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Abstract: In this paper, we apply a complete pivoting strategy to compute the left-looking version of AINV preconditioner for linear systems. As the preprocessing, the MultiLevel Nested Dissection reordering has also been applied. We have used this preconditioner as the right preconditioner for several linear systems where the coefficient matrices have been downloaded from the University of Florida Sparse Matrix Collection. Numerical experiments presented in this paper indicate the effectiveness of such a complete pivoting on the quality of left-looking version of AINV preconditioner. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the bounds of the Perron root of a nonnegative irreducible matrix . Two new methods utilizing the relationship between the Perron root of a nonnegative irreducible matrix and its generalized Perron complements are presented. The former method is efficient because it gives the bounds for only by calculating the row sums of the generalized Perron complement or even the row sums of submatrices and . And the latter gives the closest bounds (just in this paper) of . The results obtained by these methods largely improve the classical bounds. Numerical examples are given to illustrate the procedure and compare it with others, which shows that these methods are effective. [Copyright &y& Elsevier]

Abstract: This paper discusses the spectral properties of the nonsymmetric saddle point matrices of the form with A symmetric positive definite, B full rank, and C symmetric positive semidefinite. A new sufficient condition is obtained so that is diagonalizable with all its eigenvalues real and positive. This condition is weaker than that stated in the recent paper [J. Liesen, A note on the eigenvalues of saddle point matrices, Technical Report 10-2006, Institute of Mathematics, TU Berlin, 2006]. [Copyright &y& Elsevier]

Abstract: This paper presents an alternative and efficient method for solving a class of constraint parametric optimization problems using particle swarm optimization algorithm (PSO). In this paper, for the first time PSO is used for solving convex parametric programming, but PSO must be adaptive for doing it. So, for obtaining particles velocities, adaptation weight and velocity boundaries in the updating of velocities are calculated in the recursive form. Computational complexity of the PSO algorithm is decreased based on uniformity of population of particles, the uniformity of which is obtained using eigenvalue spread of covariance of population. Two simple examples are provided for showing the efficiency of the proposed method. For solving these examples, eigenvalue spread criteria applied in the PSO algorithm decrease 78% of computations. [Copyright &y& Elsevier]

Abstract: Recently, Shen et al. has published a paper concerning the properties and computations of matrix pseudospectra. However, some of the main results seem to be incomplete. As a complement of the paper, in this note we first make several examples to show the incompleteness of that results, then we give the modified results through giving more appropriate definitions replacing the one given in [Y. Shen, J. Zhao, H. Fan, Properties and computations of matrix pseudospectra, Applied Mathematics and Computation 161 (2005) 385–393]. An interesting inclusion theorem for pseudospectra is given, which provides preferable bounds for standard pseudospectra. Some other properties are also explored. [Copyright &y& Elsevier]

Some iterative methods for computing the polar decomposition are introduced. They can be used to compute the polar decomposition of rank-deficient matrix which includes both square and rectangular matrices.A very important application of the polar decomposition is about matrix sign function. Through the methods of this paper to compute the matrix sign function is convenient. Some examples are provided in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, another proof of Pell identities is presented by using the determinant of tridiagonal matrix. It is calculated via the Laplace expansion. [Copyright &y& Elsevier]

Abstract: According to the hierarchical identification principle, a hierarchical gradient based iterative estimation algorithm is derived for multivariable output error moving average systems (i.e., multivariable OEMA-like models) which is different from multivariable CARMA-like models. As there exist unmeasurable noise-free outputs and unknown noise terms in the information vector/matrix of the corresponding identification model, this paper is, by means of the auxiliary model identification idea, to replace the unmeasurable variables in the information vector/matrix with the estimated residuals and the outputs of the auxiliary model. A numerical example is provided. [ABSTRACT FROM AUTHOR]

Abstract: It is pointed out by Jiang and Wang (2005) , that the conversion formula from Bernstein into DP bases is incorrect for all even degrees and the conversion formula from DP into Bernstein bases is incorrect for every degree. Thus, in this paper we give some notes, corrections and new proofs for the relationship between these two NTP bases. [Copyright &y& Elsevier]

