply that κ ν from (25) s small, whch, together wth C1, mples that ν s computed accurately. If the condton C1 does not hold, that s, f K ν 1, remedes are gven n Remark 2 below. If nether of the condtons C2 () and C2 () holds, the remedy s to compute b n double of the workng precson as descrbed n Secton 4. Remark 2. If λ = λ 1 or λ = λ n, t can be computed from the startng matrx A as n Algorthm 1. We have two more possbltes: (a) we can compute λ by shftng to another neghborng pole provded that K ν s n ths case small (shftng to the pole d 1 nstead of d n Fgure 3 (a)), (b) f shftng to another neghborng pole s not possble (K ν 1, see Fgure 3 (b)), we can nvert A σi, where shft σ s chosen near λ, and σ / {λ, d, d 1 }. Ths results n a DPR1 matrx [ ] (A σi) 1 (D σi) 1 = + ρuu T, 0 15

16 where u = [ z T (D σi) 1 1 ] T, ρ = 1 a z T (D σi) 1 z. Egenvalues of ths matrx are zeros of ϕ (λ) = 1 + ρ n j=1 u 2 j (d j σ) λ, and the absolutely largest egenvalue s computed accurately. Eventually, ρ needs to be computed n hgher precson. 6 d d 1 d d 1 λ +1 λ(λ ) λ 1 λ +1 λ(λ ) λ 1 (a) (b) Fgure 3: Typcal stuatons from Remark 2 4. Fnal algorthm If nether of the condtons C2 () and C2 () hold, n order to guarantee that λ wll be computed wth hgh relatve accuracy, the element b from the matrx A 1 needs to be computed n hgher precson. The followng theorem mples that f 1 K b O(1/ε M ), t s suffcent to evaluate (11) n double of the workng precson. 7 8 Theorem 7. If a > 0 n (11), set and f a < 0 n (11) set P = a + z T 1 D 1 1 z 1, Q = z T 2 D 1 2 z 2, P = z T 1 D 1 1 z 1, Q = a z T 2 D 1 2 z 2. 6 Determnng whether ρ needs to be computed n hgher precson s done smlarly as determnng whether element b of A 1 needs to be computed n hgher precson, whch s descrbed n Secton 4. Further, Theorem 7 mples that t suffces to compute ρ n double of the workng precson. 7 If K b O(1/ε M ), that s, f K b = 1/ε E for some ε E < ε M, then, n vew of Theorem 7, b needs to be computed wth extended precson ε E. 8 Usage of hgher precson n conjuncton wth the egenvalue computaton for DPR1 matrces s analysed n [2], but there the hgher precson computaton s potentally needed n the teratve part. Ths s less convenent than our approach where the hgher precson computaton s used only to compute one element. 16

17 Notce that n both cases P, Q 0 and b = (P Q)/ζ 2. Let P = fl(p ) and Q = fl(q) be evaluated n standard precson, ε M. Assume that P Q and K b O(1/ε M ). If P, Q and b are all evaluated n double of the workng precson, ε 2 M, then (17) holds wth κ b O(n). We summarze the above results n one, complete algorthm, aheg. The algorthm frst checks the components of the vector z. If they are of the same order of magntude, the egenpar (λ, v) s computed by Algorthm 1. If that s not the case, the quantty K b s computed, and f K b 1, the egenpar (λ, v) s computed by Algorthm 1 but wth evaluaton of b n double of the workng precson. At the end, the quantty K ν s computed, and f K ν 1, one of the remedes from Remark 2 s appled. Algorthm 5 [λ, v] = aheg (D, z, α, k) % Computes the k-th egenpar of an rreducble arrowhead matrx % A = [dag (D) z; z α] f the quantty ( n 1 ζ k k=1 k ) / ζ from (25) s of O(n) % standard precson s enough [λ, v] = aheg basc(d, z, α, k) else compute the quantty K b from (18) f K b 1 % double precson s necessary [λ, v] = aheg basc(d, z, α, k) wth evaluaton of b n double precson else % standard precson s enough [λ, v] = aheg basc(d, z, α, k) end end compute the quantty K ν from (13) f K ν 1 apply one of the remedes from Remark 2 end 4.1. On mplementng double precson Implementaton of the double of the workng precson depends upon whether the nput s consdered to be bnary or decmal. Double standard precson n Matlab, whch assumes that nput s bnary, s obtaned by usng a combnaton of commands vpa, dgts and double [13], where - dgts(d) specfes the number of sgnfcant decmal dgts d used to do varable precson arthmetc vpa, 17

