This paper describes a new block method for solving multi-input Sylvester-observer equations that arise in the construction of the well-known Luenberger observer. The proposed method is based on the block Arnoldi process and generalizes to the multi-input case, the method proposed by Datta and Saad for the single input Sylvester-observer equation. We give new algebraic properties and show how to construct the Luenberger observer by solving a special large-scale Sylvester equation for which two unknown matrices are to be computed. The numerical tests show that the proposed approach is effective and can be used for large-scale Luenberger observer problems.
In this paper we study an elliptic equation involving the p(x)-Laplacien operateur, for that equation we prove the existence of a non trivial weak solution. The proof relies on simple variational arguments based on the Mountain-Pass theorem.
This paper is mainly focused on the solution of Sylvester matrix differential equations with time independent coefficients. We propose a new approach based on the construction of a particular constant solution which allows to construct an approximate solution of the differential equation from that of the corresponding algebraic equation. Moreover, when the matrix coefficients of the differential equation are large, we combine the constant solution approach with Krylov subspace methods for obtaining an approximate solution of the Sylvester algebraic equation, and thus form an approximate solution of the large-scale Sylvester matrix differential equation. We establish some theoretical results including error estimates and convergence as well as relations between the residuals of the differential and its corresponding algebraic Sylvester matrix equation. We also give explicit benchmark formulas for the solution of the differential equation.To illustrate the efficiency of the proposed approach, we perform numerous numerical tests and make various comparisons with other methods for solving Sylvester matrix differential equations.
In the present paper, we are concerned by weighted Arnoldi like methods for solving large and sparse linear systems that have different right-hand sides but have the same coefficient matrix. We first give detailed descriptions of the weighted Gram-Schmidt process and of a Ruhe variant of the weighted block Arnoldi algorithm. We also establish some theoretical results that links the iterates of the weighted block Arnoldi process to those of the non weighted one. Then, to accelerate the convergence of the classical restarted block and seed GMRES methods, we introduce the weighted restarted block and seed GMRES methods. Numerical experiments are reported at the end of this work in order to compare the performance and show the effectiveness of the proposed methods.
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