ETNA Electronic Transactions on Numerical Analysis. Volume 23, pp. 5-14, 2006. Copyright 2006, Kent State University. ISSN 1068-9613. Kent State University etna@mcs.kent.edu CONDITION NUMBERS OF THE KRYLOV BASES AND SPACES ASSOCIATED WITH THE TRUNCATED QZ ITERATION ALEXANDER MALYSHEV AND MILOUD SADKANE Abstract. We propose exact and computable formulas for computing condition numbers of the Krylov bases and spaces associated with the Hessenberg-Triangular reduction of a regular linear matrix pencil. Key words. condition number, Krylov spaces, QZ algorithm, generalized Arnoldi algorithm AMS subject classifications. 65F35, 15A21 1. Introduction. The algorithm [5] is the most popular algorithm for solving the dense generalized eigenvalue problem (1.1) ! $%& )*+, #" where . It is a generalization of the algorithm [2, Sec. 7.7], which solves the standard eigenproblem . The first step of the algorithm reduces, via orthogonal and , the pair to , where the matrices and matrices are respectively upper Hessenberg and upper triangular: ' (21 ' -3 (1.2) ( 455 798;: 8<798;: = 5 7 =B: 8<7 =B: = 6 .. . .-/ " -0 (*1 ' " 455 H 8A: 8 H 8A: =I>?>J> H 8A: E?FF 7@8A: ?E FF 5 7 =?: F H ?= : =I>?>J> H ?= : F > " .. .. .. .. 6 G . 7 : D .C 8 7 . : G H .: >?>?> >?>?> The Hessenberg-Triangular reduction can be recast as K -L ' ' ( #( " > (1.3) to construct the reduction (1.3) iteratively, starting from a vector M with N M N = ItPisO possible N2N = stands . The symbol for the Euclidean vector norm. The following algorithm, taken from [7], can be used to accomplish the iterative reduction. It is written in MATLAB style. A LGORITHM 1 (Generalized Arnoldi Reduction [7]). Q Q Q Q L with N M N = O ZN N = 8 [R R `W _aY\T]V 8' SM R UM TVXW 8 b & M@V - 8 [R M (#VX8 1 Y b\TVc 8 W Vb " ( 8 - Y\8 T]V^V ( OU+f >J>?> do d g eZ N cih N = VjW cihJ_ g V 8 (Xhk [R (Xh W TV - h [R - h#V g9l h1 T]V Choose Set Set for m Received January 3, 2004. Accepted for publication November 15, 2005. Recommended by D. Sorensen. University of Bergen, Department of Mathematics, Johannes Brunsgate 12, N-5008 Bergen, Norway (malyshev@mi.uib.no). Université de Brest, Département de Mathématiques, 6 avenue Victor Le Gorgeu, CS 93837, F-29238 Brest Cedex 3, France (sadkane@univ-brest.fr). 5 ETNA Kent State University etna@mcs.kent.edu 6 A. MALYSHEV AND M. SADKANE /n b #' 1 M@M n V M n WXoV M n 'Dhpb VjY eO _ N!M n N = V h M SY M@R n V H " h b\Y@8V [R H q ' h ' h MUTjV/" hk " h V YUTV b 8 M V 7 ( h1 8Bk 7 8 b -V 8 SR - 7 h T]V c hk b ( hk V hk Solve end for After steps of Algorithm 1 we obtain a truncation of (1.3) in the form r%sut v w x @' y (zy - y| {}c~y l 1y '@y z( ya"y (1.4) '*y 1 ' y y (2y 1 (zy & y (2y 1 c~y & OU t ?OU r and (zyqP( O\ t JO\ r represent the first r columns where the matrices '@y[' - o- OU r JO\ r and "2y " O\ r ?O\ r are the of ' and ( respectively. The - matrices y leading rLr submatrices of and " . The vector l y represents the r -th canonical vector and y is the identity matrix of order r . The form (1.4) is a generalization of the classical Arnoldi reduction [6, Chap. VI] and can, for example, be used to compute eigenpairs of the problem (1.1), when the sizes of and are large (see [7]). In exact arithmetic, the sets and , form bases of the Krylov spaces W 8 >?