We consider the behavior of the GMRES method for solving a linear system Ax = b when A is singular or nearly so, i.e., ill conditioned. The (near) singularity of A may or may not affect the performance of GMRES, depending on the nature of the system and the initial approximate solution. For singular A, we give conditions under which the GMRES iterates converge safely to a least-squares solution...

GMRES is a popular iterative method for the solution of linear system of equations with an unsymmetric square matrix. Range restricted GMRES (RRGMRES) is one GMRES version proposed by Calvetti et al in 2000. In this paper, a weighted implementation for RRGMRES is proposed. Numerical results prove this weighted RRGMRES is better than RRGMRES.

In this report we describe the implementations of the flexible GMRES (FGMRES) algorithm for both real and complex, single and double precision arithmetics suitable for serial, shared memory and distributed memory computers. For the sake of simplicity, flexibility and efficiency the FGMRES solvers have been implemented using the reverse communication mechanism for the matrix-vector product, the ...

The balancing domain decomposition methods by constraints are extended to solving nonsymmetric, positive definite linear systems resulting from the finite element discretization of advection-diffusion equations. A preconditioned GMRES iteration is used to solve a Schur complement system of equations for the subdomain interface variables. In the preconditioning step of each iteration, a partiall...

This paper reports the development and performance of CaMEL_Aero, our truly matrix-free, parallel and vectorized unstructured finite volume solver for compressible flows. The Jacobian-free GMRES method is used to solve the linear systems of equations inside each nonlinear Newton-Raphson iteration. Furthermore, the matrix-free Lower-Upper Symmetric Gauss Seidel (LUSGS) method is employed as a pr...

In this study, the swing-up of a double inverted pendulum is controlled by nonlinear model predictive control (NMPC). The fast computation algorithm called C/GMRES (continuation/generalized minimal residual) is applied to solve a nonlinear two-point boundary value problem over a receding horizon in real time. The goal is to swing-up and stabilize two pendulums from the downward to upright posit...

The GMRES and Arnoldi algorithms, which reduce to the CR and Lanczos algorithms in the symmetric case, both minimize p(A)b over polynomials p of degree n. The difference is that p is normalized at z 0 for GMRES and at z x for Arnoldi. Analogous "ideal GMRES" and "ideal Arnoldi" problems are obtained if one removes b from the discussion and minimizes p(/l)II instead. Investigation of these true ...

We propose a variant of GMRES, where multiple (two or more) preconditioners are applied simultaneously, while maintaining minimal residual optimality properties. To accomplish this, a block version of Flexible GMRES is used, but instead of considering blocks associated with multiple right hand sides, we consider a single right-hand side and grow the space by applying each of the preconditioners...

The sideways parabolic equation (SPE) is a model of the problem of determining the temperature on the surface of a body from the interior measurements. Mathematically it can be formulated as a noncharacteristic Cauchy problem for a parabolic partial differential equation. This problem is severely ill-posed in an L2 setting. We use a preconditioned generalized minimum residual method (GMRES) to ...