18 - vpa(x) uses varable-precson arthmetc to compute x to d decmal dgts of accuracy, - double(x) converts x to standard precson. The assgnment a1=vpa(a,32) pads the bnary representaton of a wth zeros, whch means that the decmal nterpretaton of the varable a1 may have non-zero entres after 16-th sgnfcant decmal dgt. The same effect s obtaned n Intel FORTRAN compler fort [12] by the followng program segment real(8) a real(16) a1... a1=a However, the user can assume that the true nput s gven as a decmal number, whch s, for example, assumed by extended precson computaton n Mathematca [20]. In ths case, the optons n Matlab are to ether use symbolc computaton, or to cast the nput to a strng, and then convert t to extended precson: a1=vpa(num2str(a,16),32) In ths case, the the decmal nterpretaton of the varable a1 has all zero entres after 16-th sgnfcant decmal dgt, but the bnary representaton of the varable a s, n general, padded wth non-zero entres. The same effect s obtaned n fort wrtng to and readng from a strng varable as n the followng program segment: real(8) a real(16) a1 character(25) strng... wrte(strng,*) a read(strng,*) a1 If the nput conssts of numbers for whch decmal and bnary representaton are equal (for example, ntegers, as n Example 3 below), then the two above approaches gve the same results. 5. Numercal Examples We llustrate out algorthm wth four numercally demandng examples. Examples 1 and 2 llustrate Algorthm 1, Example 3 llustrates the use of double precson arthmetc, and Example 4 llustrates and applcaton of hgher dmenson. 18

19 Example 1. In ths example both quanttes K ν from (13) and K b from (18) are for all egenvalues approxmately equal to 1, so we guarantee that all egenvalues and all components of ther correspondng egenvectors are computed wth hgh relatve accuracy by Algorthm 5 (aheg) usng only standard machne precson. Let A = The egenvalues computed by Matlab [13] routne eg, Algorthm 5 and Mathematca [20] wth 100 dgts precson, are, respectvely: λ (eg) λ (aheg) λ (Math) We see that even the tnest egenvalues λ 3 and λ 4, computed by Algorthm 5, are exact to the machne precson, whch s not true for the egenvalues computed by eg. Because of the accuracy of the computed egenvalues, the egenvectors computed by Algorthm 5 are componentwse accurate up to machne precson, and therefore, orthogonal up to machne precson. For example: v (eg) 4 v (aheg) 4 v (Math) Example 2. In ths example, despte very close dagonal elements, we agan guarantee that all egenvalues and all components of ther correspondng egenvectors are computed wth hgh relatve accuracy, wthout deflaton. Let A = 1 + 4ε M ε M ε M ε M where ε M = = For ths matrx the quanttes K ν and K b are agan of order one for all egenvalues, so Algorthm 5 uses only. 19

20 standard workng precson. The egenvalues computed by Matlab, Algorthm 5 and Mathematca wth 100 dgts precson, are, respectvely: λ (eg) λ (aheg) λ (Math) Although the egenvalue computed by Matlab appear to be accurate, they are not. Namely, λ (aheg) 2 λ (aheg) 3 = , whle λ (eg) 2 λ (eg) 3 = 0, so the egenvalues computed by Matlab do not satsfy the nterlacng property. Notce that despte of very close egenvalues, Algorthm 5 works wthout deflaton. Due to the accuracy of the computed egenvalues, the egenvectors computed by Algorthm 5 are componentwse accurate up to the machne precson, and are therefore orthogonal. For example: 9 v (eg) 3 v (aheg) 3 v (Math) Example 3. In ths example we can guarantee all egenvalues and egenvectors, componentwse wll be computed wth hgh relatve accuracy only f we use double of the workng precson when computng b from (11) n matrces A 1 2, A 1 3, A 1 4 and A 1 5. Let A = Snce, as descrbed n Secton 4.1, Mathematca uses decmal representaton of the nput, n order to obtan accurate egenvectors we need to defne ε M n Mathematca wth the output of Matlab s command vpa(eps), ε M =