>?> W`yU r S O\;f! >?>?> t y y . C 8 M 8 Span M 8 + C (1.5) and 8 8 C ) $ 8 W (1.6) Span W y y 8 ' 8 M and W 8 M 8 _ NB M respectively, with M y M 8 >?>J> M\yD 8 8 >J>?> i. C 8 y C 8 8 M M y C 8 W 8 >J>?> J. C 8 y C 8 W 8 8N=. . in a finite preUnfortunately, the generalized Arnoldi reduction of the matrix pair cision - arithmetic will not generally satisfy the relations (1.4) because the computed matrices ' , ( , and " are subject to rounding errors. The effects of rounding errors are proportional to condition numbers, which are usually determined in terms of infinitesimal perturbations. The chief advantage of infinitesimal perturbations is that they allow one to neglect second order terms. Let and be infinitesimal perturbations of and respectively, and denote by and the corresponding Hessenberg and triangular forms, where and are the computed orthogonal matrices with . In the present paper we want to analyze the difference between the exact quantities , , , and their computed counterparts , , , under variations of the perturbations and . More precisely, we are interested in the condition numbers of the Krylov spaces and and corresponding bases and . We use the arguments similar to those from [1, 3] for the Hessenberg reduction by Arnoldi’s method and those from [4] for the bidiagonal reduction by the Lanczos method. We assume that the reader is familiar with the arguments and reasoning used in these references. 8 -3 (2 1 . S { ( 8 M y y 8 = y y = = ' R W M 8 8W . 8 { 3 ' R 8 = ?> >?> = " >J>?> (# W 1 T e ' M M M T ' ( y y y y . { ( ' ' ) { ' ( 3 { { ' ( 2. Condition numbers. Following the strategy used in [1, 3, 4], perturbations of and are sought in the multiplicative form, that is and . The orthogonality of , , and implies the orthogonality of and . Since the ( ' ' ( ( ETNA Kent State University etna@mcs.kent.edu C ONDITION NUMBERS OF THE K RYLOV BASES ASSOCIATED WITH THE TRUNCATED QZ ITERATION 7 - (21 ) } { 1 B) { and are infinitesimal, the matrices and are indistinguishable from perturbations skew-symmetric matrices. Discarding quadratic terms in the identity , we obtain the equation = B) {3 ' - - {}( 1 = '{ ( 1 1 ( - { - ' 1 ' z > (2.1) If we set = ( 1 = ' ' 1 ' and ( 1 ( then and are indistinguishable from skew-symmetric matrices and satisfy, up to the first order, the equation - - > -- = { (#1 . { 1 ?. { 8 ? . S { ' In a similar way, another equation follows from " (2.2) yields , which " " 8 { " " 8 (21$ 8 ' . with The Krylov spaces y and y are the vector spaces spanned by the columnsS of)'` y { and (z' yy respectively. Their perturbations y and y are spanned by the columns of '@ y y [. { (y . The Frobenius norm of the difference and ( (2.4) '@y '@ y ? ' ' 1 '@y ? [N OU t ?OU r JN .