Computing generalized inverses using LU factorization of matrix product

An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and the Moore-Penrose inverse of a given rational matrix A is established. Classes A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R* and T*(AT*)+, where R and T are rational matrices with appropriate dimensions and corresponding rank. The proposed algorithm is based on these general representations and the Cholesky factorization of symmetric positive matrices. The algorithm is implemented in programming languages MATHEMATICA and DELPHI, and illustrated via examples. Numerical results of the algorithm, corresponding to the Moore-Penrose inverse, are compared with corresponding results obtained by several known methods for computing the Moore-Penrose inverse

Effective partitioning method for computing weighted Moore-Penrose inverse

We introduce a method and an algorithm for computing the weighted Moore-Penrose inverse of multiple-variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices. These methods and algorithms are generalizations of algorithms developed in [M.B. Tasic, P.S. Stanimirovic, M.D. Petkovic, Symbolic computation of weighted Moore-Penrose inverse using partitioning method, Appl. Math. Comput. 189 (2007) 615-640] to multiple-variable rational and polynomial matrices and improvements of these algorithms on sparse matrices. Also, these methods are generalizations of the partitioning method for computing the Moore-Penrose inverse of rational and polynomial matrices introduced in [P.S. Stanimirovic, M.B. Tasic, Partitioning method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004) 137-163; M.D. Petkovic, P.S. Stanimirovic, Symbolic computation of the Moore-Penrose inverse using partitioning method, Internat. J. Comput. Math. 82 (2005) 355-367] to the case of weighted Moore-Penrose inverse. Algorithms are implemented in the symbolic computational package MATHEMATICA

Computation of generalized inverses by using the LDL∗ decomposition

AbstractAn efficient algorithm, based on the LDL∗ factorization, for computing {1,2,3} and {1,2,4} inverses and the Moore–Penrose inverse of a given rational matrix A, is developed. We consider matrix products A∗A and AA∗ and corresponding LDL∗ factorizations in order to compute the generalized inverse of A. By considering the matrix products (R∗A)†R∗ and T∗(AT∗)†, where R and T are arbitrary rational matrices with appropriate dimensions and ranks, we characterize classes A{1,2,3} and A{1,2,4}. Some evaluation times for our algorithm are compared with corresponding times for several known algorithms for computing the Moore–Penrose inverse

Convergence rates for inverse-free rational approximation of matrix functions

This article deduces geometric convergence rates for approximating matrix functions via inverse-free rational Krylov methods. In applications one frequently encounters matrix functions such as the matrix exponential or matrix logarithm; often the matrix under consideration is too large to compute the matrix function directly, and Krylov subspace methods are used to determine a reduced problem. If many evaluations of a matrix function of the form f(A)v with a large matrix A are required, then it may be advantageous to determine a reduced problem using rational Krylov subspaces. These methods may give more accurate approximations of f(A)v with subspaces of smaller dimension than standard Krylov subspace methods. Unfortunately, the system solves required to construct an orthogonal basis for a rational Krylov subspace may create numerical difficulties and/or require excessive computing time. This paper investigates a novel approach to determine an orthogonal basis of an approximation of a rational Krylov subspace of (small) dimension from a standard orthogonal Krylov subspace basis of larger dimension. The approximation error will depend on properties of the matrix A and on the dimension of the original standard Krylov subspace. We show that our inverse-free method for approximating the rational Krylov subspace converges geometrically (for increasing dimension of the standard Krylov subspace) to a rational Krylov subspace. The convergence rate may be used to predict the dimension of the standard Krylov subspace necessary to obtain a certain accuracy in the approximation. Computed examples illustrate the theory developed

A New Algorithm to Approximate Bivariate Matrix Function via Newton-Thiele Type Formula

A new method for computing the approximation of bivariate matrix function is introduced. It uses the construction of bivariate Newton-Thiele type matrix rational interpolants on a rectangular grid. The rational interpolant is of the form motivated by Tan and Fang (2000), which is combined by Newton interpolant and branched continued fractions, with scalar denominator. The matrix quotients are based on the generalized inverse for a matrix which is introduced by C. Gu the author of this paper, and it is effective in continued fraction interpolation. The algorithm and some other important conclusions such as divisibility and characterization are given. In the end, two examples are also given to show the effectiveness of the algorithm. The numerical results of the second example show that the algorithm of this paper is better than the method of Thieletype matrix-valued rational interpolant in Gu (1997)

Algorithms for Modifying Recurrence Relations of Orthogonal Polynomial and Rational Functions when Changing the Discrete Inner Product

Often, polynomials or rational functions, orthogonal for a particular inner product are desired. In practical numerical algorithms these polynomials are not constructed, but instead the associated recurrence relations are computed. Moreover, also typically the inner product is changed to a discrete inner product, which is the finite sum of weighted functions evaluated in specific nodes. For particular applications it is beneficial to have an efficient procedure to update the recurrence relations when adding or removing nodes from the inner product. The construction of the recurrence relations is equivalent to computing a structured matrix (polynomial) or pencil (rational) having prescribed spectral properties. Hence the solution of this problem is often referred to as solving an Inverse Eigenvalue Problem. In Van Buggenhout et al. (2022) we proposed updating techniques to add nodes to the inner product while efficiently updating the recurrences. To complete this study we present in this article manners to efficiently downdate the recurrences when removing nodes from the inner product. The link between removing nodes and the QR algorithm to deflate eigenvalues is exploited to develop efficient algorithms. We will base ourselves on the perfect shift strategy and develop algorithms, both for the polynomial case and the rational function setting. Numerical experiments validate our approach.Comment: 45 pages, 19 figure