A CHOLESKY LR ALGORITHM FOR THE POSITIVE DEFINITE SYMMETRIC DIAGONAL-PLUS-SEMISEPARABLE EIGENPROBLEM

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1 A CHOLESKY LR ALGORITHM FOR THE POSITIVE DEFINITE SYMMETRIC DIAGONAL-PLUS-SEMISEPARABLE EIGENPROBLEM BOR PLESTENJAK, ELLEN VAN CAMP, AND MARC VAN BAREL Abstract. We preset a Cholesky LR algorithm with Laguerre s shift for computig the eigevalues of a positive defiite symmetric diagoal-plus-semiseparable matrix. By exploitig the semiseparable structure, each step of the method ca be performed i liear time. Key words. diagoal-plus-semiseparable matrix, LR algorithm, Laguerre s method, Cholesky decompositio AMS subject classificatios. 65F15 1. Itroductio. The symmetric eigevalue problem is a well studied topic i umerical liear algebra. Whe the origial matrix is a symmetric matrix, very ofte a orthogoal trasformatio ito a similar tridiagoal oe is applied because the eigedecompositio of a tridiagoal matrix ca be computed i O() ad such a orthogoal similarity trasformatio always exists (see, for example, [5, 9]). I [14], a orthogoal similarity reductio is preseted that reduces ay symmetric matrix ito a diagoal-plus-semiseparable (from ow o deoted by DPSS) oe with free choice of the diagoal. This trasformatio has the same order of computatioal complexity as the reductio ito tridiagoal form, oly the secod highest order term is a little bit larger. A good choice of the diagoal however, ca compesate this small delay whe computig the eigevalues ad eigevectors afterwards. Several algorithms are kow for computig the eigedecompositio of symmetric DPSS matrices, for example, i [2] ad [8] divide ad coquer techiques are used. The authors of [1, 4, ad refereces therei] focus o QR algorithms ad i [10] a implicit QR algorithm is preseted. Whe the symmetric DPSS matrix is positive defiite, also a LR algorithm, based o the Cholesky decompositio, ca be applied i order to compute the eigevalues. Such a Cholesky LR algorithm will be costructed i this paper. Therefore, we show that the DPSS structure is preserved by the Cholesky decompositio ad the LR algorithm. As a shift, Laguerre s shifts (also used for symmetric positive defiite tridiagoal matrices i [6]) are used because oe has to be sure that the shifted matrix is positive defiite agai. Exploitig the DPSS structure, oe step of the Cholesky LR algorithm, icludig the computatio of the shift, has a computatioal cost of order O(). Because two steps of the LR algorithm are equivalet to oe step of the QR algorithm (see, for example, [5]) there will be covergece towards the eigevalues. I cotrast to the QR algorithm with shifts where the eigevalues are ot computed i ay particular order, the eigevalues i the LR algorithm are computed from the smallest to the largest oe. This Versio: March 8, The research of the first author was partially supported by the Miistry of Higher Educatio, Sciece ad Techology of Sloveia, project P The research of the secod ad third author was partially supported by the Research Coucil K.U.Leuve, project OT/00/16 (SLAP: Structured Liear Algebra Package), G (ANCILA: Asymptotic analysis of the Covergece behavior of Iterative methods i umerical Liear Algebra), G (CORFU: Costructive study of Orthogoal Fuctios) ad G (RHPH: Riema-Hilbert problems, radom matrices ad Padé- Hermite approximatio), ad by the Belgia Programme o Iteruiversity Poles of Attractio, iitiated by the Belgia State, Prime Miister s Office for Sciece, Techology ad Culture, project IUAP V-22 (Dyamical Systems ad Cotrol: Computatio, Idetificatio ad Modellig). The scietific resposibility rests with the authors. 1

