Orientador: Eliana Xavier Linhares de Andrade / Banca: Roberto Andreani / Banca: Cleonice Fátima Bracciali / Resumo: O principal objetivo deste trabalho é apresentar, discutir as qualidades e desempenho e provar a convergência de um método iterativo para a solução numérica do problema de autovalores de uma matriz, que chamamos de Método Assemelhado ao de Francis (MAF). O método em questão distingue-se do QR de Francis pela maneira, mais simples e rápida, de se obter as matrizes ortogonais Qk, k = 1; 2. Apresentamos, também, uma comparação entre o MAF e os algoritmos QR de Francis e LR de Rutishauser. / Abstract: The main purpose of this work is to presente, to discuss the qualities and performance and to prove the convergence of an iterative method for the numerical solution of the eigenvalue problem, that we have called the Método Assemelhado ao de Francis (MAF)þþ. This method di ers from the QR method of Francis by providing a simpler and faster technique of getting the unitary matrices Qk; k = 1; 2; We present, also, a comparison analises between the MAF and the QR of Francis and LR of Rutishauser algorithms. / Mestre

Made available in DSpace on 2014-06-11T19:27:08Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-02-23Bitstream added on 2014-06-13T19:26:10Z : No. of bitstreams: 1 oliveira_de_me_sjrp.pdf: 1040006 bytes, checksum: 88dd8fa849febafe8d0aa9bf32892235 (MD5) / O principal objetivo deste trabalho é apresentar, discutir as qualidades e desempenho e provar a convergência de um método iterativo para a solução numérica do problema de autovalores de uma matriz, que chamamos de Método Assemelhado ao de Francis (MAF). O método em questão distingue-se do QR de Francis pela maneira, mais simples e rápida, de se obter as matrizes ortogonais Qk, k = 1; 2. Apresentamos, também, uma comparação entre o MAF e os algoritmos QR de Francis e LR de Rutishauser. / The main purpose of this work is to presente, to discuss the qualities and performance and to prove the convergence of an iterative method for the numerical solution of the eigenvalue problem, that we have called the Método Assemelhado ao de Francis (MAF)þþ. This method di ers from the QR method of Francis by providing a simpler and faster technique of getting the unitary matrices Qk; k = 1; 2; We present, also, a comparison analises between the MAF and the QR of Francis and LR of Rutishauser algorithms.

Les travaux de recherche de cette thèse concernent l'encadrement de la constante de Markov Bernstein pour la norme L2 associée aux mesures de Jacobi et Gegenbauer généralisée. Ce travail est composé de deux parties : dans la première partie, nous avons développé une généralisation de l'algorithme qd pour les matrices symétriques définies positives à largeur de bande $\ell$ et nous avons construit l'algorithme qd pour les matrices de Jacobi par blocs. Ensuite, nous l'avons généralisé aux cas des matrices par bloc à largeur de bande $\ell$. Ces algorithmes nous permettent de trouver un majorant de la constante. Enfin, nous avons développé le déterminant caractéristique d'une matrice symétrique définie positive pentadiagonale, ce qui nous permet d'obtenir un minorant de la constante en utilisant la méthode de Newton. La deuxième partie est consacrée à l'application de tous les outils développés à l'encadrement de la constante de Markov Bernstein pour la norme L2 associée à la mesure de Gegenbauer généralisée. / The aim of this thesis is to find the lower and upper bounds of the constant whichappears in the Markov Bernstein inequalities in L2 norm associated to the Jacobiand generalized Gegenbauer measures. In this work the qd algorithm is studied forobtaining some properties about the asymptotic behavior of some eigenvalues ofband matrices and block band matrices. These eigenvalues are linked to the MarkovBernstein constant. The application of all the tools developed for obtaining lowerand upper bounds of the Markov Bernstein constant in L2 norm associated to thegeneralized Gegenbauer measure is given.

用QR與LR迭代法求矩陣特徵值與特徵向量之過程中，前人曾提出位移策略以加速其收斂速度，其中最有效的是Wilkinson 移位值。在此我們希望尋求能使收斂更快速的位移值。 我們首先嘗使用一三階子矩陣之特徵值作為一次QR迭代之移位值。在此子矩陣之特徵值中，我們選擇最接近Wilkinson 移位值的特徵值為移位值，期使特徵值之收斂更快。 另一移位策略是用一較快速省功的算則先計算矩陣之特徵值，再以這些計算值作為QR迭代之位移值，來計算較為費功的特徵向量。希望能較快得到所需要的特徵值與特徵向量。 在計算特徵值之算則中，Cholesky迭代法以其計算簡單，執行速度快為我們所選擇。由程式執行結果可知這兩種算則較EISPACK 的算則分別節省了約10% 與30% 的運算量。我們比較這些策略，並將結果列於文中。 / Abstract The QR and LR algorithms are the general methods for computing eigenvalues and eigenvectors of a dense matrix. In this paper, we propose some shift strategies that can increase the efficiency of the QR algorithm by first computing the eigenvalues of the matrix (or its trailing submatrix) in a fast and economical way, and then using them as shifts to find the eigenvalues and their corresponding eigenvectors. When incorporated with QR algorithm, these kinds of shift strategies can save about 10 to 30percent of work in arithmetic operations.