Block GMRES Method Research Articles

A Seed Method for Solving Nonsymmetric Linear Systems with Multiple Right-hand Sides

We present a seed method for solving large nonsymmetric linear systems with multiple right-hand sides. The method uses a single augmented Krylov subspace corresponding to a seed system as a generator of approximations to the nonseed systems. The residual evaluate of the method is shown, and a new strategy to form a seed system which could supply information shareable among the right-hand sides is given. Numerical experiments indicate that our seed selection strategy is more efficient than two existing strategies and our method has significant time saving compared with the block GMRES method and the GMRES method with a projection process.

Tracing the buckling of a rectangular plate with the Block GMRES method

We investigate bifurcation scenario of the von Kármán equations with partially clamped boundary conditions defined in rectangular domains. First, we study how the (preconditioned) Block GMRES method can be used in the context of continuation methods for tracing solution curves of large systems of nonlinear equations. Next, we discuss linear stabilities of the von Kármán equations with partially clamped boundary conditions. In particular, we calculate the first seven eigenvalues and its associated eigenfunctions of the linearized von Kármán equations via computer algebra. The Block GMRES method is used to solve linear systems and to detect singularity along solution paths of the discrete problem. Sample numerical results are reported.

A block GMRES method augmented with eigenvectors

The GMRES method augmented with some eigenvectors is extended to a block version for solving large nonsymmetric linear systems with multiple right-hand sides. Residual bound is established which has an improved convergence rate on the standard block GMRES method. This improvement displays a good efficiency when restarting is used. Also an improved residual norm estimate is derived by using matrix-valued polynomials. Numerical experiments show that the new algorithm is more efficient, compared with the block GMRES method.

A hybrid block GMRES method for nonsymmetric systems with multiple right-hand sides

The block GMRES (BGMRES) method is a natural generalization of the GMRES algorithm for solving large non-symmetric systems with multiple right-hand sides. Unfortunately, its cost increases significantly per iteration, frequently rendering the method impractical. In this paper we propose a hybrid block GMRES method which offers significant performance improvements over BGMRES. This method uses the matrix polynomial obtained in the course of a BGMRES step and combines the advantages of the block approach with those of successful hybrid methods. We discuss the properties and several implementation variants of the method and report results from numerical experiments. We also describe how to use these techniques in order to solve multiply shifted systems with multiple right-hand sides.