2 makes it a very suitable algorithm for those applicatios where the smallest eigevalues are eeded. The paper is orgaized as follows. I 2 the cocepts used, are explaied. The preservatio of the DPSS structure uder the Cholesky decompositio ad the Cholesky LR algorithm is prove i 3. Also explicit fast algorithms for the Cholesky decompositio ad the LR algorithm are costructed. I 4 a fast computatio of Laguerre s shifts is studied. 5 focuses o the implemetatio, while umerical results are discussed i 6, followed by coclusios. 2. Prelimiaries. I this sectio we recall the defiitio of DPSS matrices ad the Gives-vector represetatio that we will use. The idea of the LR algorithm based o the Cholesky decompositio is repeated as well as Laguerre s method. DEFINITION 2.1. A matrix S is called a lower- (upper-) semiseparable matrix if every submatrix that ca be take out of the lower (upper) triagular part of the matrix S, has rak at most 1. If a matrix is lower- ad upper-semiseparable, it is called a semiseparable matrix. The sum D + S of a diagoal matrix D ad a semiseparable matrix S is called a diagoal-plussemiseparable matrix or shortly a DPSS matrix. To represet a symmetric DPSS matrix, we use the Gives-vector represetatio based o a vector f = [ f 1,..., f ] T, 1 Gives rotatios [ ci s G i = i s i c i ], i = 1,..., 1, ad a diagoal d = [d 1,...,d ] T (for more details, see, e.g., [13]). c 1 f 1 + d 1 c 2 s 1 f 1 c 1 s 2:1 f 1 s 1:1 f 1 c 2 s 1 f 1 c 2 f 2 + d 2 c 1 s 2:1 f 2 s 1:2 f 2 D + S =..... c 1 s 2:1 f 1 c 1 s 2:2 f 2 c 1 f 1 + d 1 s 1 f, 1 s 1:1 f 1 s 1:2 f 2 s 1 f 1 f + d where s a:b = s a s a 1 s b. We will deote D + S = diag(d) + Giv(c,s, f ). The above represetatio of a DPSS matrix is ot uique. Oe ca see that the parameters d 1 ad d ca be chose arbitrarily. If we chage d ito d the we ca chage f ito f = f + d d. If we chage d 1 ito d 1, the oe ca check that by takig f 1 = (c 1 f 1 + d 1 d 1 ) 2 + s 2 1 f 1 2, c 1 = (c 1 f 1 + d 1 d 1 )/ f 1, s 1 = s 1 f 1 / f 1 we get the same matrix D + S. Most ofte, however, the diagoal d is kow, so d 1 ad d are fixed. Next we recall aother importat cocept, the Cholesky LR algorithm. Let A be a symmetric positive defiite (from ow o deoted by s.p.d.) matrix. Startig from the matrix A 0 = A, a Cholesky LR algorithm geerates a sequece of similar matrices A k+1 = Vk 1 A k V k = Vk T V k, k = 0,1,..., 2

3 where V k Vk T = A k is the Cholesky decompositio of A k with V k a lower-triagular matrix. The use of a shift at each step ca speed up the covergece of the sequece A k, k = 0,1,..., towards the Schur decompositio of A. Whe applyig the Cholesky LR algorithm to a s.p.d. DPSS matrix D+S, the shift ca be icluded ito the diagoal part ad hece, whe we are able to costruct the Cholesky decompositio VV T of a arbitrary s.p.d. DPSS matrix ad the correspodig product V T V, we ca apply a step of the shifted Cholesky LR algorithm o a s.p.d. DPSS. Oe importat remark, however, is that the shift σ should be chose such that D + S σi is still positive defiite or i other words, the shift σ should be smaller tha the smallest eigevalue of D + S. To fulfill this requiremet, Laguerre s shifts are used. Let A be a s.p.d. matrix with eigevalues 0 < λ λ 1... λ 1. Let f (λ) = det(a λi) be the characteristic polyomial of A. If x is a approximatio for a eigevalue of A ad we defie S 1 (x) = S 2 (x) = i=1 i=1 1 λ i x = f (x) f (x), 1 (λ i x) 2 = f 2 (x) f (x) f (x) f 2, (x) the the ext approximatio x by Laguerre s method is give by the equatio (2.1) x = x +. S 1 (x) + ( 1)(S 2 (x) S 21 (x)) Two importat properties of Laguerre s method are that if λ is a simple eigevalue ad if x < λ the x < x < λ ad the covergece towards λ is cubic. For multiple eigevalues the covergece is liear. More details o Laguerre s method ad its properties ca be foud i, e.g., [15]. 3. Cholesky decompositio. I this sectio we show that the DPSS structure is preserved by both the Cholesky decompositio ad the LR algorithm. Eve better, if we use the Gives-vector represetatio, the we ca show that some vectors from the represetatio are ivariat to the Cholesky decompositio ad the Cholesky LR algorithm. This eables us to produce a fast algorithm for the Cholesky LR step. THEOREM 3.1. Let A be a symmetric positive defiite diagoal-plus-semiseparable matrix i the Gives-vector represetatio A = Giv(c,s, f ) + diag(d). 1. If V is a lower triagular matrix such that A = VV T is the Cholesky decompositio of A, the V ca be represeted i the Gives-vector represetatio as V = tril(giv(c,s, f )) + diag( d). 2. If B = V T V, where V is the lower triagular Cholesky factor from 1. of A, the B is agai a symmetric positive defiite diagoal-plus-semiseparable matrix with the same diagoal part as the origial matrix A: B = Giv(ĉ,ŝ, f ) + diag(d). 3

