Rui Ralha

Blocked schur algorithms for computing the matrix square root

The Schur method for computing a matrix square root reduces the matrix to the Schur triangular form and then computes a square root of the triangular matrix. We show that by using either standard blocking or recursive blocking the computation of the square root of the triangular matrix can be made rich in matrix multiplication. Numerical experiments making appropriate use of level 3 BLAS show significant speedups over the point algorithm, both in the square root phase and in the algorithm as a whole. In parallel implementations, recursive blocking is found to provide better performance than standard blocking when the parallelism comes only from threaded BLAS, but the reverse is true when parallelism is explicitly expressed using OpenMP. The excellent numerical stability of the point algorithm is shown to be preserved by blocking. These results are extended to the real Schur method. Blocking is also shown to be effective for multiplying triangular matrices.

One-sided reduction to bidiagonal form

Abstract We present an idea for reducing a rectangular matrix A to bidiagonal form which is based on the implicit reduction of the symmetric positive semidefinite matrix A t A to tridiagonal form. In other papers we have shown that a method based upon this idea may become a serious competitor (in terms of speed) for computing the singular values of large matrices and also that it is well suited for parallel processing. However, there are still some open questions related to the numerical stability of the method and these will be addressed in this paper. The algorithm, as it is at present, is not backward stable. Nevertheless, we give examples of ill-conditioned matrices for which we have been able to produce a bidiagonal form whose singular values are much more accurate than the ones computed with the standard bidiagonalization technique.

Computing the square roots of matrices with central symmetry

For computing square roots of a nonsingular matrix A, which are functions of A, two well known fast and stable algorithms, which are based on the Schur decomposition of A, were proposed by Bjork and Hammarling [A. Bjork, S. Hammarling, A Schur method for the square root of a matrix, Linear Alg. Appl. 52/53 (1983) 127–140], for square roots of general complex matrices, and by Higham [N.J. Higham, Computing real square roots of a real matrix, Linear Alg. Appl. 88/89 (1987) 405–430], for real square roots of real matrices. In this paper we further consider (the computation of) the square roots of matrices with central symmetry. We first investigate the structure of the square roots of these matrices and then develop several algorithms for computing the square roots. We show that our algorithms ensure significant savings in computational costs as compared to the use of standard algorithms for arbitrary matrices.

On computing complex square roots of real matrices

Abstract We present an idea for computing complex square roots of matrices using only real arithmetic.

On inverse eigenvalue problems for block Toeplitz matrices with Toeplitz blocks

We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum. We have implemented our ideas in a Matlab code. Numerical results produced with this code are included.

Parallel Implementation of Non Recurrent Neural Networks

The computational models introduced by neural network theory exhibit a natural parallelism in the sense that the network can be decomposed in several cellular automata working simultaneously. Following this idea, we present in this paper a parallel implementation of the learning process for two of the main non recurrent neural networks: the Multilayer Perceptron (MLP) and the Self-Organising Map of Kohonen (SOM).

Mixed Precision Bisection

We discuss the implementation of the bisection algorithm for the computation of the eigenvalues of symmetric tridiagonal matrices in a context of mixed precision arithmetic. This approach is motivated by the emergence of processors which carry out floating-point operations much faster in single precision than they do in double precision. Perturbation theory results are used to decide when to switch from single to double precision. Numerical examples are presented.

Structure-preserving Schur methods for computing square roots of real skew-Hamiltonian matrices

The contribution in this paper is two-folded. First, a complete characterization is given of the square roots of a real nonsingular skew-Hamiltonian matrix W. Using the known fact that every real skew-Hamiltonian matrix has infinitely many real Hamiltonian square roots, such square roots are described. Second, a structure-exploiting method is proposed for computing square roots of W, skew-Hamiltonian and Hamiltonian square roots. Compared to the standard real Schur method, which ignores the structure, this method requires significantly less arithmetic.

The geometric mean algorithm

Bisection (of a real interval) is a well known algorithm to compute eigenvalues of symmetric matrices. Given an initial interval [a;b], convergence to an eigenvalue which has size much smaller than a or b may be made considerably faster if one replaces the usual arithmetic mean (of the end points of the current interval) with the geometric mean. Exploring this idea, we have implemented geometric bisection in a Matlab code. We illustrate the efiectiveness of our algorithm in the context of the computation of the eigenvalues of a symmetric tridiagonal matrix which has a very large condition number.

An Efficient Parallel Algorithm for the Symmetric Tridiagonal Eigenvalue Problem

An efficient parallel algorithm, which we dubbed farmzeroinNR, for the eigenvalue problem of a symmetric tridiagonal matrix has been implemented in a distributed memory multiprocessor with 112 nodes [1]. The basis of our parallel implementation is an improved version of the zeroinNR method [2]. It is consistently faster than simple bisection and produces more accurate eigenvalues than the QR method. As it happens with bisection, zeroinNR exhibits great flexibility and allows the computation of a subset of the spectrum with some prescribed accuracy. Results were carried out with matrices of different types and sizes up to 104 and show that our algorithm is efficient and scalable.