We investigate the numerical solution of stable Sylvester equations via iterative schemes proposed for computing the sign function of a matrix. In particular, we discuss how the rational iterations for the matrix sign function can efficiently be adapted to the special structure implied by the Sylvester equation. For Sylvester equations with factored constant term as those arising in model reduction or image restoration, we derive an algorithm that computes the solution in factored form directly. We also suggest convergence criteria for the resulting iterations and compare the accuracy and performance of the resulting methods with existing Sylvester solvers. The algorithms proposed here are easy to parallelize. We report on the parallelization of those algorithms and demonstrate their high efficiency and scalability using experimental results obtained on a cluster of Intel Pentium Xeon processors.

Orientadores: Margarida Pinheiro Mello, Maria Aparecida Diniz Ehrhardt / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-22T15:49:27Z (GMT). No. of bitstreams: 1 Ferreira_DaianeGoncalves_M.pdf: 1441315 bytes, checksum: 1196d8b21c6254dbdd0e0d68266fa707 (MD5) Previous issue date: 2013 / Resumo: Métodos de Otimização de terceira ordem, embora de longa tradição, eram considerados, até passado recente, impraticáveis, devido à taxa com que o esforço computacional cresce em função da dimensão do problema. Avanços no desenvolvimento de estruturas de dados, rotinas que trabalham com estas estruturas e a exploração da esparsidade de grande parte dos problemas encontrados na prática já permitem implementações destes métodos que podem torná-los competitivos com métodos de segunda ordem. O objeto desta dissertação é a apresentação do método de Halley, um método de terceira ordem, sua implementação em MATLAB e a realização de testes computacionais, visando uma comparação empírica de sua eficiência frente ao método de Newton, o método de segunda ordem mais empregado na atualidade / Abstract: Higher order optimization methods, though of long-standing tradition, until recently have been deemed impractical, due to the rate of increase of the computational effort as a function of the size of the problem. Advances in the development of data structures, routines that work with these structures and the use of the sparsity of a vast range of practical problems have led to implementations of these methods that are competitive with second order methods. The object of this dissertation is the study of Halley's method, a thirdorder method, the development of a MATLAB implementation thereof and its testing, aiming at an empirical comparison of its efficiency against that of Newton's method, the second-order method most widely used today / Mestrado / Matematica Aplicada / Mestra em Matemática Aplicada

We investigate the numerical solution of stable Sylvester equations via iterative schemes proposed for computing the sign function of a matrix. In particular, we discuss how the rational iterations for the matrix sign function can efficiently be adapted to the special structure implied by the Sylvester equation. For Sylvester equations with factored constant term as those arising in model reduction or image restoration, we derive an algorithm that computes the solution in factored form directly. We also suggest convergence criteria for the resulting iterations and compare the accuracy and performance of the resulting methods with existing Sylvester solvers. The algorithms proposed here are easy to parallelize. We report on the parallelization of those algorithms and demonstrate their high efficiency and scalability using experimental results obtained on a cluster of Intel Pentium Xeon processors.