Article history: Received 15 October 2012 Received in revised form 25 June 2013 Accepted 15 July 2013 Available online 23 July 2013

We present an iterative solution method for the discrete high wavenumber Helmholtz equation. The basic idea of the solution method, already presented in [18], is to develop a preconditioner which is based on a Helmholtz operator with a complex-valued shift, for a Krylov subspace iterative method. The preconditioner, which can be seen as a strongly damped wave equation in Fourier space, can be a...

Given a pair of matrices and starting vectors, we present a procedure to generate the biorthonormal basis of the second-order right and left Krylov subspaces. The application is to solve the large-scale quadratic eigenvalue problems via oblique projection technique. This method can take full advantage of the sparseness of large-scale system as well as the superior convergence behavior of Krylov...

We investigate weighted Lp(0 < p <.) convergence of Hermite and Hermite– Fejér interpolation polynomials of higher order at the zeros of Freud orthogonal polynomials on the real line. Our results cover as special cases Lagrange, Hermite– Fejér and Krylov–Stayermann interpolation polynomials. © 2001 Academic Press

Motivated by a connection with block iterative methods for solving linear systems over finite fields, we consider the probability that the Krylov space generated by a fixed linear mapping and a random set of elements in a vector space over a finite field equals the space itself. We obtain an exact formula for this probability, and from it we derive good lower bounds that approach 1 exponentiall...

Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and...

An improved preconditioned CGS (PCGS) algorithm has recently been proposed, and it performs much better than the conventional PCGS algorithm. In this paper, the improved PCGS algorithm is verified as a coordinative to the left-preconditioned system; this is done by comparing, analyzing, and executing numerical examinations of various PCGS algorithms, including the most recently proposed one. We...