GMRES(k) is widely used for solving nonsymmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating ...

We consider the solution of large and sparse linear systems of equations by GMRES. Due to the appearance of unfavorable eigenvalues in the spectrum of the coefficient matrix, the convergence of GMRES may hamper. To overcome this, a deflated variant of GMRES can be used, which treats those unfavorable eigenvalues effectively. In the literature, several deflated GMRES variants are applied success...

GMRES is a popular iterative method for the solution of large linear systems of equations with a square nonsingular matrix. When the matrix is singular, GMRES may break down before an acceptable approximate solution has been determined. This paper discusses properties of GMRES solutions at breakdown and presents a modification of GMRES to overcome the breakdown.

Consider a system of linear algebraic equations with a nonsingular n by n matrix A. When solving this system with GMRES, the relative residual norm at the step k is bounded from above by the so called ideal GMRES approximation. This bound is sharp (it is attainable by the relative GMRES residual norm) in case of a normal matrix A, but it need not characterize the worstcase GMRES behavior if A i...

The GMRES method is one of the most popular iterative schemes for the solution of large linear systems of equations with a square nonsingular matrix. GMRES-type methods also have been applied to the solution of linear discrete ill-posed problems. Computational experience indicates that for the latter problems variants of the standard GMRES method, that require the solution to live in the range ...

Restarted GMRES is one of the most popular methods for solving large nonsymmetric linear systems. The algorithm GMRES(m) restarts every m iterations. It is generally thought the information of previous GMRES cycles is lost at the time of a restart, so that each cycle contributes to the global convergence individually. However, this is not the full story. In this paper, we shed light on the rela...