21 The quanttes K ν and K b are: 10 K ν K b It s clear, from the condton numbers, that the element b n each of the matrces A 1 2, A 1 3, A 1 4 and A 1 5 needs to be computed n double of the workng precson. For example, A 2 = A d 2 I = The element b = [ A 1 ] 2 computed by Algorthm 2 gves b = , 22 Matlab s routne nv yelds b = , whle computng b n double of the workng precson gves the correct value b = Egenvalues computed by Algorthm 1 (aheg basc, usng only standard workng precson), Algorthm 5 (aheg, usng double of the workng precson to compute respectve b s) and Mathematca wth 100 dgts precson, respectvely, are: λ aheg basc λ aheg λ Math The egenvectors computed by Algorthm 5 are componentwse accurate to machne precson and therefore orthogonal. Example 4. Ths example comes from the research related to decay of excted states of quantum dots n n real photon crystals [15]. In ths case - α s quantum dot transton frequency, - d s a frequency of the -th optcal mode, and 10 Algorthm 5 does not compute K ν and K b for the frst egenvalue, snce t s an absolutely largest one. 21

22 - ζ s an nteracton constant of the quantum dot wth the -th optcal mode. The sze of the matrx s changeable but, n realstc cases, t s between 10 3 and We ran a test example for n = 2501 where, typcally, d [ , ], ζ [ , ], α = For ths matrx the condton number K ν 1 for all egenvalues, and the components of the vector z are all of the same order of magntude. Therefore, condtons C1 and C2 () from Secton 3 are fulflled, so all egenvalues and all components of all egenvectors are computed wth hgh relatve accuracy by Algorthm 5 usng only standard workng precson. On the other hand about half of the egenvalues computed by the Matlab routne eg do not satsfy the nterlacng property. 6. Applcatons In ths secton we extend our results to egenvalue decompostons of Hermtan arrowhead matrces, sngular value decompostons of real trangular arrowhead matrces and egenvalue decompostons of real symmetrc dagonalplus-rank-one matrces Hermtan arrowhead matrces Let where [ D z C = z α ], D = dag(d 1, d 2,..., d n 1 ), s a real dagonal matrx of order n 1, z = [ ζ 1 ζ 2 ζ n 1 ], s a complex valued vector and α s a real scalar. Here z denotes the conjugate transpose of z. As n Secton 1, we assume that C s rreducble. The egenvalue decomposton of C s gven by C = UΛU where Λ = dag(λ 1,..., λ n ) R n n s a dagonal matrx of egenvalues, and U = [ u 1 u 2 u n ] s an untary matrx of the correspondng egenvectors. To apply Algorthm 5 to Hermtan arrowhead matrx we frst transform C to real symmetrc arrowhead matrx A by dagonal untary smlarty: 22

23 where [ A = Φ D z CΦ = z T α ( ) ζ1 Φ = dag ζ 1, ζ 2 ζ 2,..., ζ n 1 ζ n 1, 1 ], (28) We now compute the k-th egenpar (λ, v) of A by Algorthm 5, and set u = Φv. Snce we guarantee hgh relatve accuracy of the egenvalue decomposton of A computed by Algorthm 5, we also guarantee hgh relatve accuracy of the egenvalue decomposton of C. Notce that, f double precson s needed to compute b n Algorthm 5, the modules ζ n (28) need to be computed n double of the workng precson, as well. Remark 3. Smlarly, for rreducble non-symmetrc arrowhead matrx [ ] D z G = z T, α where sgn( ζ ) = sgn( ζ ), = 1,..., n 1, we defne the dagonal matrx Ψ = dag sgn( ζ 1 ) ζ1 ζ 1,..., sgn( ζ n 1 ) ζn 1, 1. ζ n 1 The matrx [ ] A = Ψ 1 D z GΨ = z T, α where ζ = ζ 1 ζ s an rreducble symmetrc arrowhead matrx. We now compute the k-th egenpar (λ, v) of A by Algorthm 5. The egenpar of G s then (λ, Ψv). set u = Φv. Snce we guarantee hgh relatve accuracy of the egenvalue decomposton of A, we also guarantee hgh relatve accuracy of the egenvalue decomposton of G. Notce that, f double precson s needed to compute b n Algorthm 5, the elements ζ need to be computed n double of the workng precson, as well Sngular value decomposton of a trangular arrowhead matrx Let B = [ D z 0 α be an rreducble upper trangular arrowhead matrx, that s, d d j for j and ζ 0 for all. The matrx [ ] A = B T D 2 Dz B = z T D α + z T, z s an rreducble symmetrc arrowhead matrx. ], 23