¢C 8 * M 8 of the basis y , which is defined can be used to measure the conditioning ` z¡ (2.3) as (see [1, 3]) v«« ¹ «« «º N N w ± ²]³z' ´;y ²]µ¶ ' y ]² ³ µ ²µ¶ «« > S§©£¥¤!¨@¦ ª «x« N 8 ¬N ­ ® ]² ·²µ¶ { ²]¸^]² µ¶ s ¯ N =N ° » s ¯ ° « In other words, @ is the smallest constant such that N 8 N = N = N = > N \ O J \ O ? N t r s¼ ¾½ NB¿N = { N?,N = (2.5) N O\ t ?OU r JN can be used to measure the conditioning of the basis Similarly, the quantity y . Because of similarities between the bases y and y we will only analyze the conditioning of zy . To this end, we will give a computable estimate of the condition number À . in (2.3) and those below the subdi$Let Á us take the components below the main diagonal Á agonal8 in (2.2). The operation of taking the components =+ below the main diagonal is denoted by and that of below the subdiagonal by . Thus, from equations (2.3), (2.2) we Á Á derive the system of linear equations Á 8 " " ^ Á 8 8 9 (2.6) = -Ã- ^ = = > (2.7) ETNA Kent State University etna@mcs.kent.edu 8 A. MALYSHEV AND M. SADKANE and are equal to zero because the matrices Note that the diagonal elements of and are real skew-symmetric. Moreover, since and , we have Ä' ÆÅi { / Ç ' 8 o and, therefore, the first column and row of are also equalÈ to8 zero. the identity M and can be described with the help of vectors ȼX!C C É¡ The of O\ >?>Jstructure > t f , and ÊUË LDC Ë XÌ#SO\ >J>?> t °O , such that 455 E FF 455 Ê!81 >J>?> E FF 55 FF 55 FF ¾ 18 Ê!=1 55 . F 5 FF FF 55 ¾ =1 Ê!Î1 .. .. FF > . . 55 .. F 5 .. .. .. .. .. . 8 = >J>?.> . 8 Ê = Ê Î >?>J.> . G 6 6 Í Ê G M O\ >?>J> t f ÈϼX!C c= 8 Similarly, with the help of vectors , the matrices and 455 5 55 8 5 8 M8 È É}ÃO\ >?>J> &O DC Ë C 8 , ÌL ¼ t , , and Ð Ë are written as ?E FFF FF F G 455 5 ?E FFF FF F > > G 55 ... 5 = .. . . .> . 8 = 6 c c cÎ 6 Ð8 Ð= ÐÎ 8 >J>?> DC = and Ê 8 >J>?> Ê DC 8 we determine the lower triangular By the aid of the vectors parts of and .. : 455 . >?>?> E FF >?>?> FF F 5 F > 5 Ñ Ñ .. .. .. .. .. . 8 = . I . G 8 Ê = Ê Î. . G 6 Í 6 Ê Ñ Ñ 1 and Ñ Ñ 1 . Then that Á ItÁ is important now to observe Á Á 8Â Å Ñ Ç & 8Â Å Ñ Ç =+Â Å Ñ - Ç & = Š- Ñ Ç & > 1" 1 1 " 1 55 >J>?> I I E FF Á 8Â Ñ Á Å " " =+Â Å Ñ -Ã- As a result, (2.8) (2.9) Equation (2.8) is equivalent to the system Ò (2.10) FF 455 55 Á Ñ Ç 8  8 9 Á Ñ Ç =+ = > c 8 H 8A: 8 Ê 8 D C Ø C 8 ÛÚX¡f! >?>?> }OU caÓ ÕÔ hÓ Ö 8 H h : Ó¥× DC Óh Ê h Ù " Ó Ó t ETNA Kent State University etna@mcs.kent.edu Á 9 Á Á QZ Â Â Ø " ÜÚ { O\ t Ú { OU t L D C Ó D C Ó and for Ú sd , × h oÝ) Ó h C Ó Â Ó]Þ L Ó h where " Ó $Ú columns are equal to zero and the lastÓ Ú columns form the identity d is the matrix whose first Ú matrix of order . Similarly, equation (2.9) is equivalent to Ò 8 7@8A: 8 !DC C = 8 8 7 =?: 8 = Ð ×8 Ê { Ê !DC C h Á C 8 à- ÁØ C 8 ÛÚ^f >?