Méthodes Sinc exponentielles doubles, méthodes de Lanczos par blocs et globale et applications

Abstract

The current thesis work is presented in two parts. The first part is devoted to the solution of the Schrödinger equation using Sinc numerical methods. The nonstationary version of this equation is treated in two dimensions using the double exponential Sinc collocation and Sinc-Galerkin methods (DESCM and DESGM). The resolution of the stationary Schrödinger equation using the DESCM and DESGM methods is also discussed in two- and three- dimensional spaces. The matrices arising from the discretization of the equation using the DESinc methods are block centrosymmetric. This property is introduced as a two-dimensional extension of the well-known centrosymmetric property. The second part of this study is aimed at resolving large sparse linear systems with multiple right-hand sides using the block and global nonsymmetric Lanczos methods (Bl-NLM and Gl-NLM) which are introduced in new versions. A new expression of the solution is presented basing on some Schur complements identities and consequently some recursive formulas are occurred to give rise to new algorithms for implementing the block and global Lanczos methods. The combination of the global nonsymmetric Lanczos process and the Schur complements is also applied for the approximation of the transfer function.