Um Novo Método Simultâneo de Sexta Ordem Tipo Ehrlich para Zeros Polinomiais Complexos

This paper presents a new iterative method for the simultaneous determination of simple polynomial zeros. The proposed method is obtained from the combination of the third-order Ehrlich iteration with an iterative correction derived from Li's fourth-order method for solving nonlinear equations. The combined method developed has order of convergence six. Some examples are presented to illustrate the convergence and efficiency of the proposed Ehrlich-type method with Li correction for the simultaneous approximation of polynomial zeros.Este artigo apresenta um novo método iterativo para a determinação simultânea de zeros polinomiais simples. O~método proposto é obtido a partir da combinação da iteração de Ehrlich de terceira ordem com uma correção iterativa derivada do método de Li de quarta ordem para a resolução de equações não lineares. O método combinado desenvolvido tem ordem de convergência seis. Alguns exemplos são apresentados para ilustrar a convergência e eficiência do método tipo Ehrlich com correção de Li proposto para a aproximação simultânea de zeros polinomiais

Linear systems and applications in high school

Orientador: Maria Aparecida Diniz EhrhardtDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Neste trabalho estudamos métodos de resolução de sistemas lineares. Os métodos diretos apresentados aqui são: eliminação de Gauss e fatoração LU. Os métodos iterativos estudados são: algoritmo de Gauss-Jacobi, algoritmo de Gauss-Seidel e algoritmo de Kaczmarz. A eficiência dos métodos iterativos foi analisada a partir de experimentos computacionaisAbstract: In this work we study methods for solving systems of linear equations. The direct methods presented here are: Gauss elimination and LU factorization. The iterative methods studied are: Gauss-Jacobi algorithm, Gauss-Seidel algorithm and Kaczmarz algorithm. The iterative methods efficiency was analized using computer experimentsMestradoMatemática em Rede NacionalMestreCAPE

KONVERGENSI MODIFIKASI METODE POTRA-PTAK MENGGUNAKAN INTERPOLASI KUADRATIK

Metode Potra-Ptak merupakan salah satu metode iterasi dengan orde konvergensi tiga untuk menentukan akar-akar persamaan nonlinier. Kecepatan sebuah metode iterasi bergantung kepada orde konvergensinya. Oleh karena itu, Pada tugas akhir ini penulis memodifikasi metode Potra- Ptak menggunakan interpolasi kuadratik guna meningkatkan orde konvergensi. Berdasarkan hasil penelitian, diperoleh bahwa modifikasi metode Potra-Ptak menghasilkan orde konvergensi enam yang melibatkan 3 evaluasi fungsi yaitu ( ) n f z , ( ) n f y , ( ) n f x dan 2 evaluasi fungsi turunan ' ( ) n f y , ' ( ) n f x dengan indeks efficiency sebesar 1.430

Dynamic programming with recursive preferences

There is now a considerable amount of research on the deficiencies of additively separable preferences for effective modelling of economically meaningful behaviour. Through analysis of observational data and the design of suitable experiments, economists have constructed progressively more realistic representations of agents and their choices. For intertemporal decisions, this typically involves a departure from the additively separable benchmark. A familiar example is the recursive preference framework of Epstein and Zin (1989), which has become central to the quantitative asset pricing literature, while also finding widespread use in applications range from optimal taxation to fiscal policy and business cycles. This thesis presents three essays which examine mathematical research questions within the context of recursive preferences and dynamic programming. The focus is particularly on showing existence and uniqueness of recursive utility processes under stationary and non-stationary consumption growth specifications, and on solving the closely related problem of optimality of dynamic programs with recursive preferences. On one hand, the thesis has been motivated by the availability of new and unexploited techniques for studying the aforementioned questions. The techniques in question primarily build upon an alternative version of the theory of monotone concave operators proposed by Du (1989, 1990). They are typically well suited to analysis of dynamic optimality with a variety of recursive preference specifications. On the other hand, motivation also comes from the demand side: while many useful results for dynamic programming within the context of recursive preferences have been obtained by existing literature, suitable results are still lacking for some of the most popular specifications for applied work, such as common parameterizations of the Epstein-Zin specification, or preference specifications that incorporate loss aversion and narrow framing into the Epstein-Zin framework, or the ambiguity sensitive preference specifications. In this connection, the thesis has sought to provide a new approach to dynamic optimality suitable for recursive preference specifications commonly used in modern economic analysis. The approach to examining the problems of dynamic programming exploits the theory of monotone convex operators, which, while less familiar than that of monotone concave operators, turns out to be well suited to dynamic maximization. The intuition is that convexity is preserved under maximization, while concavity is not. Meanwhile, concavity pairs well with minimization problems, since minimization preserves concavity. By applying this idea, a parallel theory for these two cases is established and it provides sufficient conditions that are easy to verify in applications