Abstract: In this paper, an inversion algorithm for a banded matrix is presented. The twisted decompositions of a banded matrix are given first; then the inverse of the matrix is obtained, one column at time. The method is about two times faster than the standard method based on the decomposition, as is shown with the analysis of computing complexity and the numerical experiments. [Copyright &y& Elsevier]

Abstract: In this paper, we give some new centrosymmetric splittings of a centrosymmetric matrix A and obtain corresponding iterative methods for a linear system Ax = b. These iterative schemes are particularly based on the centrosymmetric property of A and aimed at reduction of the cost of computation and storage. [Copyright &y& Elsevier]

Abstract: The symmetric solutions of the system of matrix equations A 1 XB 1 = C 1, A 2 XB 2 = C 2 are too difficult to be obtained by applying matrices decomposition. In this paper, an iterative method is applied to solve this problem. With it, the solvability of this system of matrix equations can be determined automatically, when this system of matrix equations is consistent, its solution can be obtained within finite iterative steps, and its least-norm solution can be obtained by choosing a special kind of initial iterative matrix, furthermore, its optimal approximation solution to a given matrix can be derived by finding the least-norm symmetric solution of a new system of matrix equations . Finally, numerical examples are given. [Copyright &y& Elsevier]

Abstract: The availability of explicit solutions, i.e. analytical relationships between the structural response and the design variables, allows a more direct and plain treatment of several structural problems. This paper is devoted to derive approximate explicit solutions in the framework of linear static analysis of finite element modeled structures with a given layout (fixed node positions). The proposed procedure is based on a factorization of the element stiffness matrix following the unimodal components concept, which allows a non-conventional assembly of the global stiffness matrix. The exact inversion of that matrix is a trivial task for the case of statically determinate structures, structures with few redundancies or few design variables. An approximate inverse of the stiffness matrix is herein derived for more general structural problems by resorting to the Sherman–Morrison–Woodbury formula. [Copyright &y& Elsevier]

Abstract: We propose a “fast” algorithm for the construction of a data-sparse inverse of a generalToeplitz matrix. The computational cost for inverting an N × N Toeplitz matrix equals the cost of four length-N FFTs plus an O(N)-term. This cost should be compared to the O(N log2 N) cost of previously published methods. Moreover, while those earlier methods are based on algebraic considerations, the procedure of this paper is analysis-based; as a result, its stability does not depend on the symmetry and positive-definiteness of the matrix being inverted. The performance of the scheme is illustrated with numerical examples. [Copyright &y& Elsevier]

Abstract: A central problem in the Jacobi-Davidson method is to expand a projection subspace by solving a certain correction equation. It has been commonly accepted that the correction equation always has a solution. However, it is proved in this paper that this is not true. Conditions are given to decide when it has a unique solution or many solutions or no solution. A refined Jacobi-Davidson method is proposed to overcome the possible nonconvergence of Ritz vectors by computing certain refined approximation eigenvectors from the subspace. A corresponding correction equation is derived for the refined method. Numerical experiments are conducted and efficiency of the refined method is confirmed. [Copyright &y& Elsevier]

We determine the conditions for uniqueness of the solutions of several completion problems including the positive semi-definite completion, the distance matrix completion, and the contractive matrix completion. The conditions depend on the existence of a positive semi-definite matrix satisfying certain linear constraints. Numerically, such conditions can be checked by existing computer software such as semi-definite programming routines. Some numerical examples are given to illustrate the results, and show that some results in a recent paper of Alfakih are incorrect. Discrepancies in the proof of Alfakih are explained. [Copyright &y& Elsevier]

This paper deals with H-stability of Runge–Kutta methods with variable stepsize for pantograph equations. It is shown that the Runge–Kutta methods with a regular matrix A are H-stable if and only if the modulus of stability function at infinity is less than 1. Furthermore, the same conclusion can be obtained if the Runge–Kutta methods are stiffly accurate. [Copyright &y& Elsevier]