>J> f! (2.11) Ô 7 : k Ó 8 Ð\Ó t hÖ h Óß× Ó Ê h 8Â Ó Ó Â - Ø &- .Ú { f! t Ú { O\ t DC Ó C DC Ó . where Ó 3O\;f! >J>?> t ¡O , we have a series of Let us summarize the above derivations. For r systems of linear equations E FF 45 8 E F 455 c 8 E FF 5455 H 8;8A: : = 8 DDC C =8 8 =B: = = FF 55 Ê = FF 5 c = F 55 H × DC H DC F Ê 6 ... G 6 ... G .. .. .. 6 G . . . !C 8 8 cy Êy H 8A: yA× DC y H y C 8;: yA× !DC C y y k H y : y DC y 455 E?FF 55 " Ø = FF 455 8 E?FF 5 FF 5 = F 5 .. .. . . 6 ... 8 G . .. G 6 yC Ø"2y 455 Ð 8 E FF 455 7 8A: 8 × !DC C = 88 E FF 455 Ê 8 E FF 7 =?: 8 DC = = 5 = F 5 7@8A: = !DC C Î DDC C Î 7 Î : = DC Î F 5 = F 7 B = : = Ð Ê × × .. .. .. 6 G 6 6 .8 G = . G . !C 8 7 8;: y C 8 × DC 7 =B: y C 8 × !DC C >J>?> 7 y : y C 8 DC y ÐyC Ê y y 455 E FF y 455 8 E FF F 5 = F 5 -Ø = .. . . . . 6 . - Ø y C . 8 G 6 y .C 8 G C ONDITION NUMBERS OF THE K RYLOV BASES ASSOCIATED WITH THE TRUNCATED ITERATION which we can write in the compact form Ò á +8 8 Ê á 8= â c (2.12) á =+= > Ð &á =A8 Ê : ä and 7 y : y C 8 ä Since Algorithm 1 proceeds if and only if H y yã ã , the matrix á 88 nonsingular. It follows that the system (2.12) has the solution &á è,æ c Ê ÐÕç Ðeç where Ò áå*[)á =+= á =;8 á 8+C 8 8 á 8= ;C 8 Ýá =;8 á 8C 8 8 Ï?éXê Þ (2.14) áè2&á 8C 8 8 á 8= áå { Ý á 8C 8 8 j Þ > Á Á ?é ê ?ë ê is the identity matrix of order ì y á y with ì y t r ¼Oi y y = k 8  { O Here á ât r y y = k 8 . and y (2.13) ¿&áå¢æ c is and ETNA Kent State University etna@mcs.kent.edu 10 ´ N O\ t J O\ r ?N NA y N == Þ µ Z A. MALYSHEV AND M. SADKANE NAXN = íÝ;NA 8 N == { >?>J> { Using the identity formula N OU t ?OU r JN áå¢æ c Сç = áå¢æ N " N ë ê N N ë ê s á å æ " (2.15) á å æ N N ë ê " (2.16) s If we choose áå¢æ and c NB-uN é ê ç æ NB-uN é ê ç = NB-uN é ê ç = Ð and such that c áå¢æ N N ?ë ê " = ? N u N Béê ç ÐÕç æ N N cÀ_ NB" -uN Ð _ ç we arrive at the c À_ N N?" -uN N = Ð _ ç æ N N cÀ_ N?" -uN = Ð _ ç = N 8 N N = N = > ½ NB¿N = { NB,N = æ N N = cÀ_ NB" -uN Ð _ ç = N 8 N = N = N = = ½ ?N %N = { BN N = then inequalities (2.15) and (2.16) become equalities (see [4] for a similar reasoning). We thus obtain a series of condition numbers of the orthonormal bases (not spaces!) for r OU+f >?>J> t }O 8 8 áåæ N " N ?ë ê C . : ? N u N B é ê y M ç = Ýá =+= á =;8 á +8 C 8 8 á 8 = Þ C 8 Ý;N " N á =A8 á 8+C 8 8 N?-uN B éê Þ = > (2.17) In order to derive condition numbers of the corresponding Krylov spaces y we use the P ) same arguments as in [4]. The main idea is to compare '@ y { '@y with '@y\y instead of '@y for all orthogonal matrices y of order r . The minimum î £ï¤ ' y ' y L y y #1j y is attained at y Sðòñ)ð ó 1 tion of (see, e.g., [2, p.582]). Since '*y 1 ' y , where ð ñ 1 Å '*1 ' y Ç ð ó Pô y is the singular value decomposi- õy {}' 1 ' y õy { O2 r J O# r ' y 1 '@ y y [ð/ñ y{ O\ r ?O\ r is indistinguishable