A Hybrid Optimal Control Approach to Maximum Endurance of Aircraft

Aircraft performance optimization is a field of increasing interest, especially with the prevalent use of flight management systems (FMS) on commercial aircraft, as well as the growing field of autonomous aircraft. This thesis addresses the maximum endurance performance mode. Maximizing the endurance of an aircraft has several applications in data collection, surveillance, and commercial flights. Each of these applications may be best suited for different aircraft such as fixed-wing or quad-rotor vehicles, with power plants being either fuel-burning or electric. The objectives of this thesis are to solve the maximum endurance problem using an optimal control framework for fixed-wing aircraft while developing a unified model of energy-depletion which encompasses both fuel-burning and all-electric aircraft. The unified energy-depletion model allows the results to be applied to turbojet, turbofan, turboprop, and all-electric aircraft. The problem of maximum endurance in cruise will be solved for a three-phase model of flight including climb, cruise, and descent. This problem is solved using a hybrid optimal control framework using a unified energy-depletion model. One of the advantages of using an optimal control framework is the possibility to develop analytical solutions. The results of this thesis include a general solution for maximizing the endurance of fixed-wing aircraft, as well as specific analytical solutions for each aircraft configuration wherever possible. Some benefits of analytical solutions are that they require the least amount of computation time and provide insight into the problem including sensitivities and physical dependencies. Simulations are provided to validate the results in the case of specific aircraft configurations (turbojet, turbofan, turboprop, and all-electric)

Multidelity methods for multidisciplinary system design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 211-220).Optimization of multidisciplinary systems is critical as slight performance improvements can provide significant benefits over the system's life. However, optimization of multidisciplinary systems is often plagued by computationally expensive simulations and the need to iteratively solve a complex coupling-relationship between subsystems. These challenges are typically severe enough as to prohibit formal system optimization. A solution is to use multi- fidelity optimization, where other lower-fidelity simulations may be used to approximate the behavior of the higher-fidelity simulation. Low-fidelity simulations are common in practice, for instance, simplifying the numerical simulations with additional physical assumptions or coarser discretizations, or creating direct metamodels such as response surfaces or reduced order models. This thesis offers solutions to two challenges in multidisciplinary system design optimization: developing optimization methods that use the high-fidelity analysis as little as possible but ensure convergence to a high-fidelity optimal design, and developing methods that exploit multifidelity information in order to parallelize the optimization of the system and reduce the time needed to find an optimal design. To find high-fidelity optimal designs, Bayesian model calibration is used to improve low- fidelity models and systematically reduce the use of high-fidelity simulation. The calibrated low-fidelity models are optimized and using appropriate calibration schemes convergence to a high-fidelity optimal design is established. These calibration schemes can exploit high- fidelity gradient information if available, but when not, convergence is still demonstrated for a gradient-free calibration scheme. The gradient-free calibration is novel in that it enables rigorous optimization of high-fidelity simulations that are black-boxes, may fail to provide a solution, contain some noise in the output, or are experimental. In addition, the Bayesian approach enables us to combine multiple low-fidelity simulations to best estimate the high- fidelity function without nesting. Example results show that for both aerodynamic and structural design problems this approach leads to about an 80% reduction in the number of high-fidelity evaluations compared with single-fidelity optimization methods. To enable parallelized multidisciplinary system optimization, two approaches are developed. The first approach treats the system design problem as a bilevel programming problem and enables each subsystem to be designed concurrently. The second approach optimizes surrogate models of each discipline that are all constructed in parallel. Both multidisciplinary approaches use multifidelity optimization and the gradient-free Bayesian model calibration technique, but will exploit gradients when they are available. The approaches are demonstrated on an aircraft wing design problem, and enable optimization of the system in reasonable time despite lack of sensitivity information and 19% of evaluations failing. For cases when comparable algorithms are available, these approaches reduce the time needed to find an optimal design by approximately 50%.by Andrew I. March.Ph.D