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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)

The convergence features of a preconditioned algorithm for the convection-diffusion equation based on its diffusion part are considered. Analyses of the distribution of the eigenvalues of the preconditioned matrix in arbitrary dimensions and of the fundamental parameters of convergence are provided, showing the existence of a proper cluster of eigenvalues.… (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)

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)

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)

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)

A fast full second order time-step algorithm for some recently proposed nonlinear, nonlocal active models for the inner ear is analyzed here. In particular, we emphasize the properties of discretized systems and the convergence of a hybrid direct-iterative solver for its approximate solution in view of the parameters of the continuous model. We found that… (More)