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The well-known Eneström–Kakeya Theorem states that a polynomial with real, nonnegative, monotone increasing coefficients has all its complex zeros in the closed unit disk in the complex plane. In this paper, we extend this result by showing that all quaternionic zeros of such a polynomial lie in the unit sphere in the quaternions. We also extend related results from the complex to quaternionic setting.

The Hellmann-Feynman theorem is presented together with certain allied theorems. The origin of the Hellmann-Feynman theorem in quantum physical chemistry is described. The theorem is stated with proof and with discussion of applicability and reliability. Some adaptations of the theorem to the study of the variation of zeros of special functions and orthogonal polynomials are surveyed. Possible extensions are discussed.

yes / Quantum systems with positions and momenta in Z(d) are described by the d zeros of analytic functions on a torus. The d paths of these zeros on the torus describe the time evolution of the system. A semi-analytic method for the calculation of these paths of the zeros is discussed. Detailed analysis of the paths for periodic systems is presented. A periodic system which has the displacement operator to a real power t, as time evolution operator, is studied. Several numerical examples, which elucidate these ideas, are presented.

It may sound trivial, but in this episode we’re going to learn to add and subtract…in binary. This will serve as a basis for learning about negative binary representations and the circuitry needed to perform additions in hardware.

Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked, and in the handling of singular solutions, have made probability-one homotopy methods even more practical. This thesis describes the theory behind and performance of the new code POLSYS_PLP, which consists of Fortran 90 modules for finding all isolated solutions of a complex coefficient polynomial system of equations by a probability-one homotopy method. The package is intended to be used in conjunction with HOMPACK90, and makes extensive use of Fortran 90 derived data types to support a partitioned linear product (PLP) polynomial system structure. PLP structure is a generalization of m-homogeneous structure, whereby each component of the system can have a different m-homogeneous structure. POLSYS_PLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem definition both at a high level and via hand-crafted code. Different PLP structures and their corresponding Bezout numbers can be systematically explored before committing to root finding. / Master of Science

O principal objetivo deste trabalho é apresentar técnicas de solução para equações não lineares. Especificamente, consideramos equações compostas por funções elementares, dentre elas polinomiais, racionais, trigonométricas, exponenciais e logarítmicas, e por operações algébricas de soma, subtração, multiplicação, divisão, potência e raiz. Exploramos técnicas de resolução analítica e numérica. Como não existem fórmulas resolventes de extensão geral, a técnica analítica consiste em aplicar operações elementares que nos levam a equações equivalentes (que têm a mesma solução) até que se consiga uma equação simples, de fácil resolução. Os métodos numéricos abrangem um conjunto maior de equações e obtêm uma aproximação para a solução por meio de um processo que gera uma sequência de aproximações. Entre os métodos numéricos estudados estão Bissecção, de Newton, das Secantes e do Ponto Fixo (ou Iteração Linear). Recursos Computacionais como calculadora, planilha eletrônica e o software Maxima foram utilizados com objetivo de automatizar os cálculos, tornando essa tarefa mais rápida, e também buscando extrair informações adicionais do processo de resolução como criar tabelas e traçar gráficos. Realizamos testes numéricos com equações de diversos graus de dificuldade. Observamos as vantagens, as desvantagens e as limitações de cada método e de cada recurso. / The goal of this work is to present solution techniques for nonlinear equations. Specifically, we consider equations compounded of elementary functions, among them polynomials, rational, trigonometric, exponential and logarithmic, and of algebraic operations of addition, subtraction, multiplication, division, power and root. We explore analytical and numerical resolution techniques. Since there are no general resolvent formulas, the analytic technique consists of applying elementary operations that lead to equivalent equations (which have the same solution) until a simple and easily to solve equation is obtained. Numerical methods cover a larger set of equations and obtain an approximation to the solution by a process which generates a sequence of approximations. Among the numerical methods we studied Bisection, Newton, Secant and Fixed Point (or Linear Iteration) methods. Computational resources such as calculator, spreadsheet and the software Maxima were used in order to automate calculations, making this task faster, as well as seeking for additional information from the resolution process, such as creating tables and graphics. We perform numerical tests, with equations of varying degrees of difficulty. We note the advantages, disadvantages and limitations of each method and resource.