24 When applyng Algorthm 5 to the matrx A, we must ensure that all components of A 1 n (10) are computed to hgh relatve accuracy. Ths s obvously true for elements of the vectors w and w 2. Dagonal elements, except b, are computed wth hgh relatve accuracy as dfferences of squares of orgnal quanttes, [A 1 1 ] jj = (d j d )(d j + d ), j. The element b = [A 1 ] from (11) s computed as 11 b = 1 α d 2 z T z + d + d 2 j ζ2 j. ζ2 (d j d )(d j + d ) j If double precson s needed n Algorthm 5, all entres of A need to be computed n double precson. Let B = UΣV T be the sngular value decomposton of B, where Σ = dag(σ 1,..., σ n ) are the sngular values, the columns of V are the correspondng rght sngular vectors and the columns of U are the correspondng left sngular vectors. We frst compute the k-th egenpar (λ, v) of A by Algorthm 5. Then σ = λ s the correspondng sngular value of B and v s the correspondng rght sngular vector. The value σ and all components of v are computed to almost full accuracy. From the relaton U T B = ΣV T for the k-th row we have [ ] [ ] u T D z 1:n 1 u n = σ [ ] v1:n 1 0 α T v n, whch mples u 1:n 1 = σv 1:n 1 D 1. From the relaton BV = UΣ for the k-th column we have [ ] [ ] [ ] D z v1:n 1 u1:n 1 = σ, 0 α v n whch mples u n = αv n σ. Components of u are computed by multplcaton and dvson of quanttes whch are accurate to almost full machne precson, so the are accurate to almost full machne precson, as well Dagonal-plus-rank-one matrces Let M = D + uu T, u n 11 In vew of Theorem 7, f double precson computaton s necessary, the postve and negatve parts of ths formula should be computed separately, and then added. 24

25 where D = dag(d 1,..., d n ), d 1 > d 2 > > d n, u = [ u 1 u n ] T, u 0, = 1,..., n, be a n n rreducble ordered real symmetrc dagonal-plus-rank-one (DPR1) matrx. Let Then where D = dag(d 1,..., d n 1 ), = ( D d n ) 1/2, ū = [ ] T u 1 u n 1, [ ] un L = 1 0 ū T 1. 1 [ ] D A = L 1 z ML = z T, α z = ū, α = d n + u T u, s an rreducble real symmetrc arrowhead matrx. When applyng Algorthm 5 to the matrx A, we must ensure that all components of A 1 n (10) are computed to hgh relatve accuracy. Ths s obvously true for elements of the vectors w and w 2. Dagonal elements, except b, are computed wth hgh relatve accuracy as dfferences of orgnal quanttes, and the element b = [A 1 ] from (11) s computed as b = 1 d n u T u + d +. ζ d j d j If double precson s needed n Algorthm 5, all entres of A need to be computed n double precson. Let M = QΛQ T and A = V ΛV T be the egenvalue decompostons of M and A, respectvely. Snce M s by assumpton rreducble, ts egenvalues satsfy nterlacng property λ 1 > d 1 > λ 2 > d 2 > > λ n > d n. (29) We frst compute the k-th egenpar (λ, v) of A by Algorthm 5. The value λ and all components of v are computed to almost full accuracy. The relaton V T AV = V T L 1 MLV = Λ mples that the columns of the matrx X = LV are the unnormalzed egenvectors of the matrx M. Further, snce, by (29), all egenvalues are smple, we conclude that X = QΣ, where Σ = dag(σ 1,..., σ n ) s a postve defnte matrx. Notce that QΣV T = L s, n fact, sngular value decomposton of L. ζ j 25

26 Equatng k-th columns of the equaton X = LV gves [ ] [ ] [ ] x un x = = Lv = 1 0 v ū T 1, 1 x n where x and v are parttoned accordng to L. Ths mmedately mples that x = u n 1 v. Notce that, snce all components of v are computed to almost full, accuracy, the same holds for the components of x, and t remans to compute x n accurately. Let [ ] q q = q n be the k-th column of Q and let σ = Σ kk. Equatng k-th rows of the equaton gves for the n-th element Thus, X 1 = Σ 1 Q T = V T L 1 1 q n σ = x 1 n σ 2 = v n. x n = σ 2 v n and, n order to compute x n, t s necessary to compute σ 2. From X = UΣ = LV t follows that V T L T LV = Σ 2, or, equvalently, LV = L T V Σ 2. Equatng k-th columns of ths equaton gves 1 v [ u n = v 1 ] + 1 ū v n σ 2. u n Ths gves n 1 equatons for σ 2, and we can choose the numercally most accurate one. Therefore, x n wll be computed to almost full machne precson, as are the entres of x, and t remans to normalze x and obtan q = x/σ. Remark 4. Notce that DPR1 matrces of the form D uu T cannot be reduced to symmetrc arrowhead matrx by the procedure descrbed n ths secton. By usng deas from ths paper, t s possble to derve hghly accurate algorthm for DPR1 matrces wthout pror transformaton to arrowhead form. Ths algorthm, whch s a topc of our forthcomng paper, covers more general DPR1 matrces of the form D + ρuu T, ρ R. v n 26

