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Abstract: This paper improves on generalized properties of a family of iterative methods to compute the approximate inverses of square matrices originally proposed in [1]. And while the methods of [1] can be used to compute the inner inverses of any matrix, it has not been proved that these sequences converge (in norm) to a fixed inner inverse of the matrix. In this paper, it is proved that the sequences indeed are convergent to a fixed inner inverse of the matrix which is the Moore–Penrose inverse of the matrix. The convergence proof of these sequences is given by fundamental matrix calculus, and numerical experiments show that the third-order iterations are as good as the second-order iterations. [Copyright &y& Elsevier]

In this paper we study adaptive L ( k ) QN methods, involving special matrix algebras of low complexity, to solve general (non-structured) unconstrained minimization problems. These methods, which generalize the classical BFGS method, are based on an iterative formula which exploits, at each step, an ad hoc chosen matrix algebra L ( k ) . A global convergence result is obtained under suitable assumptions on f . [ABSTRACT FROM AUTHOR]

In this paper we present a new weighted, multivariable operator mean of positive definite operators over an arbitrary Hilbert space which provides us the first generally applicable extension of the classical Kubo–Ando theory of 2-variable operator means. The construction is a weighted extension of the Bini–Meini–Poloni symmetrization process originally given for the matrix geometric mean. Here to be able to consider such an iterative procedure, we need a weighted version of every Kubo–Ando mean in two variables. Therefore we also give a new construction for two arbitrary positive operators on a possibly infinite dimensional Hilbert space that provides weighted counterparts to every (not-necessarily symmetric) Kubo–Ando mean and also agrees with the most well known weighted operator means. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present an iterative algorithm for computing the largest singular value of a nonnegative rectangular tensor. We establish the convergence of this algorithm for any irreducible nonnegative rectangular tensor. [Copyright &y& Elsevier]

Abstract: It is known that the Schur complements of doubly diagonally dominant matrices are doubly diagonally dominant. In this paper, we obtain an estimate for the doubly diagonally dominant degree on the Schur complement of strictly doubly diagonally dominant matrices. Then, as an application we obtain that the eigenvalues of the Schur complements are located in the Brauer Ovals of Cassini of the original matrices under certain conditions. As another application, we obtain an upper bound for the infinity norm on the inverse on the Schur complement of strictly doubly diagonally dominant matrices. Further, based on the derived results, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and can compute out the results faster. [Copyright &y& Elsevier]

Abstract: n this paper, we present an inexact inverse subspace iteration method for computing a few eigenpairs of the generalized eigenvalue problem . We first formulate a version of inexact inverse subspace iteration in which the approximation from one step is used as an initial approximation for the next step. We then analyze the convergence property, which relates the accuracy in the inner iteration to the convergence rate of the outer iteration. In particular, the linear convergence property of the inverse subspace iteration is preserved. Numerical examples are given to demonstrate the theoretical results. [Copyright &y& Elsevier]

Abstract: Given an real matrix A with nonnegative off-diagonal entries, the solution to , , is . The problem of identifying the initial points for which becomes and remains entrywise nonnegative is considered. It is known that such are exactly those vectors for which the iterates become and remain nonnegative, where h is a positive, not necessarily small parameter that depends on the diagonal entries of A. In this paper, this characterization of initial points is extended to a numerical test when A is irreducible: if becomes and remains positive, then so does ; if fails to become and remain positive, then either becomes and remains negative or it always has a negative and a positive entry. Due to round-off errors, the latter case manifests itself numerically by converging with a relatively small convergence ratio to a positive or a negative vector. An algorithm implementing this test is provided, along with its numerical analysis and examples. The reducible case is also discussed and a similar test is described. [Copyright &y& Elsevier]

Abstract: In this paper, we consider an algebraic multiplicative Schwarz iteration scheme for solving the linear complementarity problem that involves an -matrix. We show that the sequence generated by the multiplicative Schwarz iteration scheme converges to the unique solution of the problem without any restriction on the initial point. For different overlapping sizes, the convergence rate of the proposed method is analyzed in an algebraic setting. Moreover, we establish monotone convergence of the proposed method under appropriate conditions. Numerical results show that efficiency can be achieved by the multiplicative Schwarz iteration scheme. [Copyright &y& Elsevier]

Abstract: Preconditioning techniques for linear systems are widely used in order to speed up the convergence of iterative methods. If the linear system is generated by the discretization of an ill-posed problem, preconditioning may lead to wrong results, since components related to noise on input data are amplified. Using basic concepts from the theory of inverse problems, we identify a class of preconditioners which acts as a regularizing tool. In this paper we study relationships between this class and previously known circulant preconditioners for ill-conditioned Hermitian Toeplitz systems. In particular, we deal with the low-pass filtered optimal preconditioners and with a recent family of superoptimal preconditioners. We go on to describe a set of preconditioners endowed with particular regularization properties, whose effectiveness is supported by several numerical tests. [Copyright &y& Elsevier]