BiCG CGS BiCGStab BiCG CGS 5),6) BiCGStab M Minimum esidual part CGS BiCGStab BiCGStab 2 PBiCG PCGS α β 3 BiCGStab PBiCGStab PBiCG 4 PBiCGStab 5 2. Bi

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1 BiCGStab BiCGStab PBiCGStab BiCG CGS CGS PBiCGStab BiCGStab M PBiCGStab An improvement in preconditioned algorithm of BiCGStab method Shoji Itoh, 1 aahiro Katagiri, 1 aao Saurai, 2 Mitsuyoshi Igai, 3 Satoshi hshima, 1 Hisayasu Kuroda 4 and Ken Naono 2 An improved preconditioned BiCGStab algorithm (improved PBiCGStab) is proposed. ational preconditioned algorithm of CGS has been constructed, by applying the derivation procedure of the CGS to the preconditioned BiCG. In order to extend this approach to the BiCGStab, minimum residual part of the BiCGStab must be considered logically. his proposed algorithm is also more rational than the conventional typical PBiCGStab mathematically. Numerical results show advantages of this improved PBiCGStab. 1. Ax b (1) BiCGStab PBiCGStab Preconditioned BiCGStab method 17) PBiCGStab 1 Information echnology Center, he University of oyo 2 Central esearch aboratory, Hitachi, td. 3 SI Hitachi USI Systems Co., td. 4 Graduate School of Science and Engineering, Ehime University 8) 6) BiCGStab BiCGStab CGS 14) CGS PCGS BiCG PBiCG 2),10) CGS 9) PBiCGStab 117

2 BiCG CGS BiCGStab BiCG CGS 5),6) BiCGStab M Minimum esidual part CGS BiCGStab BiCGStab 2 PBiCG PCGS α β 3 BiCGStab PBiCGStab PBiCG 4 PBiCGStab 5 2. BiCG CGS (1) A x b (2) BiCG 2),10) BiCG CGS 14) M BiCGSAB 17) 1 α β BiCG CGS 1 BiCG CGS BiCG CGS α β 1 2 PBiCG PCGS BiCGStab K K U (1) A K K K Ã x b, Ã K x K x, AK, b K b (3) (3) (1) (3) Ã (3) (1) K K, K I (4) I K I, K K (5) Ã K A, x x, b K b, Ã AK, x Kx, b b (1) 1),3),18) 2.2 BiCG BiCG 2),10) A (1) (2) r 0 b Ax 0 r 0 b A x 0 r (A)r 0, (6) r (A )r 0 (7) 118

3 p P (A)r 0, (8) p P (A )r 0 (9) BiCG i, r j 0 (i j), (10) i, Ap j 0 (i j) (11) (6) (9) (z) (z) α zp (z), 0(z) 1, P (z) (z) + β P (z), P 0 (z) 1 3),10),14),15) p r + β p, p r + β p, r +1 r α Ap, r +1 r α A p (10)(11) u, v 4),15) (u, v) BiCG α β α, r, Ap, (12) +1 β, r +1, r. (13) BiCG BiCG (1) (2) Ă x [ b, [ A Ă, x A x x, b CG 13),18) Ă K [ K K K K [ Ă K Ă K [ Ã Ã x K x, b K b. [ K K AK K [ [ b b K K A K, (14) K p r [ [ K p p K p, r K r K p K r K r (1) (2) BiCG PBiCG r K K K r K b ( K b ( K AK ) (Kx), A K (15) ) ( K x ). (16) r r 1 r b Ax, (17) r b A x. (18) i, r j K r i K r j i K r j 0 (i j), (19) p i, Ã p j K p i, ( ) ( ) K AK Kp j i, Ap j 0 (i j). (20) (12)(13) α PBiCG, r r p, Ã p K r p, Ap, (21) β PBiCG +1, r +1 r +1, r K r +1 r. K r (22) (21)(22) (4) (5) α PBiCG β PBiCG 2.3 CGS CGS BiCG A BiCG 14) (1)(2) (6)(7) (8)(9) BiCG α β 1 (1) r r K (b Ax) 5) 7). 119

