In this report we describe the implementations of the GMRES 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 GMRES solvers have been implemented using the reverse communication mechanism for the matrixvector product, the preconditioning and ...

In this paper, a two-stage iterative algorithm is proposed to improve the convergence of Krylov based iterative methods, typically those of GMRES variants. The principle of the proposed approach is to build an external iteration over the Krylov method, and to frequently store its current residual (at each GMRES restart for instance). After a given number of outer iterations, a least-squares min...

ML(n)BiCGStab is a Krylov subspace method for the solution of large, sparse and non-symmetric linear systems. In theory, it is a method that lies between the well-known BiCGStab and GMRES/FOM. In fact, when n = 1, ML(1)BiCGStab is BiCGStab and when n = N, ML(N)BiCGStab is GMRES/FOM where N is the size of the linear system. Therefore, ML(n)BiCGStab is a bridge that connects the Lanczos-based BiC...

In this paper we study nonmonotone globalization techniques, in connection with finite-difference inexact Newton-GMRES methods, for solving large-scale systems of nonlinear equations in the case that the Jacobian matrix is not available. We first define a globalization scheme, which combines nonmonotone watchdog rules and nonmonotone derivative-free line searches, and we prove its global conver...

In the Generalized Minimal Residual Method (GMRES), the global all-to-all communication required in each iteration for orthogonalization and normalization of the Krylov base vectors is becoming a performance bottleneck on massively parallel machines. Long latencies, system noise and load imbalance cause these global reductions to become very costly global synchronizations. In this work, we prop...

One of the most powerful tools for solving large and sparse systems of linear equation is iterative methods. Their significant advantages like low memory requirements and good approximation properties make them very popular, and they are widely used in applications throughout science and engineering. Residual Algorithm for solving large-scale nonsymmetric linear system of equation which symmetr...

The block GMRES (BGMRES) method is a natural generalization of the GMRES algorithm for solving large nonsymmetric 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 th...

The sideways Heat equation (SHE) 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: the solution does not depend continuously on the data. We use a preconditioned Generalized Mi...