27 Appendx A. Proofs Proof of Theorem 1. Let µ and λ be defned by (14). Then λ fl (d + µ) = (d + µ) (1 + ε 1 ). By smplfyng the equalty (d + µ (1 + κ µ ε M )) (1 + ε 1 ) = λ (1 + κ λ ε M ) and usng λ = µ + d, we have d ε 1 + µ (κ µ ε M + ε 1 ) = λκ λ ε M. Takng absolute value gves κ λ d + µ λ ( κ µ + 1). Proof of Theorem 2. or () The assumpton sgn (d ) = sgn (µ) mmedately mples d + µ λ () The assumptons mply that ether = d + µ d + µ = 1. 0 < d +1 < λ < d, µ < 0, d < λ < d 1 < 0, µ > 0. In the frst case λ s closest to the pole d and d + µ λ d d d d + d d 1 2 d d d 3d = d d d 1 2 d d d +1 Here we used the nequaltes µ 1 2 d d +1 and λ 1 2 d + d +1 for the frst nequalty, d d +1 > 0 and d + d +1 > 0 for the second nequalty and d +1 > 0 for the fourth nequalty, respectvely. The proof for the second case s analogous. 27

28 Proof of Theorem 3. Let x and x be defned by (12) and (21), respectvely. The theorem obvously holds for x n = x n = 1. For x we have ( x = fl ζ ) ζ = µ µ (1 + κ µ ε M ) (1 + ε 1) = x (1 + ε x ). By usng (16) and (21), the frst order approxmaton gves For j / {, n}, by solvng the equalty x j = ε x ( κ µ + 1) ε M. ζ j ((d j d ) (1 + ε 1 ) µ (1 + κ µ ε M )) (1 + ε 2 ) (1 + ε 3) = ζ j d j λ (1 + ε x) for ε x, usng (16) and λ = µ + d, and gnorng hgher order terms, we have Therefore, then If ε x = (d j d ) (ε 1 + ε 2 + ε 3 ) µ (κ µ ε M + ε 2 + ε 3 ). d j λ ε x d j d + µ d j λ ( κ µ + 3) ε M. To complete the proof we need to analyze two cases. If d j d + µ d j λ sgn (d j d ) = sgn µ, = d j d µ d j λ sgn (d j d ) = sgn µ, = d j λ d j λ = 1. then, snce d s pole closest to λ, we have µ 0.5 d j d and d j d + µ d j λ d 3 j d + µ d j d µ 2 d j d 1 2 d j d = 3. Fnally, the theorem follows by nsertng ths nto (A.1). (A.1) Proof of Theorem 4. For the non-zero computed elements of the matrx A 1 from (10) and (11) we have: fl( [ A 1 ]jj ) = 1 (d j d ) (1 + ε 1 ) (1 + ε 2), j / {, n}, fl( [ A 1 ]j ) = fl([ A 1 fl([a 1 ] n ) = fl([a 1 ] n ) = 1 (1 + ε 6 ), ζ ]j ) = ζ j (d j d ) (1 + ε 3 ) ζ (1 + ε 4 ) (1 + ε 5), j / {1, n}, 28

29 where ε k ε M for all ndces k. The frst statement of the theorem now follows by usng standard frst order approxmatons. Smlar analyss of the formula (11) yelds fl([a 1 ] ) = b = b + δb, where δb 3 ζ 2 ( a + z T 1 D 1 1 z 1 + z T 2 D 1 2 z 2 Ths, n turn, mples (17) wth κ b δb 1 b ε M = 3 a + z1 T D1 1 z 1 + z2 T D 1 a + z T 1 D1 1 z 1 + z2 T D 1 ) ε M. 2 z 2 2 z 2 = 3K b, (A.2) where K b s defned by (18). Proof of Theorem 5. Let or, equvalently, ( ) A 1 = A 1 + δa 1, ν ν = δa 1. From ths, (24), and Theorem 4, we have Snce A 1 νκ ν ε M δa A 1 2 max{1, K b }ε M. 2 n A 1 2, we have κ ν 3 n max{1, K b }. (A.3) For the second part of the proof, we may assume that the element a = α d s not perturbed. More precsely, we represent the perturbaton n a as perturbatons n D 1 and D 2 by pre- and post-multplcaton of the matrx à = fl(a ) wth the dagonal matrx [ ] (1 + ε a ) 1/2 I n (1 + ε a ) 1/2. Ths ntroduces addtonal errors n D 1 and D 2 bounded by ε M. Therefore, we can wrte νκ ν ε M δa A δb. (A.4) By modfyng (A.2) to accommodate lack of perturbaton n a, we have δb 6 ζ 2 ( z T 1 D 1 1 z 1 + z T 2 D 1 2 z 2 ) ε M. 29

