手法を比較

	GMRES	共役勾配法
分野	数値解析	数値解析
系統	Machine learning	Machine learning
提唱年	1986	1952
提唱者	Youcef Saad and Martin H. Schultz	Magnus Hestenes and Eduard Stiefel
種類	Iterative linear solver for non-symmetric systems	Iterative linear solver
原典	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.	Hestenes, M. R., & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49(6), 409–436.
別名	GMRES(m), restarted GMRES, Krylov-GMRES	CG method, Krylov subspace method
関連	1	1
概要	GMRES (Generalized Minimal Residual) is an iterative method for solving large sparse non-symmetric or nonsymmetric linear systems Ax = b, developed by Saad and Schultz in 1986. It builds an orthonormal Krylov basis using Arnoldi's method and solves a least-squares problem to minimize residual at each iteration.	The Conjugate Gradient (CG) Method is an iterative algorithm for solving large sparse symmetric positive-definite linear systems Ax = b, developed by Hestenes and Stiefel in 1952. It is one of the most widely used iterative solvers in scientific computing because it converges in at most n iterations for an n × n matrix and typically requires far fewer.