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Some iterativemethods are introduced and demonstrated for finding the matrix sign function. It is analytically shown that the new schemes are asymptotically stable. Convergence analysis along with the error bounds of the main proposed method is established. Different numerical experiments are employed to compare the behavior of the new schemes with the existing matrix iterations of the same type. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new matrix approach for solving second order linear partial differential equations (PDEs) under given initial conditions has been proposed. The basic idea includes integrating from the considered PDEs and transforming them to the associated integro-differential equations with partial derivatives. Therefore, Bernoulli operational matrices of differentiation and integration together with the completeness of Bernoulli polynomials can be used for transforming integro-differential equations to the corresponding algebraic equations. A rigorous error analysis in the infinity norm is given provided that the known functions and the exact solution are sufficiently smooth and bounded. A numerical example is included to demonstrate the validity and the applicability of the technique. The results confirm the theoretical prediction. [Copyright &y& Elsevier]

Abstract: This paper presents a direct solution technique for solving the generalized pantograph equation with variable coefficients subject to initial conditions, using a collocation method based on Bernoulli operational matrix of derivatives. Only small dimension of Bernoulli operational matrix is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for generalized pantograph equations. [Copyright &y& Elsevier]

A new and efficient numerical approach is developed for solving nonlinear Lane-Emden type equations via Bernoulli operational matrix of differentiation. Thefundamental structure of the presented method is based on the Taumethod togetherwith the Bernoulli polynomial approximations in which a new operational matrix is introduced. After implementation of our scheme, the main problem would be transformed into a system of algebraic equations such that its solutions are the unknown Bernoulli coefficients. Also, under severalmild conditions the error analysis of the proposed method is provided. Several examples are included to illustrate the efficiency and accuracy of the proposed technique and also the results are compared with the differentmethods. All calculations are done in Maple 13. [ABSTRACT FROM AUTHOR]

Abstract: This article develops an efficient direct solver for solving numerically the high-order linear Fredholm integro-differential equations (FIDEs) with piecewise intervals under initial-boundary conditions. A Bernoulli matrix approach is implemented for solving linear and nonlinear FIDEs with piecewise intervals under initial boundary conditions. The main characteristic behind this approach is that it reduces such problem to those of solving a system of algebraic equations. A small number of Bernoulli polynomials is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving FIDEs with piecewise intervals. [Copyright &y& Elsevier]

In this paper we propose a new projection method to solve both large-scale continuous-time matrix Riccati equations and differential matrix Riccati equations. The new approach projects the problem onto an extended block Krylov subspace and gets a low-dimensional equation. We use the block Golub–Kahan procedure to construct the orthonormal bases for the extended Krylov subspaces. For matrix Riccati equations, the reduced problem is then solved by means of a direct Riccati scheme such as the Schur method. When we solve differential matrix Riccati equations, the reduced problem is solved by the Backward Differentiation Formula (BDF) method and the obtained solution is used to build the low rank approximate solution of the original problem. Finally, we give some theoretical results and present numerical experiments. [ABSTRACT FROM AUTHOR]