from an orthogonal matrix, we can take y ð o ô à and ó In fact, the leading rr y . This leads to the following interpretation. submatrix of is cut off. The rest of represents the distance between the Krylov spaces, i.e., only last t r components of each vector Ó contribute to the difference between the (2.18) ETNA Kent State University etna@mcs.kent.edu C ONDITION NUMBERS OF THE K RYLOV BASES ASSOCIATED WITH THE TRUNCATED QZ ITERATION 11 ö y , whose columns are the coordinate vectors Krylov spaces. Let us introduce the matrix corresponding to the contributing components. Then the condition numbers of the Krylov spaces are determined for as lÓ OU+f! >?>J> t °O r . C 8 * M 8 z ö y á =+= á =;8 á 8C 8 8 á 8 = C 8 Ý+N " N á =;8 á 8+C 8 8 pNB-uN BéXê Þ = > (2.19) y The condition numbers of the Krylov bases y and corresponding Krylov spaces y , r eO\;f! >?>?> t }O , can be derived analogously by the aid of the expression for Ê in (2.13): 8 8 á è,æ N " N ë ê C ) $ : B N } N B é ê (2.20) y W ç = > 8 8 áèæ N " N ?ë ê C ) $ NB-uN ?éXê ç = y W (2.21) ö y 3. Numerical examples. In this section we illustrate behavior of the condition numbers of Krylov bases and subspaces given in (2.17) and (2.19) on some examples. E XAMPLE 3.1. is the identity matrix, and is the upper-Hessenberg Toeplitz matrix 455 O 5 ø ÷¢ 6 ÷ ÷ O O .. ø ZO . >J>?> O E FF >J>?> O F > .. .. G . ø O. The matrix is singular if and only if . Our aim here is to show the effect of conditioning of on the computed basis and subspace condition numbers, when the parameter varies. The Hessenberg and triangular forms obtained by Algorithm 1 are of order . The starting vector has all its components equal to before normalization. Table 3.1 summarizes the information obtained on the computed matrices , , and for different is also given. parameters . The standard condition number ø O ú .*÷@[N?÷j' N y = N\( . 2y ÷ " AC y 8 N = y M r oOpù ø TABLE 3.1 Numerical results from Algorithm 1 (Example 3.1) ø ?N y '#1 '@y N = NB y (2y 1 ( y N = NB- y (21y ' y N = N " y (21 y ÷ ' y N = NB ÷ ' y y ( y " y N = BN ' y ( y - y c y l 1 N = y ú . ÷ ú " y ú ðÏ÷@ û > ù\ù,8OJC 88ü f > ù ý OJC 8 f > ù OJC 8 ù > OJÿOJC þ> ûUþ OJ C 88ü f > Oif,OJC = f > ù ý Oi = f > û Oi ÿ >aý Oi 8 û > ý\þ 4OJC 88ü ý> f ý OJC 8 ý> O û OJC 8 O > þ OJC Î O > Oif,OJ C 88Îü > O2OJC û > f ý Oi Î f > þ Oi Î û > ÿ OOi Î ù > û f,2OJC 88ü þ> f\f,OJC 8 f > OJC 8+8 O > \OJC 8+8 O > \OJ C ù > \OJC 8ü O > U,Oi ÿ > ûUû Oi û > OOi û > ù~ÿ,1.3OJC 88ü þ> û OJC 8 f > ÿ ý OJC ü O > ù þ Oi!C ü O > ù û Oi C û > ,OJC 8 8ü f > ý\þ Oi 8+Î8 O > þ f,Oi ý> f!O2Oi 8+8 ( y ' y ( y - y {uc y l 1 ' y ÷'y ( y" y y ÷ 8 Ý ¡ : y ÝC * M 8 ÞiÞ û > ù\f,0Oi!C O > ù\f,Oi!C O > û Oi!C O > f!O2Oi!C O > ý ÿOi C f > û O2Oi!C f > \,OJ f > ù~ÿ,OJ The results show that the orthogonality of and is well maintained along with the first equality of (1.4), i.e., . However, the relations and