Restarted GMRES methods augmented with approximate eigenvectors are widely used for solving large sparse linear systems. Recently a new scheme of augmenting with error approximations is proposed. The main aim of this paper is to develop a restarted GMRES method augmented with the combination of harmonic Ritz vectors and error approximations. We demonstrate that the resulted combination method c...

This paper studies convergence properties of the block gmres algorithm when applied to nonsymmetric systems with multiple right-hand sides. A convergence theory is developed based on a representation of the method using matrix-valued polynomials. Relations between the roots of the residual polynomial for block gmres and the matrix "-pseudospectrum are derived, and illustrated with numerical exp...

We present a polynomial preconditioner for solving large systems of linear equations. The is derived from the minimum residual (the GMRES polynomial) and more straightforward to compute implement than many previous preconditioners. Our current implementation this using its roots naturally stable methods computing same polynomial. further stability control added roots, allows high degree polynom...

The generalized minimum residual method (GMRES) is well known for solving large nonsymmetric systems of linear equations. It generally uses restarting, which slows the convergence. However, some information can be retained at the time of the restart and used in the next cycle. We present algorithms that use implicit restarting in order to retain this information. Approximate eigenvectors determ...

A robust multilevel preconditioner based on the hybridizable discontinuous Galerkin method for the Helmholtz equation with high wave number is presented in this paper. There are two keys in our algorithm, one is how to choose a suitable intergrid transfer operator, and the other is using GMRES smoothing on coarse grids. The multilevel method is performed as a preconditioner in the outer GMRES i...

The Generalized Minimum Residual (GMRES) method is a popular Krylov subspace projection method for solving a nonsymmetric linear system of equations. On modern computers, communication is becoming increasingly expensive compared to arithmetic operations, and a communication-avoiding variant (CA-GMRES) may improve the performance of GMRES. To further enhance the performance of CAGMRES, in this p...

We present a variant of the GMRES algorithm which allows changes in the precon-ditioning at every step. There are many possible applications of the new algorithm some of which are brieey discussed. In particular, a result of the exibility of the new variant is that any iterative method can be used as a preconditioner. For example, the standard GMRES algorithm itself can be used as a preconditio...

In this paper, we consider a family of algorithms, called IDR, based on the induced dimension reduction theorem. IDR is efficient short recurrence methods introduced by Sonneveld and Van Gijzen for solving large systems nonsymmetric linear equations. These generate residual vectors that live in sequence nested subspaces. We present IDR(s) method give two improvements its convergence. also defin...

The GMRES algorithm of Saad and Schultz [SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856–869] is an iterative method for approximately solving linear systems , with initial guess residual . employs the Arnoldi process to generate Krylov basis vectors (the columns ). It well known that this can be viewed as a factorization matrix at each iteration. Despite loss orthogonality, unit roundoff conditi...