4 , r (A )r 0, (A)r (A)r 0, (23), Ap P (A )r 0, AP (A)r 0 0, AP 2 (A)r 0. (24) r CGS (A)r 2 0, p CGS P 2 (A)r 0 r r α BiCG β BiCG, Ap, r 0, rcgs 0, ApCGS +1, r +1 r, r 0, rcgs +1 0, rcgs α CGS, (25) β CGS (26) BiCG CGS α β CGS (6)(8) PCGS CGS Ã K AK, x K x, b K b, p CGS K p CGS, r CGS K r CGS, r 0 K r 0 (27) CGS α 0 rcgs 0 Ã pcgs r 0 K rcgs r 0, ( ) ( ) K AK K pcgs 0 rcgs 0 (28) AK p CGS 1),3),17) CGS β (4)(5) (28) (27) K r K b ( ) ( K A K K x ) r K ( b A x ) (29) (18) K BiCG 1 α β ω PBiCG PCGS PBiCGStab ω (18) K r K b ( K A X ) ( X x ) (30) 1 (14) Ã (23)(24) α PCGS α PBiCG β PBiCG β PCGS (25)(26) 9) PCGS PBiCG CGS Ã K AK, x K x, b K b, p CGS K p CGS, r CGS K r CGS, r 0 K r 0 (31) 9) K r K b ( K A K ) ( K x ) (18) Ã (31) (25)(26) α PBiCG, r 0 p, Ã p rcgs 0 Ã pcgs r 0 K rcgs r 0, (K 0, K r CGS AK 0, K Ap CGS )(K p CGS ) α PCGS (21) β PBiCG β PCGS 3. BiCGStab PBiCG PCGS PBiCGStab 1 1 X r K r X K K 9) 120

5 PBiCG PCGS (3)(4)(5) 5),6) PBiCGStab PBiCG PCGS α β PBiCGStab ω PBiCGStab 3.1 BiCGStab BiCGStab ω S (z) (1 ω z) (1 ω 2 z) (1 ω 0 z) (32) BiCGStab S (A ) s S (A )r 0 BiCG r (32) s s lc(s ) ( r lc( ) + d r + + d 1r 1 + ) d0r 0 (33) 1 lc leading coefficient lc( +1 ) lc( ) α, lc(s +1 ) lc(s ) ω BiCG (33) r (10) s, r lc(s ) r lc( ) + d r + lc(s ) lc( ), r + d 1 r 1 + d 0r 0, r 1 r c, r, r c s, r c S (A )r 0, (A)r 0 c 0, S (A) (A)r 0. (34) BiCG p r +β p (11), Ap r, Ap + β p, Ap, Ap (35) r i, Ap j p i β ip i, Ap j 0 (i j) s Ap (33) 1 d i (i 1,, 0) r i lc(s )/lc( ) r (10) (11), Ap c s, Ap c S (A )r 0, AP (A)r 0 c 0, AS (A)P (A)r 0. (36) r SAB S (A) (A)r 0, (37) p SAB S (A)P (A)r 0 (38) (34) (36) α BiCG, r r 0, Ap rsab r 0 ApSAB α SAB, (39) r r β BiCG 0, rsab +1 +1, r +1, r α ω r 0 rsab β SAB (40) BiCG BiCGStab α β 15) CGS (40) ω (As, s ) (As, As ) (41) (u, v) BiCG CGS u, v (41) BiCGStab r SAB +1 M r SAB +1 s ω As. (42) BiCGStab (37)(38) BiCGStab à K AK, x K x, b K b, p SAB K p SAB, s K s, r SAB K r SAB, r 0 K r 0 (43) 3),17) α 0, rsab 0, à psab r 0, K rsab r 0, ( ) ( ) K AK K psab 0 rsab 0 α PSAB AK p SAB (44) PCGS ω ) (à s, s ω (à s, à s (45) ) (4)(5) 121

6 (42) r SAB +1 s ω Ã s (46) (b) (l) (r) ( ) ( ) K r SAB +1 K s ω b K AK K s, (47) ( K r SAB +1 K s ω l K A ) ( ) K s, (48) ( r SAB +1 s ω ) r AK s (49) ω ( K ω b AK s, K s ) ( ), K AK s, K AK s ( ) K AK s, K s ω l (K AK s, K AK s ), ( ) AK s, s ω r (AK s, AK s ) (45) (47) (49) ω b ω l ω r ω ω (47) (49) r SAB +1 s ω ( AK ) s (50) ω b ω l ω r β PSAB ω 1 PBiCGStab ω PBiCGStab M ω b ω l ω r 3.2 BiCGStab ω (47) (49) r SAB +1 (5) 18) (1) (AK )(Kx) b. (51) BiCGStab 1),3),17),18) (43) p SAB p SAB, s s, r SAB r SAB, r 0 r 0. (52) BiCGStab 1 3) SAB Algorithm 1. BiCGStab : x 0, r 0 b Ax 0, r 0, r 0 0, e.g., r 0 r 0, β 0, For 0, 1, 2,, until convergence, Do: p r + β (p ω AK p ), α r 0, r r 0, AK p, s r α AK p, (AK s, s ) ω (AK s, AK s ), x +1 x + α K p + ω K s, r +1 s ω AK s, β α ω r 0, r +1 r 0, r, End Do Alg.1 K p K s BiCGStab (1) (2) BiCG (14) (16) (K A )x K b (53) PBiCG (53) 1 1) 7) 7), 5) 6) 9) 122