deteriorate, when gets ill-conditioned. Figure 3.1 shows the behavior of the condition numbers of the Krylov bases and Krylov spaces . We observe that the condition numbers of the bases increase with the dimension of the bases and that the condition numbers of the corresponding spaces increase and decrease but are always smaller than those of the bases. (2y 1 ÷ ' y " y 8 Ý @y Ý ¢C * M 8 ÞiÞ 8 8 8ü 8 8 8 8 8ü ETNA Kent State University etna@mcs.kent.edu 12 A. MALYSHEV AND M. SADKANE ÷ ú . 2÷@ SO > þ N" C 8N= "2yy ú )á 8+8 ú ú " y" y clearly influences the computed condition numbers (see the The ill-conditioning of case in Table 3.1). It can be more practical to monitor the condition number instead of . This quantity is always available. Since , large means essentially that the matrix in (2.12) is ill-conditioned. The computed condition numbers might therefore be large or even inaccurate. One may wonder whether the “non-normality” influences, in a way, the computed condenote a matrix of eigenvectors of and define dition numbers. Let ø N "2y N = ú á 8+8 ð÷ #÷ ÷XN = N~)ð÷@;C 8 N = if ÷ is diagonalizable ú )ð÷ K NJðÏ > otherwise ð ÷ be used to quantify the departure from normality of ÷ . The smaller The ú ðÏfactor ÷@ , theú closer can 2÷ to a normal matrix. Table 3.1 and Figure 3.1 show that the computed ð ÷ factors ú , @ and do not seem to be related. Example 1, α = 8 4 10 Example 1, α = 4 5 10 κb κb 3 4 10 10 condition number condition number κ κ 2 3 10 10 1 10 2 2 4 6 8 10 dimension of the Krylov basis 12 14 10 16 Example 1, α = 2 9 10 2 4 6 8 10 dimension of the Krylov basis 14 16 Example 1, α = 1.3 15 10 κb 12 κb 14 10 8 10 condition number 7 10 13 κ 10 12 10 6 10 11 10 5 10 10 0 2 4 6 8 10 dimension of the Krylov basis 12 14 16 10 0 2 4 6 8 10 dimension of the Krylov basis 12 14 Example 1, α = 0 3 10 κb κ condition number condition number κ 2 10 1 10 2 4 6 8 10 dimension of the Krylov basis 12 14 16 F IG . 3.1. Condition numbers of the Krylov bases and spaces (Example 3.1). 16 ETNA Kent State University etna@mcs.kent.edu C ONDITION NUMBERS OF THE K RYLOV BASES ASSOCIATED WITH THE QZ ITERATION TRUNCATED 13 TABLE 3.2 Numerical results from Algorithm 1 (Example 3.2) )*+, þ > OOi!C 888 f > ý Oi C f > O OJ!C û > û fOi!C 8= f > OOi!C 8= f > ÿUÿ,Oi!C 8 Î ü > OJ,OJ BN matrix '*1 pair N= y NB y ( y 1 (' y y N = N?- y (21y '@y N = N "2²y· ê (2y1 ê Àê '@² y N = ²]C · y ² µ µ ²]¸ ê C ê ^ê C ê ê ² µ ²¸^² µ ú "2y . +, þ > f*ª Oi!C 88 þ > þUþ Oi C 8= > f*Oi!C û > ù þ Oi!C 8= f > Oi!C 8= ù > aû Oi!C 8ü > O2OJ# " .*+ þ > UOJ!