Topological insulators and topological crystalline insulators are materials that have a bulk band structure that is gapped, but that also have toplogically protected non-gapped surface states. This implies that the bulk is insulating, but that the material can conduct electricity on some of its surfaces. The robustness of these surface states is a consequence of time-reversal symmetry, possibly in combination with invariance under other symmetries, like that of the crystal itself. In this thesis we review some of the basic theory for such materials. In particular we discuss how topological invariants can be derived for some specific systems. We then move on to do band structure calculations using the tight-binding method, with the aim to see the topologically protected surface states in a topological crystalline insulator. These calculations require the diagonalization of block tridiagonal matrices. We finish the thesis by studying the properties of such matrices in more detail and derive some results regarding the distribution and convergence of their eigenvalues.

我們將原本求只有實根的多項式問題轉換為利用QR方法求一個友矩陣(companion matrix)或是對稱三對角(symmetric tridiagonal matrix)的特徵值問題，在數值測試中顯示出利用傳統演算法去求多項式的根會比求轉換過後矩陣特徵值的方法較沒效率。 / Given a polynomial pn(x) of degree n with real roots, we transform the problem of finding all roots of pn (x) into a problem of finding the eigenvalues of a companion matrix or of a symmetric tridiagonal matrix, which can be done with the QR algorithm. Numerical testing shows that finding the roots of a polynomial by standard algorithms is less efficient than by computing the eigenvalues of a related matrix.

Submitted by Emanuel Varela Cardoso (emanuel.varela@ufcg.edu.br) on 2018-05-23T00:29:07Z No. of bitstreams: 1 ALUIZIO FREIRE DA SILVA JUNIOR – DISSERTAÇÃO (PPGEP) 2015.pdf: 8240920 bytes, checksum: 03c678880f1d1648e407849134c89aa4 (MD5) / Made available in DSpace on 2018-05-23T00:29:07Z (GMT). No. of bitstreams: 1 ALUIZIO FREIRE DA SILVA JUNIOR – DISSERTAÇÃO (PPGEP) 2015.pdf: 8240920 bytes, checksum: 03c678880f1d1648e407849134c89aa4 (MD5) Previous issue date: 2015-07-31 / O presente trabalho tem como objetivo desenvolver ferramentas numéricas e computacionais tendo vista descrever processos difusivos em sólidos com formas cilíndricas. Para isto a equação de difusão, considerando os casos de um cilindro infinito e de um cilindro finito, foi discretizada via método dos volumes finitos com uma formulação totalmente implícita, admitindo uma condição de contorno do terceiro tipo. Para as soluções numéricas obtidas pelas discretizações, foram desenvolvidos softwares na plataforma Windows, utilizando a linguagem de programação Fortran. As soluções desenvolvidas foram validadas pela comparação com resultados fornecidos por soluções analíticas. Os testes realizados indicaram coerência nos resultados fornecidos pelas soluções numéricas. Além disso, a fim de obter os parâmetros físicos dos processos de transferência de massa, um otimizador foi desenvolvido e acoplado às soluções numéricas. Foram realizados testes com o otimizador desenvolvido tendo em vista analisar a capacidade deste em encontrar os valores ótimos de um processo de transferência de massa. Os testes indicaram que o otimizador tem capacidade para obter os parâmetros necessários ao estudo deste trabalho, conseguindo chegar a região que contém os valores ótimos para os parâmetros, mesmo quando considerados valores iniciais distantes destes valores ótimos. A partir dos dados obtidos em experimentos de desidratação osmótica de banana (cortada em pedaços de 10 mm) realizados em combinações de 40 e 70°C de temperatura e 40 e 60 °Brix de concentração, foram realizadas otimizações a fim de obter expressões para descrição das difusividades efetivas de água e sacarose e valores para o coeficiente de transferência convectiva de massa. Os resultados obtidos para as difusividades de água e sacarose estão de acordo com a literatura. Os valores fornecidos pelo otimizador para o coeficiente de transferência convectiva de massa indicaram uma condição de contorno do primeiro tipo. Foram realizadas otimizações a partir dos dados da secagem complementar das amostras osmoticamente desidratadas, e os resultados obtidos para a difusividade de água foram compatíveis com os encontrados na literatura. Foi concluído pelas otimizações que as altas concentrações da desidratação osmótica influenciaram a condição de contorno da secagem complementar. / This study aims to develop numerical and computational tools to describe diffusion processes in solids with cylindrical shapes. For this the diffusion equation, considering the case of an infinite cylinder and a finite cylinder, was discretized via finite volume method with a fully implicit formulation, assuming a boundary condition of the third kind. For the numerical solutions obtained by discretization, software has been developed on the Windows platform using the Fortran programming language. The solutions developed were validated by comparison with results provided by analytical solutions. The tests showed consistency in the results provided by the numerical solutions. Furthermore, in order to obtain the physical parameters of the mass transfer process, was developed an optimizer which was coupled with numerical solutions. Tests were performed with the optimizer developed in order to analyze the capacity of finding the optimal values of a mass transfer process. The tests indicated that the optimizer is able to obtain the parameters necessary for the study of this work, reaching the region containing the optimal values for the parameters, even when initial values were considered far from the optimal values. From the data obtained in banana (cut into pieces of 10 mm) osmotic dehydration experiments performed by combining temperature of 40 and 70 ° C and concentration of 40 and 60 ° Brix, optimizations were carried out to obtain expressions for describing the effective diffusivity of water and sucrose and values for convective mass transfer coefficient. The results obtained for the diffusivities of water and sucrose are in agreement with the literature. The values supplied by the optimizer for the mass convective transfer coefficient indicated a boundary condition of the first kind. Optimizations were carried out from the complementary drying data of osmotically dehydrated samples, and the results obtained for the diffusivity of water were consistent with those found in the literature. It was concluded by the optimizations that high concentrations of osmotic dehydration influenced the boundary condition of the complementary drying.

