1 線形方程式の解法の選択
2 参考文献および参考書の記述

解法原著論文[2][14][27][23]
CG 法[10]14–17187–19437–47148–15331–35
CR 法[22]―194―――
MINRES 法[12]17–18―84–91――
GMRES 法, GMRES(m)?[15]19–21164–17265–84173–18157–63
GCR 法, GCR(m)?, ORTHOMIN(m)?[5]―194–196―164–17363–70
FOM 法, FOM(m)?[13]―159–161―――
DQGMRES(m) 法[16]―172–177―――
GMRES-DR(m; k) 法[11]―――――
Look-Back GMRES(m) 法[?]―――――
Bi-CG 法[6]21–23222–22495–98181–19038–41
Bi-CR 法[18], [19]―――――
QMR 法[8]23–25224–22898–102――
CGS 法[20]25–27229–231102–106―46–47
Bi-CGSTAB 法[26]27–28231–234133–138190–19347–49
Bi-CGSTAB2 法[9]――138–141―53–55
Bi-CGSTAB(l) 法[17]――138–141195–201―
GPBi-CG 法[28]――141–144193–19451–53
CRS 法, Bi-CRSTAB 法, GPBi-CR 法[1]―――――
TFQMR 法[7]―234–240―――
QMRCGSTAB 法[3]―――――
QMRCGSTAB(l) 法[25]―――――
IDR(s) 法[21]―――――
GBi-CGSTAB(s; l) 法[24]―――――
Jacobi 法, Gauss-Seidel 法, SOR 法―7–12103–106―63–86―
AOR 法――――――
ADI 法――124–126―92–106―
減速定常反復法――――――
Chebyshev 加速――――86–92―
CGNE 法[4]18253–254――35–36
CGNR 法[10]18252–253――35–36
Cimmino-NR 法――249–251―――
幾何的/代数的マルチグリッド法?――407–449―106–136―
COCG 法―――107–111162―
COCR 法――――162―
QMR-SYM 法―――111–113――
Uzawa 法――254–257―――
FFT に基づく高速解法?――――――

参考文献 †

[1] Kuniyoshi Abe, Tomohiro Sogabe, Seiji Fujino and Shao-Liang Zhang, A product-type Krylov subspace method based on conjugate residual method for nonsymmetric coefficient matrices, IPSJ Transactions on Advanced Computing Systems 2007; 48(SIG 8):11–21, (in Japanese).

[2] Richard Barrett, Michael W. Berry, Tony F. Chan, James Demmel, June Donato, Jack Dongarra, Victor Eijkhout, Roldan Pozo, Charles Romine and Henk A. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM: Philadelphia, PA, 1993.

[3] Tony F. Chan, Efstratios Gallopoulos, Valeria Simoncini, Tedd Szeto and Charles H. Tong, A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems, SIAM Journal on Scientific Computing 1994; 15(2):338–347.

[4] Edward J. Craig, The N-step iteration procedures, Journal of Mathematics and Physics 1955; 34:64–73.

[5] Stanley C. Eisenstat, Howard C. Elman and Martin H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations, SIAM Journal on Numerical Analysis 1983; 20(2):345–357.

[6] Roger Fletcher, Conjugate gradient methods for indefinite systems, In: Lecture Notes in Mathematics, G. Alistair Watros (ed.), vol. 506, Springer-Verlag: New York, NY, 1976; 73–89.

[7] Roland W. Freund, A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM Journal on Scientific Computing 1993; 14(2):470–482.

[8] Roland W. Freund and No¨el M. Nachtigal, QMR: a quasi-minimal residual method for non- Hermitian linear systems, Numerische Mathematik 1991; 60(1):315–339.

[9] Martin H. Gutknecht, Variants of BiCGStab for matrices with complex spectrum, SIAM Journal on Scientific Computing 1993; 14(5):1020–1033.

[10] Magnes R. Hestenes and Eduard Stiefel, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards 1952; 49(6):409–436.

[11] Ronald B. Morgan, GMRES with deflated restarting, SIAM Journal on Scientific Computing 2002; 24(1):20–37.

[12] Christopher C. Paige and Michael A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM Journal on Numerical Analysis 1975; 12(4):617–629.

[13] Yousef Saad, Krylov subspace methods for solving large unsymmetric linear systems, Mathematics of Computation 1981; 37(155):105–126.

[14] Yousef Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM: Philadelphia, PA, 2003.

[15] Yousef Saad and Martin H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing 1986; 7(3):856–869.

[16] Yousef Saad and Kesheng Wu, DQGMRES: a direct quasi-minimal residual algorithm based on incomplete orthogonalization, Numerical Linear Algebra with Applications 1996; 3(4):329–343.

[17] Gerard L. G. Sleijpen and Diederik R. Fokkema, BiCGStab(l) for linear equations involving unsymmetric matrices with complex spectrum, Electronic Transactions on Numerical Analysis 1993; 1:11–32.

[18] Tomohiro Sogabe, Masaaki Sugihara and Shao-Liang Zhang, An extension of the conjugate residual method for solving nonsymmetric linear systems, Transactions of the Japan Society for Industrial and Applied Mathematics 2005; 15(3):445–459, (in Japanese).

[19] Tomohiro Sogabe, Masaaki Sugihara and Shao-Liang Zhang, An extension of the conjugate residual method to nonsymmetric linear systems, Journal of Computational and Applied Mathematics 2009; 226(1):103–113.

[20] Peter Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing 1989; 10(1):36–52.

[21] Peter Sonneveld and Martin B. van Gijzen, IDR(s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM Journal on Scientific Computing 2008; 31(2):1035–1062.

[22] Eduard Stiefel, Relaxationsmethoden bester strategie zur l¨osung linearer gleichungssysteme, Commentarii Mathematici Helvetici 1952; 29(1):157–179.

[23] Masaaki Sugihara and Kazuo Murota, Theoretical Numerical Linear Algebra, Iwanami Press: Tokyo, 2009, (in Japanese).

[24] Masaaki Tanio and Masaaki Sugihara, GBi-CGSTAB(s;L): IDR(s) with higher-order stabilization polynomials, Journal of Computational and Applied Mathematics 2010; 235(3):765–784.

[25] Charles H. Tong, A family of quasi-minimal residual methods for nonsymmetric linear systems, SIAM Journal on Scientific Computing 1994; 15(1):89–105.

[26] Henk A. van der Vorst, Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing 1992; 13(2):631–644.

[27] Henk A. van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge University Press: New York, NY, 2003.

[28] Shao-Liang Zhang, GPBi-CG: generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems, SIAM Journal on Scientific Computing 1997; 18(2):537–551.

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