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GMRES and Integral Operators [PDF]
SIAM Journal on Scientific Computing, 1996 The purpose of this paper is to show how the generalized minimal residual (GMRES) method can be modified to incorporate Nyström interpolation at a small cost in both computational effort and algorithmic complexity. The result is an algorithm that has the convergence property of Broyden's method.
Carl T. Kelley, Z. Q. Xue
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SIAM Journal on Scientific Computing, 2023 The GMRES algorithm of Saad and Schultz (1986) is an iterative method for approximately solving linear systems $A{\bf x}={\bf b}$, with initial guess ${\bf x}_0$ and residual ${\bf r}_0 = {\bf b} - A{\bf x}_0$. The algorithm employs the Arnoldi process to generate the Krylov basis vectors (the columns of $V_k$).
Stephen J. Thomas +4 more
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Properties of Worst-Case GMRES [PDF]
SIAM Journal on Matrix Analysis and Applications, 2013 In the convergence analysis of the GMRES method for a given matrix $A$, one quantity of interest is the largest possible residual norm that can be attained, at a given iteration step $k$, over all unit norm initial vectors. This quantity is called the worst-case GMRES residual norm for $A$ and $k$.
Vance Faber, Jörg Liesen, Petr Tichý
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FIVE-PRECISION GMRES-BASED ITERATIVE REFINEMENT ∗ [PDF]
, 2023 GMRES-based iterative refinement in three precisions (GMRES-IR3) uses a low precision LU factorization to accelerate the solution of a linear system without compromising numerical stability or robustness.
Amestoy, Patrick +5 more
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CoRRAccepted by Journal of Foundations of Computational ...
Pierre Matalon, Nicole Spillane
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, 1994 \begin{abstract} \noindent The GMRES algorithm minimizes $\norm{p(A)b}$ over polynomials $p$ of degree $n$ normalized at $z=0$. The ideal GMRES problem is obtained if one considers minimization of $\norm{p(A)}$ instead.
Kim-Chuan, Toh
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Adaptive version of Simpler GMRES [PDF]
Numerical Algorithms, 2009 The authors propose and theoretically analyze a stable version of simpler generalized minimal residual (GMRES) algorithm, based on an adaptive choice of the Krylov subspace basis at a given iteration step. They show that this adaptive choice of direction vectors keeps the basis well-conditioned and that the condition number grows at most linearly with ...
Pavel Jiránek, Miroslav Rozlozník
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Five-precision GMRES-based Iterative Refinement [PDF]
, 2021 GMRES-based iterative refinement in three precisions (GMRES-IR3) uses a low precision LU factorization to accelerate the solution of a linear system without compromising numerical stability or robustness.
Amestoy, Patrick +7 more
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A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
Open Mathematics, 2018 In this paper, we propose a relaxed block splitting preconditioner for a class of complex symmetric indefinite linear systems to accelerate the convergence rate of the Krylov subspace iteration method and the relaxed preconditioner is much closer to the ...
Huang Yunying, Chen Guoliang
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Advances in Electrical and Electronic Engineering, 2014 The paper deals with the solution of large non-symmetric two-by-two block linear systems with a singular leading submatrix. Our algorithm consists of two levels.
Radek Kucera +4 more
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