7 2 BiCGStab r :AK :r b Ax K r :K A :r b A x r :K A :r K ( b A x ) r :A K :r b A x K r K b (K A )x (54) r K r (55) (18) (52) r r (56) (29) (55) (56) 2 A K (34) (36) S (A ) S (z) ω α PSAB β PBiCG β PSAB (55) α PBiCG CGS 6),9) CGS α β (51) BiCGStab p SAB Kp SAB, s s, r SAB r SAB, r 0 K r 0 (57) (39)(40) α PBiCG, r 0 p, Ã p rsab 0 Ã psab r 0 rsab r 0, (AK ) (Kp SAB ) 0 K r SAB 0 K Ap SAB α PSAB, (58) β PBiCG +1, r +1, r α ω r 0 rsab +1 r 0 rsab 0 K r SAB +1 0 K r SAB α ω α ω 0, rsab +1 0, rsab β PSAB (59) BiCGStab α β BiCG (21)(22) BiCGStab Algorithm 2. BiCGStab : x 0, r 0 b Ax 0, r 0, r 0 0, e.g., r 0 K r 0, β 0, For 0, 1, 2,, until convergence, Do: p K r + β ( p ω K Ap ), α r 0, K r r 0, K Ap, (60) s r α Ap, K s K r α K Ap, (61) (AK s, s ) ω (AK s, AK s ), x +1 x + α p + ω K s, r +1 s ω AK s, β α ω r 0, K r +1 r 0, K r, (62) End Do Alg.2 3 K s, K r, K Ap K s (61) K r K Ap (62) 0 (60) 4. PBiCGStab im Davis s collection 16) Matrix Maret 12) 1.0 (1) 123

8 is ibrary of Iterative Solvers for inear Systems 11) is Maefile x 0 0 0, r 0 0 Alg. 1 r 0 r0 Alg. 2 r 0 K r 0 r 2 / b r DE Precision 7400 Intel Xeon E5420, 2.5GHz, MEM:16GB Cent S (Kernel ) Intel icc 10.1 IU(0) BiCGStab 3 N NNZ Conventional(Alg.1) Improved(Alg.2) (Iter.) log 10 () [sec (ime) 1 jpwh 991 Alg.1 Alg.2 IU(0)-BiCGStab 8) 9) cryg10000 olm5000 Alg.1 Alg.2 cryg2500 Alg.1 Alg.2 Iter PBiCG Algorithm s relative residual 2-norm (log scale) Algorithm s relative residual 2-norm (log scale) Algorithm s relative residual 2-norm (log scale) e-06 1e-08 1e-10 1e-12 1e e-06 1e-08 1e-10 1e-12 1e e-06 1e-08 1e-10 1e-12 cryg Iteration number (cryg2500) cryg10000 Conventional Improved Iteration number (cryg10000) fs_760_3 Conventional Improved Conventional Improved 1 BiCGStab Alg.1 Alg.2 1e Iteration number 4 (fs 760 3) 124

9 3 Conventional(Alg.1) Improved(Alg.2) Matrix N NNZ Iter. ime Iter. ime cryg e e-2 cryg No convergence e-1 fs e e-2 fs e e-2 jpwh Breadown e-3 memplus e e-1 olm No convergence e-2 raefsy e e0 watt e e-2 Algorithm s relative residual 2-norm (log scale) e-06 1e-08 1e-10 1e-12 1e-14 olm Iteration number (olm5000) Conventional Improved PGPBiCG PBiCGStab(l) BiCGStab Alg.1 Alg.2 Xabclib 19) (S.I.) PCGS e B BiCGStab PBiCG CGS PCGS PBiCG PCGS α β M ω M PBiCGStab PCGS BiCG CGS BiCGStab PBiCGStab M 1) Barrett,., et al., emplates for the solution of linear systems: Building Blocs for Iterative Methods, SIAM, (1994). emplates (1996). 2) Fletcher,., Conjugate Gradient Methods for Indefinite Systems, Numerical Analysis Dundee 1975, ed. by Watson, G., ecture Notes in Mathematics, 506, Springer-Verlag, pp (1976). 3) (1996). 4) II (1997). 5) (2009). 6) 5 Vol.15 pp ) : 125

10 (ACS) Vol.3, No.2 pp ) Itoh, S. and Sugihara, M., Systematic performance evaluation of linear solvers using quality control techniques, Software Automatic uning From Concepts to State-of-the-Art esults (eds. Naono, K., eranishi, K., Cavazos, J. and Suda,.), pp , Springer, ) CGS Vol.2011-HPC-130, No.46, pp.1 10 (2011). 10) anczos, C., Solution of Systems of inear Equations by Minimized Iterations, J. of es. Nat. Bur. of Standards, 49, pp (1952). 11) 12) 13) BCG CGS 613 pp (1987). 14) Sonneveld, P., CGS, A fast anczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 10(1), pp (1989). 15) (2009). 16) 17) Van der Vorst, Hen A., BI-CGSAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13(2), pp , ) Van der Vorst, Hen A., Iterative Krylov Methods for arge inear Systems, Cambridge University Press, (2003). 19) Xabclib project: 126