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In mathematics, the generalized minimal residual method (usually abbreviated GMRES) is an iterative method for the numerical solution of a nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector. The GMRES method was developed by Yousef Saad and Martin H. Schultz in 1986.〔Saad and Schultz〕 GMRES is a generalization of the MINRES method developed by Chris Paige and Michael Saunders in 1975. GMRES also is a special case of the DIIS method developed by Peter Pulay in 1980. DIIS is also applicable to non-linear systems. == The method == Denote the Euclidean norm of any vector ''v'' by . Denote the system of linear equations to be solved by : The matrix ''A'' is assumed to be invertible of size ''m''-by-''m''. Furthermore, it is assumed that ''b'' is normalized, i.e., that ||''b''|| = 1. The ''n''th Krylov subspace for this problem is : GMRES approximates the exact solution of ''Ax'' = ''b'' by the vector ''x''''n'' ∈ ''K''''n'' that minimizes the Euclidean norm of the residual ''r''''n'' = ''Ax''''n'' − ''b''. The vectors ''b'', ''Ab'', …, ''A''''n''−1''b'' might be almost linearly dependent, so instead of this basis, the Arnoldi iteration is used to find orthonormal vectors : which form a basis for ''K''''n''. Hence, the vector ''x''''n'' ∈ ''K''''n'' can be written as ''x''''n'' = ''Q''''n''''y''''n'' with ''y''''n'' ∈ R''n'', where ''Q''''n'' is the ''m''-by-''n'' matrix formed by ''q''1, …, ''q''n. The Arnoldi process also produces an (''n''+1)-by-''n'' upper Hessenberg matrix with : Because columns of are orthogonal, we have : where : is the first vector in the standard basis of R''n''+1, and : being the first trial vector (usually zero). Hence, can be found by minimizing the Euclidean norm of the residual : This is a linear least squares problem of size ''n''. This yields the GMRES method. On the ''n''-th iteration: # calculate with the Arnoldi method; # find the which minimizes ||''r''''n''||; # compute ; # repeat if the residual is not yet small enough. At every iteration, a matrix-vector product ''Aq''''n'' must be computed. This costs about 2''m''2 floating-point operations for general dense matrices of size ''m'', but the cost can decrease to O(''m'') for sparse matrices. In addition to the matrix-vector product, O(''n'' ''m'') floating-point operations must be computed at the ''n''th iteration. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Generalized minimal residual method」の詳細全文を読む スポンサード リンク
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