4 Proof. 1. We use iductio. As we geerate A from the top to the bottom, the followig relatio holds betwee A k ad A k+1, where A 1 = [ f 1 + d 1 ] ad A = A. If we write [ ] Bk a A k = k, f k + d k where B k R (k 1) (k 1) ad a k R k 1, the B k c k a k s k a k A k+1 = c k a T k c k f k + d k s k f k. s k a T k s k f k f k+1 + d k+1 If a T k [ ] Wk 0 V k =, v T k where W k R (k 1) (k 1) ad v k R k 1, is the Cholesky factor of A k the oe ca see that the Cholesky factor of A k+1 has the form W k 0 0 V k+1 = c k v T k 0. s k v T k It is easy to see that there exist f k ad d k such that [ ] Wk 0 V k = f k + d k v T k ad W k 0 0 V k+1 = c k v T k c k f k + d k 0. s k v T k s k f k I the last step, whe k = 1, oe ca also choose appropriate f ad d for the right bottom elemet of V. Hece, the Gives trasformatios from A appear i the Gives-vector represetatio of the Cholesky factor V as well. 2. From 1. we kow that A =VV T with V a osigular, lower-semiseparable ad lower triagular matrix. A is also a s.p.d. DPSS matrix, so A = D + S. Hece, This implies: D + S = VV T. V T V = V T (D + S)V T = V T DV T +V T SV T = D 1 + S 1. The matrix D 1 is a upper triagular matrix with the diagoal D as diagoal elemets. All the submatrices of the lower triagular part of S 1 have rak at most 1. So, D 1 + S 1 ca be rewritte as D 1 + S 1 = D + Ŝ, 4

5 where all the submatrices of the lower triagular part of Ŝ have rak at most 1. Because of symmetry, also the submatrices of the upper triagular part of Ŝ have rak at most 1 ad hece, Ŝ is a semiseparable matrix. This fiishes the proof that V T V = D + Ŝ is agai a symmetric DPSS matrix with the same diagoal part as the origial matrix A. The fact that the Gives trasformatios used i A ad i the Cholesky factor V are the same, simplifies the computatio of V. The same is true for the fact that the diagoal part of A is ivariat uder the LR algorithm. This will be exploited ow. The Cholesky factor of A has the form V = c 1 f 1 + d 1 c 2 s 1 f 1 c 2 f 2 + d c 1 s 2:1 f 1 c 1 s 2:2 f 2 c 1 f 1 + d 1 s 1:1 f 1 s 1:2 f 2 s 1 f 1 f + d where s a:b = s a s a 1 s b. By comparig the elemets of A ad VV T we get equatios for the vectors f ad d. As we kow all Gives rotatios, it is eough to compare the elemets o the diagoal ad the mai subdiagoal. Hece, we get the followig equatios, (3.1) (3.2) k 1 c k f k + d k = j=1 k 1 c k+1 s k f k = j=1 (c k s k 1 s j f j ) 2 + (c k f k + d k ) 2, k = 1,...,, c k c k+1 s k (s k 1 s j f j ) 2 +c k+1 s k f k (c k f k + d k ), k = 1,..., 1, where we assume that c = 1. If we deote the we ca write (3.1) ad (3.2) as k 1 q k := j=1 (s k 1 s k 2 s j f j ) 2, (3.3) (3.4) c k f k + d k = c 2 k q k + (c k f k + d k ) 2, k = 1,...,, c k+1 s k f k = c k c k+1 s k q k + c k+1 s k f k (c k f k + d k ), k = 1,..., 1. The solutio of (3.3) ad (3.4) for f k ad d k is (3.5) (3.6) f k = d k = f k c k q k dk + c k ( f k c k q k ), d k dk + c k ( f k c k q k ), k = 1,...,, k = 1,...,, where we assume that c = 1 ad q 1 = 0. 5

6 For later use, let us defie the commo factors i the umerator ad the deomiator of (3.5) ad (3.6) as follows: z k = f k c k q k, y k = d k + c k z k. Oe ca see from (3.1) that y k is i fact the diagoal elemet of V because (3.7) c k f k + d k = d k + c k ( f k c k q k ) = d k + c k z k = y k. As i the stadard Cholesky algorithm, a egative or zero value uder the square root appears if A is ot positive defiite, so this is a way to check whether A is positive defiite or ot. Let us remark that f ad d are ot uiquely determied. We choose the values (3.5) ad (3.6) because of cosistecy. From the above equatios we ca obtai a algorithm that computes the Cholesky factorizatio of a s.p.d. DPSS matrix i 11 + O(1) flops. ALGORITHM 3.2. A algorithm for the Cholesky decompositio VV T = A of a s.p.d. DPSS matrix A = Giv(c,s, f ) + diag(d). The result are vectors f ad d such that V = tril(giv(c,s, f )) + diag( d). I the algorithm we assume that c = 1. fuctio [ f, d] = Cholesky(c, s, f, d) c = 1 q 1 = 0 for k = 1,..., : z k = f k c k q k y k = d k + c k z k f k = z k /y k d k = d k /y k q k+1 = s 2 k (q k + f k 2) Next we study how to costruct the product V T V i a efficiet way. The product B = V T V is agai a s.p.d. DPSS matrix. A short calculatio shows that the diagoal ad subdiagoal elemets of B are equal to (3.8) (3.9) b kk = (c k f k + d k ) 2 + (s k f k ) 2, b jk = s k s k+1 s j 1 f k ( f j + c j d j ), where k = 1,...,, j > k, ad we assume that c = 1 which implies that s = 0. Let us deote B = Giv(ĉ,ŝ, f ) + diag(d). From the equality ad (3.9) it follows that ŝ 2 k f 2 k = b 2 jk j=k+1 (3.10) ŝ 2 k f 2 k = f 2 k p k, k = 1,..., 1 6

