Claude Brezinski

Claude Brezinski, Michela Redivo-Zaglia

The method of alternation projections (MAP) is an iterative procedure for finding the projection of a point on the intersection of closed subspaces of an Hilbert space. The convergence of this method is usually slow, and several methods for its acceleration have already been proposed. In this work, we consider a special MAP, namely Kaczmarz' method for solving systems of linear equations. The convergence of this method is discussed. After giving its matrix formulation and its projection properties, we consider several procedures for accelerating its convergence. They are based on sequence transformations whose kernels contain sequences of the same form as the sequence of vectors generated by Kaczmarz' method. Acceleration can be achieved either directly, that is without modifying the sequence obtained by the method, or by restarting it from the vector obtained by acceleration. Numerical examples show the effectiveness of both procedures.

Claude Brezinski, Michela Redivo-Zaglia

In this paper, we consider the particular case of the general rational Hermite interpolation problem where only the value of the function is interpolated at some points, and where the function and its first derivatives agree at the origin. Thus, the interpolants constructed in this way possess a Padé--type property at 0. Numerical examples show the interest of the procedure. The interpolation procedure can be easily modified to introduce a partial knowledge on the poles and the zeros of the function to approximated. A strategy for removing the spurious poles is explained. A formula for the error is proved in the real case. Applications are given.

Claude Brezinski, Paolo Novati, Michela Redivo-Zaglia

For the solution of full-rank ill-posed linear systems a new approach based on the Arnoldi algorithm is presented. Working with regularized systems, the method theoretically reconstructs the true solution by means of the computation of a suitable function of matrix. In this sense the method can be referred to as an iterative refinement process. Numerical experiments arising from integral equations and interpolation theory are presented. Finally, the method is extended to work in connection with the standard Tikhonov regularization with a right hand side contaminated by noise.

Claude Brezinski, Yi He, Xing-Biao Hu et al.

In this paper, we give a multistep extension of the epsilon-algorithm of Wynn, and we show that it implements a multistep extension of the Shanks' sequence transformation which is defined by ratios of determinants. Reciprocally, the quantities defined in this transformation can be recursively computed by the multistep epsilon-algorithm. The multistep epsilon-algorithm and the multistep Shanks' transformation are related to an extended discrete Lotka-Volterra system. These results are obtained by using the Hirota's bilinear method, a procedure quite useful in the solution of nonlinear partial differential and difference equations.