Orientador: Vanessa Avansini Botta Pirani / Resumo: O presente trabalho tem como objetivo principal estudar o comportamento dos zeros de alguns tipos de polinômios auto-recíprocos gerados a partir de polinômios quaseortogonais de Chebyshev de ordens um e dois. Os zeros dos polinômios auto-recíprocos que construímos estão ligados aos zeros de polinômios quase-ortogonais. Os polinômios quaseortogonais podem ser obtidos a partir de uma sequência de polinômios ortogonais. Neste trabalho, usaremos os polinômios de Chebyshev para obter polinômios quase-ortogonais e usaremos resultados sobre o comportamento de zeros desses polinômios para obter informações sobre o comportamento dos zeros de polinômios auto-recíprocos. / Abstract: The main objective of this work is to study the behavior of the zeros of some classes of self-reciprocal polynomials related to Chebyshev quasi-orthogonal polynomials of order one and two. The zeros of self-reciprocal polynomials are linked to the zeros of quasiorthogonal polynomials, which can be obtained from a sequence of orthogonal polynomials. In this work we use the Chebyshev polynomials to obtain classes of quasi-orthogonal polynomials and from results on the behavior of their zeros, we obtain information about the zeros of some classes of self-reciprocal polynomials. / Mestre

Made available in DSpace on 2014-06-11T19:27:08Z (GMT). No. of bitstreams: 0 Previous issue date: 2003-02-24Bitstream added on 2014-06-13T20:08:13Z : No. of bitstreams: 1 botta_va_me_sjrp.pdf: 571155 bytes, checksum: 6e200c838e03e019c93da99a37b1515f (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / O principal objetivo deste trabalho é realizar um estudo sobre polinômios algébricos e trigonométricos que possuem somente zeros reais. O Teorema de Hermite nos dá condições necessárias e su cientes para que isto aconteça. São discutidas questões relacionadas à localização dos zeros, onde a Regra de Sinais de Descartes teve grande importância. Além disso, alguns teoremas clássicos sobre zeros de polinômios algébricos e trigonométricos são apresentados. / The main purpose of this work is to study algebraic and trigonometric poly- nomials that have only real zeros. The Hermite Theorem gives necessary and su cient conditions for this to be true. Questions concerning the locations of the zeros are discussed, where the Descarte's Rule of Signs is of great impor- tance. Furthermore, some classical theorems concerning zeros of algebraic and trigonometric polynomials are presented.

O principal objetivo deste trabalho é apresentar técnicas de solução para equações não lineares. Especificamente, consideramos equações compostas por funções elementares, dentre elas polinomiais, racionais, trigonométricas, exponenciais e logarítmicas, e por operações algébricas de soma, subtração, multiplicação, divisão, potência e raiz. Exploramos técnicas de resolução analítica e numérica. Como não existem fórmulas resolventes de extensão geral, a técnica analítica consiste em aplicar operações elementares que nos levam a equações equivalentes (que têm a mesma solução) até que se consiga uma equação simples, de fácil resolução. Os métodos numéricos abrangem um conjunto maior de equações e obtêm uma aproximação para a solução por meio de um processo que gera uma sequência de aproximações. Entre os métodos numéricos estudados estão Bissecção, de Newton, das Secantes e do Ponto Fixo (ou Iteração Linear). Recursos Computacionais como calculadora, planilha eletrônica e o software Maxima foram utilizados com objetivo de automatizar os cálculos, tornando essa tarefa mais rápida, e também buscando extrair informações adicionais do processo de resolução como criar tabelas e traçar gráficos. Realizamos testes numéricos com equações de diversos graus de dificuldade. Observamos as vantagens, as desvantagens e as limitações de cada método e de cada recurso. / The goal of this work is to present solution techniques for nonlinear equations. Specifically, we consider equations compounded of elementary functions, among them polynomials, rational, trigonometric, exponential and logarithmic, and of algebraic operations of addition, subtraction, multiplication, division, power and root. We explore analytical and numerical resolution techniques. Since there are no general resolvent formulas, the analytic technique consists of applying elementary operations that lead to equivalent equations (which have the same solution) until a simple and easily to solve equation is obtained. Numerical methods cover a larger set of equations and obtain an approximation to the solution by a process which generates a sequence of approximations. Among the numerical methods we studied Bisection, Newton, Secant and Fixed Point (or Linear Iteration) methods. Computational resources such as calculator, spreadsheet and the software Maxima were used in order to automate calculations, making this task faster, as well as seeking for additional information from the resolution process, such as creating tables and graphics. We perform numerical tests, with equations of varying degrees of difficulty. We note the advantages, disadvantages and limitations of each method and resource.