7 where p k = (s k s k+1 s j 1 ) 2 ( f j + c j d j ) 2. j=k+1 For p k, k = 1,...,1, we ca apply the recursio p k = s 2 k (p k+1 + ( f k+1 + c k+1 dk+1 ) 2) that starts with p = 0. From (3.8) we obtai (3.11) ĉ k f k = (c k f k + d k ) 2 + (s k f k ) 2 d k. By applyig the relatio (3.7) we simplify (3.11) ito (3.12) ĉ k f k = c k z k + (s k f k ) 2 ad reduce the possibility of cacellatio. From (3.10) ad (3.12) we ca compute the vectors ĉ, ŝ, ad f. ALGORITHM 3.3. A algorithm for the product B = V T V, where V = tril(giv(c,s, f )) +diag( d) is the lower triagular Cholesky factor of a s.p.d. DPSS matrix A = Giv(c,s, f ) +diag(d). The vector z was already computed i Algorithm 3.2. The result are vectors ĉ, ŝ, ad f such that B = Giv(ĉ,ŝ, f ) + diag(d). fuctio [ĉ, ŝ, f ] = VTV(c,s, f,z) c = 1 f = ( f + d ) 2 d p = 0 for k = 1,...,2,1 ) p k = s 2 k (p k+1 + ( f k+1 + c k+1 dk+1 ) 2 [ĉ k, ŝ k, f k ] = Gives(c k z k + s 2 f k k 2, f k pk ) The fuctio [c, s, f ] = Gives(x, y) i Algorithm 3.3 returs the Gives trasformatio such that [ ][ ] [ ] c s x f =. s c y 0 A stable implemetatio that guards agaist overflow requires 7 flops (see, for example, [5]). Note that some quatities such as f k 2 ad s2 k already appear i Algorithm 3.2, so we have to compute them oly oce. As a result a efficiet implemetatio of Algorithm 3.3 requires 16 + O(1) flops ad oe step of the Cholesky LR algorithm without shifts ca be performed i 27 + O(1) flops. Let us remark that i Algorithm 3.3 we do ot care about the sig of ŝ k as the eigevalues are ivariat to the sig of ŝ k, k = 1,..., 1. 7

8 4. Computatio of Laguerre s shift. As idicated i (2.1), for Laguerre s shift we eed to compute S 1 ad S 2. It is easy to see that ad S 1 (σ) = S 2 (σ) = i=1 i=1 1 λ i σ = Tr((A σi) 1 ) 1 (λ i σ) 2 = Tr((A σi) 2 ). So, if A σi = VV T is the Cholesky decompositio of the s.p.d. DPSS matrix A σi ad W = V 1, the ad S 1 (σ) = Tr(W T W ) = W 2 F S 2 (σ) = Tr(W T WW T W) = Tr(WW T WW T ) = WW T 2 F. The aim is to compute S 1 ad S 2 i a stable ad efficiet way. Let us assume that W = tril(giv( c, s, f )) + diag( d). We will later show that the algorithm derived uder the above assumptio is correct also whe W is ot DPSS. Oe ca check that W is ot DPSS whe d i = 0 for some i = 2,..., 1. I the ext lemmas ad remark, we will show that S 1 ad S 2 ca be computed i a efficiet way. LEMMA 4.1. If A = Giv(c,s, f ) + diag(d) is a symmetric diagoal-plus-semiseparable matrix the where we assume that c = 1. Proof. As A is symmetric, A 2 F = k f k + d k ) k=1(c s 2 k f k 2, A 2 F = k=1 a 2 kk k=1 If follows from the structure of A that a kk = c k f k + d k ad a 2 jk = s2 k f k 2. j=k+1 k=1 j=k+1 a 2 jk. Based o Lemma 4.1, we ca derive the followig expressios for S 1 ad S 2 : LEMMA 4.2. If W = tril(giv( c, s, f )) + diag( d) is a lower osigular triagular matrix, such that c k 0 for k = 2,..., 1, the (4.1) WW T 2 F = k=1 (WW T ) 2 kk k=1 ( (WW T ) 2 ) k+1,k, c k+1 8