30 By nsertng ths bound nto (A.4) and rearrangng, we have ( n 1 z T κ ν D1 1 z 1 + z T 2 D2 1 z ) 2. (A.5) ν ζ 2 Snce A 1 2 = ν = max A 1 x x 2 =1 2 A 1 e 2 k 1 = (d k d ) 2 + ζk 2 ζ 2 (d k d ) 2 ζ k ζ d k d, by smply dvdng each term ζ 2 k ζ 2 d k d n (A.5) wth the correspondng quotent we obtan ζ k ζ d k d, ( n n 1 1 κ ν 6 + ζ k=1 k The bound (25) now follows from (A.3) and (A.6). Proof of Theorem 6. ) ζ k. (A.6) We frst prove the bound (27). Snce ν = fl( ν) s computed by bsecton, from (22) we have ν = ν(1 + κ bs ε M ). Ths and (24) mply ν = ν(1 + κ ν ε M )(1 + κ bs ε M ). Snce µ = fl(1/ ν), the bound (27) follows by gnorng hgher order terms. The bound (26) now follows by nsertng (27) nto Theorems 1 and 2. Proof of Theorem 7. Let the assumptons of the theorem hold. Let b be computed n double of the workng precson, ε 2 M, and then stored n the standard precson. The standard floatng-pont error analyss wth neglectng hgher order terms gves P ( ) ( ) 1 + κ P ε 2 M Q 1 + κq ε 2 M ( 1 + ζ 2 κ1 ε 2 ) P Q M = ζ 2 (1 + κ b ε M ) b (1 + κ b ε M ), 30

31 where κ P, κ Q (n + 1) and κ 1 3. Solvng the above equalty for κ b, neglectng hgher order terms, and takng absolute values gves κ b P + Q P Q (n + 4) ε M K b (n + 4)ε M. Snce, by assumpton, K b O(1/ε M ), ths mples κ b O(n), as desred. References [1] E. Anderson et al., LAPACK Users Gude, SIAM 3rd ed., Phladelpha, (1999). [2] J. L. Barlow, Error analyss of update methods for the symmetrc egenvalue problem, SIAM J. Matrx Anal. Appl., 14 (1993) [3] M. Bxon and J. Jortner, Intramolecular radatonless transtons, J. Chem. Physcs, 48 (1968) [4] C. F. Borges, W. B. Gragg, A parallel Dvde - and - Conquer Method for the Generalzed Real Symmetrc Defnte Trdagonal Egenproblem, n Numercal Lnear Algebra and Scentfc Computng, L. Rechel, A. Ruttan and R. S. Varga, eds., de Gruyter, Berln (1993) [5] J. R. Bunch and C. P. Nelsen, Rank-one modfcaton of the symmetrc egenproblem, Numer. Math., 31 (1978) [6] J. J. M. Cuppen, A dvde and conquer method for the symmetrc trdagonal egenproblem, Numer. Math., 36 (1981) [7] J. Dongarra and D. Sorensen, A fully parallel algorthm for the symmetrc egenvalue problem, SIAM J. Sc. Statst. Comput., 8 (1987) [8] J. W. Gadzuk, Localzed vbratonal modes n Ferm lquds, general Theory, Phys. Rev. B, 24 (1981) [9] D. Goldberg, What Every Computer Scentst Should Know About Floatng-Pont Arthmetc, ACM Computng Surveys, 23:1 (1991) [10] G. H. Golub and C. F. Van Loan, Matrx Computatons, The John Hopkns Unversty Press, Baltmore, 3rd ed. (1996). [11] M. Gu and S. C. Esenstat, A dvde-and-conquer algorthm for the symmetrc trdagonal egenproblem, SIAM J. Matrx Anal. Appl., 16 (1995)

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