GMRES法
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2023/09/11 01:36 UTC 版)
数学において、GMRES法（GMRESほう、generalized minimal residual method）は、連立一次方程式の数値解を求めるための反復法の一種である[1]。残差をクリロフ部分空間において最小化することにより、近似解を計算する。ベクトルの計算にはアーノルディ法[2]が用いられる。ヨセフ・サードとマルティン・H・シュルツにより、1986年に開発された[3]。
- ^ a b Black, Noel and Moore, Shirley. "Generalized Minimal Residual Method." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. mathworld
.wolfram .com /GeneralizedMinimalResidualMethod .html - ^ W. E. Arnoldi (1951), "The principle of minimized iterations in the solution of the matrix eigenvalue problem," Quarterly of Applied Mathematics, volume 9, pages 17–29.
- ^ Saad, Y., & Schultz, M. H. (1986). GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on scientific and statistical computing, 7(3), 856-869.
- ^ Greenbaum, A., Pták, V., & Strakoš, Z. E. K. (1996). Any nonincreasing convergence curve is possible for GMRES. SIAM journal on matrix analysis and applications, 17(3), 465-469.
- ^ Trefethen & Bau, Thm 35.2
- ^ Baglama, J., Calvetti, D., Golub, G. H., & Reichel, L. (1998). Adaptively preconditioned GMRES algorithms. SIAM Journal on Scientific Computing, 20(1), 243-269.
- ^ Burrage, K., & Erhel, J. (1998). On the performance of various adaptive preconditioned GMRES strategies. Numerical linear algebra with applications, 5(2), 101-121.
- ^ Frayssé, V., Giraud, L., & Gratton, S. (1998). A set of Flexible-GMRES routines for real and complex arithmetics. Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique TR/PA/98/20.
- ^ Lanczos, C. (1950). "An iteration method for the solution of the eigenvalue problem of linear differential and integral operators". Journal of Research of the National Bureau of Standards. 45 (4): 255–282.
- ^ Black, Noel and Moore, Shirley. "Minimal Residual Method." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. mathworld
.wolfram .com /MinimalResidualMethod .html - ^ Paige, C. and Saunders, M. "Solution of Sparse Indefinite Systems of Linear Equations." SIAM J. Numer. Anal. 12, 617-629, 1975.
- ^ Fong, D. C. L., & Saunders, M. (2012). CG versus MINRES: An empirical comparison. Sultan Qaboos University Journal for Science [SQUJS], 17(1), 44-62.
- ^ Black, Noel and Moore, Shirley. "Biconjugate Gradient Method." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. mathworld
.wolfram .com /BiconjugateGradientMethod .html - ^ Black, Noel and Moore, Shirley. "Conjugate Gradient Squared Method." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. mathworld
.wolfram .com /ConjugateGradientSquaredMethod .html - ^ Van der Vorst, H. A. (1992). 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, 13(2), 631-644.
- ^ Stoer and Bulirsch, §8.7.2
[続きの解説]
「GMRES法」の続きの解説一覧
- 1 GMRES法とは
- 2 GMRES法の概要
- 3 解法の拡張
- 4 参考文献
- GMRES法のページへのリンク