9 (4.2) W 2 F = k=1 (WW T ) kk, where we assume that c = 1. Proof. WW T is a s.p.d. DPSS matrix. As a cosequece of poit 1. of Theorem 3.1, the Gives trasformatios of the represetatio of W are preserved i the product WW T. Hece, there exist two vectors x,y R such that WW T = Giv( c, s,x) + diag(y). Applyig Lemma 4.1 ad the relatios fiishes the proof. s k x k = (WW T ) k+1,k c k+1 for k = 1,..., 1, c k x k + y k = (WW T ) k,k for k = 1,..., REMARK 4.3. The formula (4.1) of Lemma 4.2 ca be geeralized such that the coditio c k 0 for k = 2,..., 1 is o loger required. If we deote by t(k) the smallest idex j, j > k, such that c j 0, the (4.3) ( ) WW T 2 F = (WW T ) 2 1 (WW kk + 2 T 2 ) t(k),k. k=1 k=1 c t(k) Sice c = 1, we always have k < t(k) ad (4.3) is well defied. I additio to d i 0 for k = 1,...,, such that W is a DPSS, let us assume from ow o also that c k 0 for k = 2,..., 1 i the Cholesky factor V. Uder this assumptios it follows from Lemma 4.2 that oly the Gives trasformatios of W ad the diagoal ad subdiagoal elemets of WW T are required for computig S 1 ad S 2. Oe ca check that ad (WW T ) kk = c 2 k k 1( ) 2 sk 1 s i f i + ( ck f k + d k ) 2 i=1 (WW T k 1 ) 2 ) k+1,k = c k+1 c k s k ( sk 1 s i f i + ck+1 s k f k ( c k f k + d k ). i=1 Because V is a lower triagular matrix ad W = V 1, the diagoal ad subdiagoal elemets of W are of the form: (4.4) (4.5) w kk = c k f k + d k = y 1 k, k = 1,...,, w k+1,k = c k+1 s k f k = c k+1s k f k y k y k+1, k = 1,..., 1, where y k = c k f k + d k is the diagoal elemet of V computed i Algorithm 3.2. If we defie r k = i=1 k 1 ) 2 ( sk 1 s i f i the we ca write (WW T ) kk = c 2 k r k + y 2 k 9

10 ad (WW T ) k+1,k c k+1 = c k s k r k + s k f k y k. For r k, k = 1,...,, we use the recursio r k+1 = s 2 k r k + s 2 f k k 2 that starts with r 1 = 0. From the relatios (4.4) ad (4.5) it follows that i order to compute the diagoal ad the subdiagoal elemets of WW T, it is eough to kow the Gives rotatios ad the diagoal ad the subdiagoal elemets of W. The followig lemma, which follows from the results i [3], helps us to compute the ecessary elemets of W. LEMMA 4.4. Let V = tril(giv(c,s, f )) + diag( d) be a osigular lower triagular matrix such that d i 0 for i = 1,...,. The W = V 1 ca be represeted i the Gives-vector represetatio as W = tril(giv( c, s, f )) +diag( d), where d 1 i = d i for i = 1,...,. Hece, the diagoal elemets of W ca be writte as (4.6) w kk = c k f k + d 1 k If we rearrage the equatios (4.5) ad (4.6) ito ad (4.7) = y 1 k, k = 1,...,. c k f k = c k f k d k y k. s k f k = c k+1s k f k c k+1 y k y k+1, the it follows that c k ad s k form a Gives trasformatio such that [ ][ ] [ ] ck s k ck y k+1 c k+1 =. s k c k c k+1 s k dk 0 Agai, for k = 1 we assume that c = c = 1. Oe ca see by iductio that c k 0 for k = 1,...,2 because we assumed that c k 0 for k = 2,..., 1 ad y k+1 = 0 would cotradict the fact that A is s.p.d. Now we ca write a algorithm for the computatio of WW T 2 F ad W 2 F. I the algorihtm ξ k deotes (WW T ) k+1,k / c k+1 ad ω k deotes the diagoal elemet (WW T ) kk. These are the values that appear i equatios (4.1) ad (4.2) for S 1 ad S 2. We use β k for the itermediate result (4.7). A careful implemetatio of the algorithm, where the values that appear i Algorithms 3.2 ad 3.3 are computed oly oce, requires 31 + O(1) flops. ALGORITHM 4.5. A algorithm that computes S 1 = W 2 F ad S 2 = WW T 2 F, where W = V 1 ad V = tril(giv(c,s, f )) + diag( d) is the Cholesky factor of a s.p.d. DPSS matrix A = Giv(c,s, f ) + diag(d), ad y = diag(v ). I the algorithm we assume c k 0 for k = 2,..., 1 ad c = c = 1. fuctio [S 1, S 2 ] = ivtrace(c,s, f, d,y) c = c = 1 for k = 1,...,2,1 : [ c k, s k ] = Gives(c k c k+1 y k+1,c k+1 s k dk ) 10

