Liner Algebr nd its Applictions 287 (1999) 181±189 Stbility of block LDL T fctoriztion of symmetric tridigonl mtrix Nichols J. Highm 1 Deprtment of Mthemtics, University of Mnchester, Mnchester M13 9PL, UK Received 30 September 1997; ccepted 20 April 1998 Submitted by D.D. Olesky Dedicted to Ludwig Elsner on the occsion of his 60th birthdy Abstrct For symmetric inde nite tridigonl mtrices, block LDL T fctoriztion without interchnges is shown to hve excellent numericl stbility when pivoting strtegy of Bunch is used to choose the dimension (1 or 2) of the pivots. Ó 1999 Elsevier Science Inc. All rights reserved. AMS clssi ction: 65F05; 65G05 Keywords: Tridigonl mtrix; Symmetric inde nite mtrix; Digonl pivoting method; LDL T fctoriztion; Growth fctor; Numericl stbility; Rounding error nlysis; LAPACK; LINPACK 1. Introduction Liner systems involving symmetric inde nite tridigonl mtrices rise in number of situtions. For exmple, Asen's method with prtil pivoting [1] produces fctoriztion PAP T ˆ LTL T of symmetric mtrix A, where P is permuttion mtrix, L is unit lower tringulr, nd T is tridigonl. To solve liner system Ax ˆ b using Asen's method it is necessry to solve system with coe cient mtrix T. A recent ppliction tht produces liner systems with symmetric tridigonl coe cient mtrices is Lnczos-bsed trust region method for unconstrined optimiztion of Gould et l. [8]. 1 E-mil: highm@m.mn.c.uk. 0024-3795/99/$ ± see front mtter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4-3 7 9 5 ( 9 8 ) 1 0 0 7 4-5
182 N.J. Highm / Liner Algebr nd its Applictions 287 (1999) 181±189 Symmetric tridigonl liner systems re most commonly solved by Gussin elimintion with prtil pivoting (GEPP) or by LDL T fctoriztion without pivoting. Neither method is completely stisfctory. GEPP destroys the symmetry, nd therefore cnnot be used to determine the inerti, while n LDL T fctoriztion yields the inerti of A directly from the digonl of D, but cn fil to exist nd its computtion cn be numericlly unstble when it does exist. A method tht promises to combine the bene ts of GEPP nd LDL T fctoriztion ws proposed by Bunch [3], but hs received little ttention in the literture. Bunch's ide is to compute block LDL T fctoriztion without interchnges, with prticulr strtegy for choosing the pivot size (1 or 2) t ech stge of the fctoriztion. Bunch's method requires less storge but slightly more computtion thn GEPP (see [3] for the detils). The purpose of this work is to exmine the numericl stbility of block LDL T fctoriztion with Bunch's pivoting strtegy. In Section 2 we de ne the pivoting strtegy nd explin how Bunch's derivtion of it yields bound of order 1 for the growth fctor. In Section 3 we show tht klk=kak cn be rbitrrily lrge nd explin why numericl stbility is therefore not consequence of error nlysis for generl block LU fctoriztion. We prove normwise bckwrd stbility of the method in Section 3, mking use of results of Highm [10] on the stbility of generl block LDL T fctoriztion. 