More recently, Beik and Salkuyeh [F. P. A. Beik and D. K. Salkuyeh, On the global Krylov subspace methods for solving general coupled matrix equations, Computers and Mathematics with Applications, 62 (2011) 4605–4613] have presented the Gl-FOM and Gl-GMRES algorithms for solving the general coupled linear matrix equations. In this paper, two new algorithms called weighted Gl-FOM (WGl-FOM) and w...

This paper presents, a preconditioned version of global FOM and GMRES methods for solving Lyapunov matrix equations AX + XA = −BTB. These preconditioned methods are based on the global full orthogonalization and generalized minimal residual methods. For constructing effective preconditioners, we will use ADI spiliting of above lyapunov matrix equations. Numerical experiments show that the solut...

In the present paper, we propose the global full orthogonalization method (Gl-FOM) and global generalized minimum residual (Gl-GMRES) method for solving large and sparse general coupled matrix equations

In this paper, we consider both local and global convergence of the Newton algorithm to solve nonlinear problems when GMRES is used to invert the Jacobian at each Newton iteration. Under weak assumptions, we give a suucient condition for an inexact solution of GMRES to be a descent direction in order to apply a backtracking technique. Moreover, we extend this result to a nite diierence scheme c...

We present a new minimal residual method, called global R-linear GMRES, to solve the R-linear matrix equations X + AXB = C and X + AXB = C, where C, X ∈ Cm×n, X denotes the complex conjugate of X, X its complex conjugate transpose, and A, B are complex matrices with appropriate dimensions. We show that the new method requires fewer matrix-matrix products than the global GMRES method applied to ...

Many problems in science and engineering field require the solution of shifted linear systems with multiple right hand sides and multiple shifts. To solve such systems efficiently, the implicitly restarted global GMRES algorithm is extended in this paper. However, the shift invariant property could no longer hold over the augmented global Krylov subspace due to adding the harmonic Ritz matrices...

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...

Steepest descent preconditioning is considered for the recently proposed nonlinear generalized minimal residual (N-GMRES) optimization algorithm for unconstrained nonlinear optimization. Two steepest descent preconditioning variants are proposed. The first employs a line search, while the second employs a predefined small step. A simple global convergence proof is provided for the NGMRES optimi...

In this paper, we propose an efficient parallel implementation of the GMRES method for GPU clusters. This implementation requires us to parallelize the GMRES algorithm between CPUs of the cluster. Hence, all parallel and intensive computations on local data are performed on GPUs and reduction operations to compute global results are carried out by CPUs. The performances of our parallel GMRES so...

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...