11 r 1 = 0 for k = 1,..., 1 β k = c k+1 s k f k /( c k+1 y k y k+1 ) ω k = c 2 k r k + y 2 k ξ k = c k s k r k + β k /y k r k+1 = s 2 k r k + β 2 k ω = r + y 2 S 1 = k=1 ω k S 2 = k=1 ω2 k k=1 ξ 2 k What remais to be cosidered is the case that W is ot a DPSS matrix. If d k = 0 for some k = 2,..., 1 the W has a zero block W(k + 1 :,1 : k 1), see, e.g., [7, Lemma 2.5] ad it is ot a DPSS matrix aymore. However, Algorithm 4.5, that was derived uder the assumptio that W is a DPSS matrix, returs correct values for W 2 F ad WW T 2 F i such case as well. There are o divisios by d k i the algorithm that could cause problems. We oly use d k to compute y k. If we chage d k i V the oe ca see that as log as V is osigular, W 2 F ad WW T 2 F are cotiuous fuctios of d k. So, the algorithm is correct also i the limit whe d k = 0. Aother restrictio i Algorithm 4.5 is the assumptio c k 0 for k = 2,..., 1. Whe this assumptio is ot valid, we ca still compute S 1 ad S 2 if we apply formula (4.3) from Remark 4.3. Oe ca see that i the kth colum of W we eed the elemets w kk ad w t(k),k. Because w jk = 0 for k < j < t(k), Laguerre s shift ca still be computed i O() flops. 5. Implemetatio. I this sectio we discuss some details o the implemetatio of the algorithm preseted i the previous sectios. The software ca be dowloaded freely at: First we discuss how to deflate. If s k is small eough for some k = 1,..., 1, the we decouple the problem ito two smaller problems with matrices A(1 : k,1 : k) ad A(k + 1 :,k + 1 : ). I the special case whe s 1 is small eough, we take f + d as a approximatio of a eigevalue of A ad cotiue with vectors c(1 : 2), s(1 : 2), f (1 : 1), ad d(1 : 1). As iitial shift for the smaller problem we take f + d. Aother importat problem that ca appear durig the implemetatio is the shift. If a shift i the QR algorithm is by chace a exact eigevalue the we ca immediately extract this eigevalue ad cotiue with the smaller problem. This is ot true i the Cholesky LR algorithm where shifts σ k have to be strictly below the smallest eigevalue λ, otherwise the Cholesky factorizatio does ot exist. Without the Cholesky factorizatio we ca ot compute A k+1 = Vk 1 A k V k ad deflate. I umerical computatios, eve whe σ k < λ, the Cholesky factorizatio ca fail if the differece is too small. This ca cause a problem as usually Laguerre s shifts coverge faster to the smallest eigevalue tha the elemets A k (,). A good strategy is to isert a factor τ close, but smaller, to 1 ito (2.1) ad use σ k+1 = σ k + τ S 1 (σ k ) + ( 1)(S 2 (σ k ) S1 2(σ k)) as a shift i the ew iteratio. Based o our umerical experimets we suggest the value τ = If it happes ayway that the shift is so large that the Cholesky factorizatio fails, we first reduce the shift by the factor τ = ad if the ew shift is still too large, we start agai with the shift 0. 11

12 The computatio of Laguerre s shift requires more tha half of the operatios i oe step of the Cholesky LR algorithm. We ca save work by usig the same shift oce the shift improvemet is small eough. Our umerical experimets show a speed up up to 15% if we stop improvig the shift after (σ k+1 σ k )/σ k The eigevalues should be computed from the smallest to the largest oe, however, it might happe that s 1 is so small that we deflate, ad the extracted eigevalue is ot the smallest oe. This causes a problem i the ext phase as we use the extracted eigevalue as iitial shift ad this shift is too large. The strategy from the previous paragraph overcomes this problem ad the shift goes to zero after two usuccessful Cholesky factorizatios. At the ed of 4 we proposed a modificatio of Algorithm 4.5 that hadles the case c k = 0 for some k = 2,..., 1. Without this modificatio we get zero divided by zero i such a situatio. I practice we ca implemet a simpler solutio. If we perturb c k ito wheever c k = 0 the a small c k results i a small c k. These two quatities avoid the zero divided by zero problem i Algorithm 4.5 ad we ed up with accurate results. 6. Numerical results. The followig umerical results were obtaied with Matlab 7.0 ruig o a Petium4 2.6 GHz Widows XP operatig system. We compared the Cholesky LR algorithm with a Matlab implemetatio of the implicit QR algorithm for DPSS matrices [10] ad with the Matlab fuctio eig. Exact eigevalues were computed i Mathematica 5 usig variable precisio. For all umerical examples i this sectio the cutoff criterio for both Cholesky LR ad implicit QR is With the maximum relative error we deote max 1 i λ i λ i λ i where λ i, i = 1,...,, are the exact eigevalues of the test matrix ad λ i, i = 1,...,, the computed oes. EXAMPLE 6.1. I our first example we use radom s.p.d. DPSS matrices of the form A = diag(1,...,) + triu(uv T,1) + triu(uv T,1) T + αi, where u ad v are vectors of uiformly distributed radom etries o [0,1], obtaied by the Matlab fuctio rad, ad the shift α is such that the smallest eigevalue of A is 1. The coditio umbers of these matrices are approximately. The exact eigevalues of A are computed i Mathematica usig variable precisio. Before usig eig we compute all the elemets of A accurately i double precisio, so that the iitial data for all three methods are of full precisio. The compariso is ot completely fair as i eig we first have to reduce the matrix to the tridiagoal form where additioal umerical errors could occur. The results i Table 6.1 show that the Cholesky LR method is competitive i accuracy with the other two methods. I most cases, especially for larger matrices, it is slightly more accurate tha the implicit QR method. The compariso with eig shows that by exploitig the structure we ca get more accurate results. I eig some accuracy is lost i the reductio to the tridiagoal form. Oe step of the Cholesky LR method has approximately the same complexity as oe step of the implicit QR method, but although Cholesky LR requires roughly 3.5 times more steps tha the implicit QR method, it rus much faster. This is due to a more efficiet Matlab implemetatio. The same holds for eig which rus faster tha Cholesky LR although it has O( 3 ) complexity while the complexity of Cholesky LR is O( 2 ). The differece i umber of steps is also due to the fact that i implicit QR we may choose the shift more freely as i Cholesky LR, where the shifted matrix must remai positive defiite. EXAMPLE 6.2. We use the same costructio of the test matrices as i Example 6.1. For = 200 we geerate 25 radom matrices ad compare the accuracy of the eigevalues computed by the 12