2. Block LDL T fctoriztion nd the choice of pivot Consider the computtion of block LDL T fctoriztion without interchnges of symmetric tridigonl mtrix A 2 R nn. In the rst stge of the fctoriztion we choose n integer s ˆ 1 or 2 nd prtition A ˆ s s n s E C T : 2:1 n s C B If E is singulr for both choices of s then 11 ˆ 21 ˆ 0, but 21 ˆ 0 mens tht the rst row nd column is lredy in digonl form nd we cn skip to the next stge of the fctoriztion. Therefore, we cn ssume tht E is nonsingulr. Then we cn fctorize I s 0 E 0 Is E 1 C T A ˆ : 2:2 CE 1 I n s 0 B CE 1 C T 0 I n s This process cn be repeted recursively on the n s n s Schur complement S ˆ B CE 1 C T :
N.J. Highm / Liner Algebr nd its Applictions 287 (1999) 181±189 183 The result is fctoriztion A ˆ LDL T ; 2:3 where L is unit lower tringulr nd D is block digonl with ech digonl block hving dimension 1 or 2. While the fctoriztion lwys exists, whether it cn be computed in numericlly stble wy depends on the choice of pivots. Bunch's strtegy [3] for choosing the pivot size s t ech stge of the fctoriztion is fully de ned by describing the choice of the rst pivot. Algorithm 1 (Bunch's pivoting strtegy). This lgorithm determines the pivot size, s, for the rst stge of block LDL T fctoriztion pplied to symmetric tridigonl mtrix A 2 R nn. r :ˆ mxfj p ij j: i; j ˆ 1: ng (compute once, t the strt of the fctoriztion) :ˆ 5 1 =2 0:62 if rj 11 j P 2 21 s ˆ 1 else s ˆ 2 end Bunch excludes 11 from the mximiztion de ning r; we nd it more nturl to include it, becuse it increses the probbility tht 1 1 pivot will be chosen, while hving no e ect on the nlysis below. Bunch's choice of pivot cn be explined by considering element growth in the fctoriztion [3]. Since A is tridigonl, the mtrix C in (2.1) hs the form C ˆ s 1;s e 1 e T s, for unit vectors e 1 2 R n s, e s 2 R s. Hence the Schur complement S ˆ B 2 s 1;s et s E 1 e s e 1 e T 1 ; 2:4 which shows tht only the (1,1) element of the Schur complement di ers from the corresponding element of A. We now exmine the possible element growth in this position. Consider rst the cse s ˆ 1. We hve s 11 ˆ 22 2 21 = 11: Hence, from the conditions in Algorithm 1, js 11 j 6 r r : The choice s ˆ 2 is mde when rj 11 j < 2 21 : 2:5
184 N.J. Highm / Liner Algebr nd its Applictions 287 (1999) 181±189 For s ˆ 2 we therefore hve det E ˆ 11 22 2 21 6 j 11 22 j 2 21 6 2 21 j 22j=r 2 21 6 1 2 21 < 0; 2:6 since < 1. Hence E is inde nite. We hve E 1 ˆ 1 22 21 2:7 det E 21 11 nd so, from (2.4), s 11 ˆ 33 2 32 11=det E. Hence, using (2.5), 2 32 j 11j js 11 j 6 j 33 j 1 2 21 6 r r2 1 r ˆ r 1 : We hve obtined bounds depending only on nd r for the size of the (1,1) element of the Schur complement. This element is not subsequently modi ed nd becomes digonl element of D. It follows tht growth in ny prticulr element tkes plce over single stge of the fctoriztion nd is not cumultive. The vlue of cn therefore be determined by equting the mximl element growth for n s ˆ 1 step with tht for n s ˆ 2 step. Hence we set r r ˆ r 1 ; p which is qudrtic in hving the positive root :ˆ 5 1 =2. With so chosen, the growth fctor q n for the fctoriztion stis es q n :ˆ mx i;j jd ij j mx i;j j ij j 6 1 2 p 5 3 2:62: 3. Error nlysis Tht the growth fctor is nicely bounded does not, by itself, imply tht computtion of the block LDL T fctoriztion is numericlly stble process; see [10] for discussion in the cse of block LDL T fctoriztion of generl symmetric mtrices. From results on block LU fctoriztion [6], numericl stbility could be deduced if we could show tht klk=kak is suitbly bounded. We therefore exmine the size of the block CE 1 of L in (2.2). For s ˆ 1 we hve kce 1 k 1 ˆ j 21j j 11 j 6 r j 21 j ; nd the bound is shrp. It follows tht klk=kak cn be rbitrrily lrge. A prmetrized exmple is given by
