In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite 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.[1] It is a generalization and improvement of the MINRES method due to Paige and Saunders in 1975.[2][3] The MINRES method requires that the matrix is symmetric, but has the advantage that it only requires handling of three vectors. GMRES is a special case of the DIIS method developed by Peter Pulay in 1980. DIIS is applicable to non-linear systems.
^Saad, Youcef; Schultz, Martin H. (1986). "GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems". SIAM Journal on Scientific and Statistical Computing. 7 (3): 856–869. doi:10.1137/0907058. ISSN 0196-5204.
^Paige and Saunders, "Solution of Sparse Indefinite Systems of Linear Equations", SIAM J. Numer. Anal., vol 12, page 617 (1975) https://doi.org/10.1137/0712047
^Nifa, Naoufal (2017). Solveurs performants pour l'optimisation sous contraintes en identification de paramètres [Efficient solvers for constrained optimization in parameter identification problems] (Thesis) (in French).
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