[pt] A identificação algorítmica de matrizes de Jacobi a partir de variáveis espectrais é um tema tradicional de análise numérica. Uma nova representação, as coordenadas bidiagonais, naturalmente exigiu que fosse considerado um novo algoritmo. O algoritmo é apresentado e confrontado com as técnicas habituais. / [en] Algorithms relating Jacobi matrices and spectral variables are standard objects in numerical analysis. The recent discovery of bidiagonal coordinates led to the search of an appropriate algorithm for these new variables. The new algorithm is presented and compared with previous techniques.

This project deals with computation of eigenvalues and eigenvectors of Hermitian positive-semidefinite complex square matrix of order 4. The target is an implementation of computation in language VHDL to field-programmable gate array of type Xilinx Zynq-7000. This master project deals with algorithms used for computation of eigenvalues and eigenvectors of positive-semidefinite symmetric real square and positive-semidefinite complex Hermitian matrix and the analysis of algorithms by AnalyzeAlgorithm program assembled for this purpose. The closing part of this project describes implementation of the computation into field-programmable gate array with use of IP core Xilinx® Floating-Point \linebreak Operator and SVAOptimalizer, SVAInterpreter and SVAToDSPCompiler programs.

Nous nous intéressons dans cette thèse, à l'étude de certaines méthodes numériques de type krylov dans le cas symplectique, en utilisant la technique de blocs. Ces méthodes, contrairement aux méthodes classiques, permettent à la matrice réduite de conserver la structure Hamiltonienne ou anti-Hamiltonienne ou encore symplectique d'une matrice donnée. Parmi ces méthodes, nous nous sommes intéressés à la méthodes d'Arnoldi symplectique par bloc que nous appelons aussi bloc J-Arnoldi. Notre but essentiel est d’étudier cette méthode de façon théorique et numérique, sur la nouvelle structure du K-module libre ℝ²nx²s avec K = ℝ²sx²s où s ≪ n désigne la taille des blocs utilisés. Un deuxième objectif est de chercher une approximation de l'epérateur exp(A)V, nous étudions en particulier le cas où A est une matrice réelle Hamiltonnienne et anti-symétrique de taille 2n x 2n et V est une matrice rectangulaire ortho-symplectique de taille 2n x 2s sur le sous-espace de Krylov par blocs Km(A,V) = blockspan {V,AV,...,Am-1V}, en conservant la structure de la matrice V. Cette approximation permet de résoudre plusieurs problèmes issus des équations différentielles dépendants d'un paramètre (EDP) et des systèmes d'équations différentielles ordinaires (EDO). Nous présentons également une méthode de Lanczos symplectique par bloc, que nous nommons bloc J-Lanczos. Cette méthode permet de réduire une matrice structurée sous la forme J-tridiagonale par bloc. Nous proposons des algorithmes basés sur deux types de normalisation : la factorisation S R et la factorisation Rj R. Dans une dernière partie, nous proposons un algorithme qui généralise la méthode de Greville afin de déterminer la pseudo inverse de Moore-Penros bloc de lignes par bloc de lignes d'une matrice rectangulaire de manière itérative. Nous proposons un algorithme qui utilise la technique de bloc. Pour toutes ces méthodes, nous proposons des exemples numériques qui montrent l'efficacité de nos approches. / We study, in this thesis, some numerical block Krylov subspace methods. These methods preserve geometric properties of the reduced matrix (Hamiltonian or skew-Hamiltonian or symplectic). Among these methods, we interest on block symplectic Arnoldi, namely block J-Arnoldi algorithm. Our main goal is to study this method, theoretically and numerically, on using ℝ²nx²s as free module on (ℝ²sx²s, +, x) with s ≪ n the size of block. A second aim is to study the approximation of exp (A)V, where A is a real Hamiltonian and skew-symmetric matrix of size 2n x 2n and V a rectangular matrix of size 2n x 2s on block Krylov subspace Km (A, V) = blockspan {V, AV,...Am-1V}, that preserve the structure of the initial matrix. this approximation is required in many applications. For example, this approximation is important for solving systems of ordinary differential equations (ODEs) or time-dependant partial differential equations (PDEs). We also present a block symplectic structure preserving Lanczos method, namely block J-Lanczos algorithm. Our approach is based on a block J-tridiagonalization procedure of a structured matrix. We propose algorithms based on two normalization methods : the SR factorization and the Rj R factorization. In the last part, we proposea generalized algorithm of Greville method for iteratively computing the Moore-Penrose inverse of a rectangular real matrix. our purpose is to give a block version of Greville's method. All methods are completed by many numerical examples.