N.J. Highm / Liner Algebr nd its Applictions 287 (1999) 181±189 185 A ˆ 1=2 ; 0 6 1; 1=2 2 for which the rst pivot is 1 1 nd 1 0 L ˆ 1=2 1 For s ˆ 2, ; D ˆ 0 0 1 kce 1 k 1 ˆ j 32 jke T 2 E 1 k 1 6 j 32 j j 21j j 11 j 1 2 21 6 j 32j 1 1 j 21 j ; r ; klk 1 =kak 1 1=2 =2: using (2.5) gin. This bound is shrp nd gin it esy to construct prmetrized exmple in which kce 1 k 1 =kak 1 cn be rbitrrily lrge. We conclude tht numericl stbility does not follow from results on generl block LU fctoriztion. Highm [10] proves the following generl result. We employ the usul model of oting point rithmetic fl x op y ˆ x op y 1 d ; jdj 6 u; op ˆ ; ; =; where u is the unit roundo. Absolute vlues of mtrices nd inequlities between mtrices re to be interpreted componentwise. Theorem 3.1. Let block LDL T fctoriztion with ny pivoting strtegy be pplied to symmetric mtrix A 2 R nn to yield the computed fctoriztion PAP T ^L ^D^L T, where P is permuttion mtrix nd D hs digonl blocks of dimension 1 or 2. Let ^x be the computed solution to Ax ˆ b obtined using the fctoriztion. Assume tht for ll liner systems Ey ˆ f involving 2 2 pivots E the computed solution ^x stis es E DE ^y ˆ f ; jdej 6 cu O u 2 jej; 3:1 where c is constnt. Then, P A DA 1 P T ˆ ^L ^D^L T ; A DA 2 ^x ˆ b; where jda i j 6 p n u jaj P T j^ljj ^Djj^L T jp O u 2 ; i ˆ 1: 2; 3:2 with p liner polynomil.
186 N.J. Highm / Liner Algebr nd its Applictions 287 (1999) 181±189 For our tridigonl A we cn set the polynomil p in Theorem 3.1 to be of zero degree, nd we hve P ˆ I. However, to verify tht the theorem is pplicble, we hve to check condition (3.1). It su ces to consider the rst stge of the fctoriztion. Suppose, rst, tht GEPP is used to solve Ey ˆ f. For 2 2 pivot E to be selected we must hve rj 11 j < 2 21 6 j 21jr; which implies j 11 j < j 21 j < j 21 j: Hence GEPP interchnges rows 1 nd 2 of E nd fctorizes " PE ˆ 21 22 ˆ 1 0 # 21 22 11 11 21 21 1 0 21 11 22 ˆ LU: 21 From ([9], Theorem. 9.4), we hve the bckwrd error result PE DE ^y ˆ Pf ; jdej 6 6u O u 2 j^ljj ^Uj: Now, using (2.5), j 21 j j 22 j jljjuj 6 j 11 j 11 22 21 21 11 22 6 j 21j j 22 j : 21 j 11 j 2 1 j 21 j p Hence jljjuj 6 p 5 PjEj. It follows tht (3.1) holds with c ˆ 6 5. Another wy to solve the liner systems Ey ˆ f is by the use of the explicit inverse, s is done in LINPACK [7] nd LAPACK [2] in their implementtions of block LDL T fctoriztion with the pivoting strtegyof Bunch nd Kufmn [4] for generl symmetric mtrices. The formul used in LINPACK nd LA- PACK is suitble here too 1 22 y ˆ 21 1 11 21 21 22 21 1 1 11 f : 3:3 21 It is not hrd to show tht condition (3.1) holds when the formul (3.3) is used; the proof is very similr to tht in [10] for the pivoting strtegy of Bunch nd Kufmn. We hve now estblished tht Theorem 3.1 is pplicble. To deduce stbility of the fctoriztion we hve to show tht jljjdjjl T j is suitbly bounded in norm (we hve replced the computed L nd D by their exct counterprts, which ffects only the second order term of (3.2)). We write