13 TABLE 6.1 Compariso of the Cholesky LR method, implicit QR for DPSS matrices, ad eig from Matlab o radom s.p.d. DPSS matrices of sizes = 50 to = 500 ad small coditio umbers. The colums are: t: ruig time i secods; steps: umber of LR (QR) steps; error: the maximum relative error of the computed eigevalues. Cholesky LR Implicit QR eig t steps error t steps error t error Cholesky LR method, implicit QR for DPSS matrices, ad eig. Agai, the exact eigevalues of A are computed i Mathematica usig variable precisio. FIG Compariso of the Cholesky LR method, implicit QR for s.p.d. DPSS matrices, ad eig from Matlab o 25 radom s.p.d. matrices of size = 200. log10 of the maximum relative error Implicit QR Matlab eig Cholesky LR idex Results, ordered by the maximum relative error of the Cholesky LR method, are show i Figure 6.1. We ca see that the most accurate method for this particular class of matrices is the Cholesky 13

14 LR algorithm. The results from eig are comparable while the results of the implicit QR are slightly worse i geeral. EXAMPLE 6.3. I this example we use s.p.d. matrices A = Qdiag(1 : )Q T, where Q is a radom orthogoal matrix, obtaied i Matlab as orth(rad()). As i the previous examples we compare the Cholesky LR method, implicit QR for DPSS matrices, ad eig from Matlab. The differece from the previous examples is that ow we have to reduce the matrix ito a similar DPSS matrix before we ca apply Cholesky LR or implicit QR. We do this usig the algorithm of [14], where we choose the diagoal elemets as radom umbers distributed uiformly o [0, 1]. There is a coectio betwee the Laczos method ad the reductio ito a similar DPSS matrix [11] which causes that the largest eigevalues of A are approximated by the lower right diagoal elemets of the DPSS matrix. This is ot good for the Cholesky LR method where the smallest eigevalues are computed first. Therefore, we apply a method that reverses the directio of the colums ad rows of the DPSS matrix i liear time [12, Chapter 2, 8.1]. TABLE 6.2 Compariso of the Cholesky LR method, implicit QR for DPSS matrices, ad eig from Matlab o radom s.p.d. matrices of sizes = 500 to = 2000 with the exact eigevalues 1,...,. The colums are: t: ruig time i secods (time for LR ad QR does ot iclude reductio ito a DPSS matrix); steps: umber of LR (QR) steps; error: the maximum relative error of the computed eigevalues. Cholesky LR Implicit QR eig t steps error t steps error t error The results i Table 6.2 show that the eigevalues of a s.p.d. matrix ca be computed accurately usig a reductio ito a DPSS matrix followed by the Cholesky LR method or the implicit QR method. For larger matrices, the Cholesky LR algorithm teds to be slightly more accurate tha the implicit QR. Sice both methods use the same reduced DPSS matrices, this implies that Cholesky LR is more accurate tha implicit QR. The computatioal times are hard to compare because of differet implemetatios ad because the time for eig icludes the reductio to the tridiagoal form while the reductio to DPSS matrices is excluded from the times of the Cholesky LR ad the implicit QR method. EXAMPLE 6.4. We use the same costructio of the test matrices as i Example 6.3. For = 1000 we geerate 25 radom matrices ad compare the accuracy of the eigevalues computed by the Cholesky LR method, implicit QR for DPSS matrices, ad eig. For the reductio ito a similar DPSS matrix we use the same approach as i Example 6.3. Results are show i Figure 6.2. Similar to the previous examples, the Cholesky LR method is comparable with eig ad usually gives slightly better results tha the implicit QR method. Similar tests o matrices with multiple eigevalues ad with eigevalues λ i = 2 i,i = 1,...,, were performed. The results obtaied by the three algorithms also i these cases are comparable. 7. Coclusios. We have preseted a versio of the Cholesky LR algorithm that exploits the structure of positive defiite DPSS matrices. We propose to combie the method with Laguerre s 14

