Encontrados 11 documentos, a visualizar página 1 de 2
Blocked schur algorithms for computing the matrix square root
Deadman, Edvin; Higham, Nicholas J.; Ralha, RuiThe Schur method for computing a matrix square root reduces the matrix to Schur triangular form and then computes a square root of the triangular matrix. We show that by using either a 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...
On computing complex square roots of real matrices
Liu Zhongyun; Zhang Yulin; Santos, Jorge M. F.; Ralha, RuiWe present an idea for computing complex square roots of matrices using only real arithmetic.
Structure-preserving schur methods for computing square roots of real skew-hami...
Liu Zhongyun; Zhang Yulin; Ferreira, Carla; Ralha, RuiThe 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 Hamilt...
The geometric mean algorithm
Ralha, RuiBisection (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 ...
Reliable eigenvalues of symmetric tridiagonals
Ralha, RuiFor the eigenvalues of a symmetric tridiagonal matrix T, the most accurate algorithms deliver approximations which are the exact eigenvalues of a matrix whose entries differ from the corresponding entries of T by small relative perturbations. However, for matrices with eigenvalues of different magnitudes, the number of correct digits in the computed approximations for eigenvalues of size smaller than ‖T‖₂ depen...
On inverse eigenvalue problems for block Toeplitz matrices with Toeplitz blocks
Zhang Yulin; Liu Zhongyun; Ferreira, Carla; Ralha, RuiWe 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 ma...
Aventuras numéricas no cálculo do e
Ralha, Rui; Ralha, Elfrida; Gomes, Paula Cristina da CostaFaz-se uma análise dos erros de arredondamento no cálculo de aproximações do número de Neper com a expressão (1+1/n)^n.
Perturbation splitting for more accurate eigenvalues
Ralha, RuiLet $T$ be a symmetric tridiagonal matrix with entries and eigenvalues of different magnitudes. For some $T$, small entrywise relative perturbations induce small errors in the eigenvalues, independently of the size of the entries of the matrix; this is certainly true when the perturbed matrix can be written as $\widetilde{T}=X^{T}TX$ with small $; ; X^{T}X-I; ; $. Even if it is not possible to express in this w...
Computing the square roots of matrices with central symmetry
Liu Zhongyun; Zhang Yulin; Ralha, RuiFor 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 Bj¨ork and Hammarling [3], for square roots of general complex matrices, and by Higham [10], for real square roots of real matrices. In this paper we further consider (the computation of) the square roots of matrices with cen...
Parallel bidiagonalization of a dense matrix
Campos, Carlos; Guerrero, David; Hernandez, Vicente; Ralha, RuiA new stable method for the reduction of rectangular dense matrices to bidiagonal form has been proposed recently. This is a one-sided method since it can be entirely expressed in terms of operations with (full) columns of the matrix under transformation. The algorithm is well suited to parallel computing and, in order to make it even more attractive for distributed memory systems, we introduce a modification w...
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