ª C 88= O > þ f,OJ C 8 O > þUþ OJ!C ª þ > ÿ ý OJ!C 8= 8ü > ùUù,OJ!C f > ù OJ!C !8 O >ÿ ù áe-%$ û O\ áe-%$ O û t BN N = P f > ù\f DOJ Î = OJ Î NB¿N = f > OJÿ ú .,Ï þ> ÿ\ÿ `OJ ù~ r O M 'y ( ú " y & N?,N = ª BN ª N = û > O\O \Oi Î ú ) ª f > O NB ª N = û > þ ÿ ú ª ) ª z{ f û O&\ ú )2[ÿ > ù E XAMPLE 3.2. The matrix is and the matrix is from the & MHD set1 . These matrices are of order and arise in the modal(analysis of dissipative ' )' magnetohydrodynamics. The matrix is unsymmetric, , *' # . The matrix is symmetric, , . The Hessenberg and triangular forms obtained from Algorithm 1 are of order . As in the previous example, the starting vector has all its components equal to before normalization. Table 3.2 summarizes the information obtained on the computed matrices , , and . The condition number is also given. For more comparisons, the table also shows the results obtained with the well-conditioned matrices and . +' , , , . Example 2, matrix pair = (A , B) 15 10 κ Example 2, matrix pair = (A0 , B) 13 10 b κb κ 14 10 - y y " y { BN %N = 12 10 13 10 12 11 10 condition number condition number 10 11 10 10 10 9 10 10 κ 9 10 10 8 10 8 10 7 10 6 10 7 0 5 10 15 20 25 30 dimension of the Krylov basis 35 40 45 50 10 0 5 10 15 20 25 30 dimension of the Krylov basis 35 40 45 50 Example 2, matrix pair = (A , B0) 4 10 κb κ 3 condition number 10 2 10 1 10 0 10 0 5 10 15 20 25 30 dimension of the Krylov basis 35 40 45 50 F IG . 3.2. Condition numbers of the Krylov bases # and spaces (Example 3.2) From Table 3.2 and Figure 3.2, it is clear that the ill-conditioning of ú "y , is responsible for the large condition numbers. We also see that 1 see http://math.nist.gov/MatrixMarket/ , also shown in ETNA Kent State University etna@mcs.kent.edu 14 (y 1 ( y , y í- y-, '`y , (yp"2y , leads to less accuracy in the approximation - the less accuracy is more pronounced when both and are used. In conclusion, the numerical experiments indicate that when the truncated reduction is good, i.e., when the relations (1.4) are accurately satisfied, the ill-conditioning of is re2( y 1 ' y the use of - the use of A. MALYSHEV AND M. SADKANE leads to less accuracy in the approximations sponsible for large basis and space condition numbers. REFERENCES [1] J.-F. C ARPRAUX , S. K. G ODUNOV, AND S. V. K UZNETSOV , Condition number of the Krylov bases and subspaces, Linear Algebra Appl., 248 (1996), pp. 136–160. [2] G. H. G OLUB AND C. F. VAN L OAN , Matrix Computations, Third editon, The Johns Hopkins Press, Baltimore, MD., 1996. [3] S .V. K UZNETSOV , Perturbation bounds of the Krylov bases and associated Hessenberg forms, Linear Algebra Appl., 265 (1997), pp. 1–28. [4] A. M ALYSHEV AND M. S ADKANE , Condition numbers for Lanczos bidiagonalization with complete reorthogonalization, Linear Algebra Appl., 371 (2003), pp. 315–331. [5] C. B. M OLER AND G. W. S TEWART , An algorithm for generalized matrix eigenvalue problems, SIAM J. Numer. Anal., 10 (1973), pp. 241–256. [6] Y. S AAD , Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester, 1992. [7] D. C. S ORENSEN , Truncated QZ methods for large scale generalized eigenvalue problems, Electron. Trans. Numer. Anal., 7 (1998), pp. 141–162, http://etna.math.kent.edu/vol.7.1998/pp141-162.dir/pp141-162.html.