N.J. Highm / Liner Algebr nd its Applictions 287 (1999) 181±189 187 I jej I jl T jljjdjjl T j ˆ 21 j jl 21 j jl S j jd S j jl T S " j jej jejjl T 21 ˆ j # jl 21 jjej jl 21 jjejjl T 21 j jl SjjD S jjl T S j ; 3:4 where L 21 nd E re from the rst stge of the fctoriztion. We rst bound F :ˆ jl 21 jjej jcjje 1 jjej: For s ˆ 1 we hve, trivilly, kf k 1 ˆ j 21 j 6 r. For s ˆ 2, using (2.5)±(2.7), je 1 1 j 22 j j 21 j j11 j j 21 j jjej 6 1 2 21 j 21 j j 11 j j 21 j j 22 j 1 j 22 jj 11 j 2 21 2j 22 jj 21 j ˆ 1 2 21 2j 21 jj 11 j 2 21 j 11jj 22 j 2 3 6 1 j 22 j 1 2 j 22j r j 21 j 4 5 1 2 j 11j j 21 1 j 22j j r 2 3 6 1 1 2 j 22j j 21 j 4 5: 3:5 1 1 2 j 11j j 21 j Hence kf k 1 6 j 32 jke 1 e T 2 je 1 jjejk 1 6 j 32j 1 2 j 11j j 21 j 1 6 r 1 2 j 21j r 1 6 3 1 r 1 Now we bound G :ˆ jl 21 jjejjl T 21j. For s ˆ 1, < 8r: 3:6 kgk 1 ˆ 2 21 j 11 j 6 r < 2r: For s ˆ 2, G 6 jcjje 1 jjejje 1 jc T j ˆ 2 32 et 2 je 1 jjejje 1 j e 2 e 1 e T 1 : We bound the (2,2) element of je 1 jjejje 1 j strting with (3.5) nd nd tht 2 32 kgk 1 6 1 2 2 21 3 j 11 j 6 3 2 32 1 2 r 3 6 r < 16r: 3:7 2 1
188 N.J. Highm / Liner Algebr nd its Applictions 287 (1999) 181±189 We hve now bounded ll the terms in (3.4) except the term jl S jjd S jjl T S j. But L S nd D S re block LDL T fctors of the Schur complement of D in A, nd every Schur complement S stis es ksk M 6 q n kak M 6 2:62kAk M ; 3:8 where kak M ˆ mxj ij j: i;j From the bounds (3.6)±(3.8) nd the structure of the (2,2) block in (3.4) we deduce tht kjljjdjjl T jk M 6 16 2:62kAk M < 42kAk M : The following result summrizes the stbility of block LDL T fctoriztion with Bunch's pivoting strtegy. Theorem 3.2. Let block LDL T fctoriztion with the pivoting strtegy of Algorithm 1 be pplied to symmetric tridigonl mtrix A 2 R nn to yield the computed fctoriztion A ^L ^D^L T, nd let ^x be the computed solution to Ax ˆ b obtined using the fctoriztion. Assume tht ll liner systems Ey ˆ f involving 2 2 pivots E re solved by GEPP or by using the explicit inverse formul (3.3). Then A DA 1 ˆ ^L ^D^L T ; A DA 2 ^x ˆ b; where kda i k M 6 cukak M O u 2 ; i ˆ 1: 2; 3:9 with c constnt. 4. Conclusions Theorem 3.2 shows tht block LDL T fctoriztion with the pivoting strtegy of Algorithm 1 is normwise bckwrd stble wy to fctorize symmetric tridigonl mtrix A nd to solve liner system Ax ˆ b. Block LDL T fctoriztion therefore provides n ttrctive lterntive to GEPP for solving such liner systems. Since the inerti of A is the sme s tht of the block digonl fctor D, the fctoriztion lso provides normwise bckwrd stble wy to compute the inerti. However, for computing inertis of symmetric tridigonl mtrices stndrd LDL T fctoriztion without pivoting hs the stronger componentwise reltive form of bckwrd stbility ([5], Lemm 5.3), nd so is preferble in the bisection method for computing eigenvlues, for exmple.
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