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In this paper we consider the problem of preconditioning symmetric positive definite matrices of the form Aα = A + αI where α > 0. We discuss how to cheaply modify an existing sparse approximate inverse preconditioner for A in order to obtain a precon-ditioner for Aα. Numerical experiments illustrating the performance of the proposed approaches are… (More)

In this paper, a recently introduced block circulant preconditioner for the linear systems of the codes for ordinary differential equations (ODEs) is investigated. Most ODE codes based on implicit formulas, at each integration step, need the solution of one or more unsymmetric linear systems that are often large and sparse. Here, the boundary value methods,… (More)

SUMMARY Implicit time-step numerical integrators for ordinary and evolutionary partial diierential equations need, at each step, the solution of linear algebraic equations that are unsymmetric and often large and sparse. Recently, a block preconditioner based on circulant approximations for the linear systems arising in the boundary value methods (BVMs) was… (More)

The solution of ordinary and partial differential equations using implicit linear multi-step formulas (LMF) is considered. More precisely, boundary value methods (BVMs), a class of methods based on implicit formulas will be taken into account in this paper. These methods require the solution of large and sparse linear systemsˆM x = b. Block-circulant… (More)

The numerical solution of large and sparse nonsymmetric linear systems of algebraic equations is usually the most time consuming part of time-step integrators for differential equations based on implicit formulas. Preconditioned Krylov subspace methods using Strang block circulant pre-conditioners have been employed to solve such linear systems. However, it… (More)

We consider real-valued preconditioned Krylov subspace methods for the solution of complex linear systems, with an emphasis on symmetric (non-Hermitian) problems. Different choices of the real equivalent formulation are discussed, as well as different types of block preconditioners for Krylov subspace methods. Numerical experiments illustrating the… (More)

We revisit real-valued preconditioned iterative methods for the solution of complex linear systems, with an emphasis on symmetric (non-Hermitian) problems. Different choices of the real equivalent formulation are discussed, as well as different types of block preconditioners for Krylov subspace methods. We argue that if either the real or the symmetric part… (More)

Solution of sequences of complex symmetric linear systems of the form A j x j = b j , complex parameters arise in a variety of challenging problems. This is the case of time dependent PDEs; lattice gauge computations in quantum chromodynamics; the Helmholtz equation; shift-and-invert and Jacobi–Davidson algorithms for large-scale eigenvalue calculations;… (More)

Newton-Krylov methods, combination of Newton-like methods and Krylov subspace methods for solving the Newton equations, often need adequate preconditioning in order to be successful. Approximations of the Jacobian matrices are required to form preconditioners and this step is very often the dominant cost of Newton-Krylov methods. Therefore, working with… (More)