15 FIG Compariso of the Cholesky LR method, implicit QR for DPSS matrices, ad eig from Matlab o 25 radom s.p.d. matrices of size = 1000 with the exact eigevalues 1,..., log10 of the maximum relative error Implicit QR Matlab eig Cholesky LR idex shifts. It seems atural to compare the method to the implicit QR for DPSS matrices [10]. I Cholesky LR the eigevalues are computed from the smallest to the largest eigevalue, therefore the method is very appropriate for applicatios where oe is iterested i few of the smallest or the largest eigevalues. If the complete spectrum is computed, Cholesky LR is more expesive tha implicit QR, but, as it teds to be slightly more accurate, it presets a alterative. The proposed method combied with the reductio to DPSS matrices [14] ca also be applied to a geeral s.p.d. matrix. REFERENCES [1] Bii, D.A., Gemigai, L., Pa, V.: QR-like algorithms for geeralized semiseparable matrices. Tech. Report 1470, Departmet of Mathematics, Uiversity of Pisa, 2003 [2] Chadrasekara, S., Gu, M.: A divide ad coquer algorithm for the eigedecompositio of symmetric blockdiagoal plus semi-separable matrices. Numer. Math. 96, (2004) [3] Delvaux, S., Va Barel, M.: Structures preserved by matrix iversio. Report TW 414, Departmet of Computer Sciece, K.U.Leuve, Leuve, Belgium, December 2004 [4] Fasio, D.: Ratioal Krylov matrices ad QR-steps o Hermitia diagoal-plus-semiseparable matrices. To appear i Numer. Liear Algebra Appl.; also available from ftp://ftp.dimi.uiud.it/pub/fasio/bari.ps [5] Golub, G.H., Va Loa, C.F.: Matrix Computatios, 3rd Editio. The Johs Hopkis Uiversity Press, Baltimore, 1996 [6] Grad, J., Zakrajšek, E.: LR algorithm with Laguerre shifts for symmetric tridiagoal matrices. Comput. J. 15, (1972) [7] Fiedler, M., Vavří, Z.: Geeralized Hesseberg matrices. Liear Algebra Appl. 380, (2004) [8] Mastroardi, N., Va Camp, E., Va Barel, M.: Divide ad coquer type algorithms for computig the eigedecompositio of diagoal plus semiseparable matrices. Techical Report 7 (5/2003), Istituto per le Applicazioi del 15

16 Calcolo M. Picoe, Cosiglio Nazioale delle Ricerche, Rome, Italy, To appear i Numerical Algorithms. [9] Parlett, B.N.: The symmetric eigevalue problem. Classics i Applied Mathematics, Pretice-Hall, Eglewood Cliffs, N.J., 1980 [10] Va Camp, E., Delvaux, S., Va Barel, M., Vadebril, R., Mastroardi, N.: A implicit QR-algorithm for symmetric diagoal-plus-semiseparable matrices, Report TW 419, Departmet of Computer Sciece, K.U.Leuve, Leuve, Belgium, March [11] Va Camp, E., Va Barel, M., Vadebril, R., Mastroardi, N.: Orthogoal similarity trasformatio of a symmetric matrix ito a diagoal-plus-semiseparable oe with free choice of the diagoal. Structured Numerical Liear Algebra Problems: Algorithms ad Applicatios, Cortoa, Italy, September 19-24, cortoa04/program.htm [12] Vadebril, R.: Semiseparable matrices ad the symmetric eigevalue problem. PhD, K.U.Leuve, Leuve, May 2004 [13] Vadebril, R., Va Barel, R., Mastroardi, N.: A ote o the represetatio ad defiitio of semiseparable matrices. Report TW 393, Departmet of Computer Sciece, K.U.Leuve, Leuve, Belgium, May To appear i Numer. Liear Algebra Appl. [14] Vadebril, R., Va Camp, R., Va Barel, M., Mastroardi, N.: Orthogoal similarity trasformatio of a symmetric matrix ito a diagoal-plus-semiseparable oe with free choice of the diagoal. Report TW 398, Departmet of Computer Sciece, K.U.Leuve, Leuve, Belgium, August 2004 [15] Wilkiso, J.: Algebraic eigevalue problem. Numerical Mathematics ad Scietific Computatio, Oxford Uiversity Press, Oxford,