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We present an iterative method based on the Hermitian and skew-Hermitian splitting (HSS) for solving the continuous Sylvester equation. By using the HSS of the coefficient matricesAandB, we establish a method which is practically inner/outer iterations, by employing a conjugate gradient on the normal equations (CGNR)-like method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent splitting of the coefficient matrices. Via this method, a Sylvester equation with coefficient matricesand(which are the skew-Hermitian part ofAandB, respectively) is solved iteratively by a CGNR-like method. Convergence conditions of this method are studied and numerical examples show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods. [ABSTRACT FROM PUBLISHER]

We present a shifted nested iterative method for solving systems of linear equations with a coefficient matrix that contains a dominant skew-Hermitian part. This new scheme is practically the inner/outer iterations, which employs the CGNR method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent splitting of the coefficient matrix. Convergence properties of the new scheme are studied in depth and possible choices of the shift parameter are discussed. Moreover, an adapted version of the method is used for ill-posed problems and image restoration. At the last, numerical examples are used to further examine the effectiveness and robustness of the new method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an approximate approach based on Bernoulli operational matrices has been presented to obtain the numerical solution of first-order matrix hyperbolic partial differential equations under the given initial conditions. After using the operational matrices, we use the completeness of Bernoulli basis in which the main problem reduces to a system of algebraic equations. The solutions of this algebraic system are the coefficients of the truncated double Bernoulli series which are defined in the interval [0, 1]. Also, convergence analysis associated to the presented idea is provided under several mild conditions. Several numerical examples are considered to show the efficiency of the technique. [Copyright &y& Elsevier]

Abstract: In this paper, we present a nested splitting conjugate gradient (NSCG) iteration method for solving a class of matrix equations with nonsymmetric coefficient matrices. This method is actually inner/outer iterations, which employs a CG-like method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent and symmetric positive definite splitting of the coefficient matrices. Convergence conditions of this method are studied in depth and numerical experiments show the efficiency of this method. Moreover, we show that the use of the quasi-Hermitian splitting as a preconditioner can induce an accurate, robust and effective preconditioned Krylov subspace method. [Copyright &y& Elsevier]

This paper contributes a new matrix method for the solution of high-order linear complex differential equations with variable coefficients in rectangular domains under the considered initial conditions. On the basis of the presented approach, thematrix forms of the Bernoulli polynomials and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown Bernoulli coefficients are determined and thus the approximate solutions are obtained. Also, an error analysis based on the use of the Bernoulli polynomials is provided under several mild conditions. To illustrate the efficiency of our method, some numerical examples are given. [ABSTRACT FROM AUTHOR]

Abstract: The power method is an algorithm for computing the largest eigenvalue of matrix A in absolute value. To find the other eigenvalues one can apply the power method to the matrix for some shift σ. This scheme is called the inverse iteration method. Both of these two methods produce a convergence sequence and the limit is approximated by one of the iterates. In the chosen iterate, it may be difficult to estimate the global error, consisting of the truncation error and the round-off error. In this paper, by using the CESTAC method and the CADNA library, we propose a method for computing the optimal iterate, the iterate for which the global error is minimal. In the proposed method the accuracy of the computed eigenvalue may also be estimated. Some numerical examples are given to show the efficiency of the method. [Copyright &y& Elsevier]

Abstract: This paper is concerned with approaches to compute a factored sparse approximate inverse for block tridiagonal and block pentadiagonal matrices. Recurrence formulas are developed for computing sparse approximate inverse factors of these matrices using bordering technique. Resulting factored sparse approximate inverse is used as a preconditioner for the conjugate gradient method (PCG). As an application these formulas are simplified for computing the preconditioner for solving Lyapanuv matrix equations by PCG method. Numerical experiments on linear system, arising from discretization of partial differential equations are presented. [Copyright &y& Elsevier]

Abstract: In this paper, we propose two new algorithms based on modified global Arnoldi algorithm for solving large Sylvester matrix equations AX + XB = C where , , X and . These algorithms are based on the global FOM and GMRES algorithms and we call them by Global FOM-Sylvester-Like (GFSL) and Global GMRES-Sylvester-Like (GGSL) algorithms, respectively. Some theoretical results and numerical examples are also given. [Copyright &y& Elsevier]

Abstract: This paper presents a new method for obtaining a matrix M which is an approximate inverse preconditioner for a given matrix A, where the eigenvalues of A all either have negative real parts or all have positive real parts. This method is based on the approximate solution of the special Sylvester equation AX + XA =2I. We use a Krylov subspace method for obtaining an approximate solution of this Sylvester matrix equation which is based on the Arnoldi algorithm and on an integral formula. The computation of the preconditioner can be carried out in parallel and its implementation requires only the solution of very simple and small Sylvester equations. The sparsity of the preconditioner is preserved by using a proper dropping strategy. Some numerical experiments on test matrices from Harwell–Boing collection for comparing the numerical performance of the new method with an available well-known algorithm are presented. [Copyright &y& Elsevier]