Optimal solution of nonlinear equations

Journal ArticleWe survey recent worst case complexity results for the solution of nonlinear equations. Notes on worst and average case analysis of iterative algorithms and a bibliography of the subject are also included

A family of root-finding methods with accelerated convergence

AbstractA parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given

Finding All Solutions to a System of Polynomial Equations

Given a polynomial equation of degree d over the complex domain, the Fundamental Theorem of Algebra tells us that there are d solutions, assuming that the solutions are counted by multiplicity. These solutions can be approximated by deforming a standard n th degree equation into the given equation, and following the solutions through the deformation. This is called the homotopy method. The Fundamental Theorem of Algebra can be proved by the same technique. In this paper we extend these results and methods to a system of n polynomial equations in I complex variables. We show that the number of solutions to such a system is the product of the degrees of the equations (assuming that infinite solutions are included and solutions are counted by multiplicity)*. The proof is based on a homotopy, or deformation, from a standard system of equations with the same degrees and known solutions. This homotopy provides a computational method of approximating all solutions. Computational results demonstrating the feasibility of this method are also presented

Computer-Aided Design of Switched-Capacitor Filters

This thesis describes a series of computer methods for the design of switched-capacitor filters. Current software is greatly restricted in the types of transfer function that can be designed and in the range of filter structures by which they can be implemented. To solve the former problem, several new filter approximation algorithms are derived from Newton's method, yielding the Remez algortithm as a special case (confirming its convergency properties). Amplitude responses with arbitrary passband shaping and stopband notch positions are computed. Points of a specified degree of tangency to attenuation boundaries (touch points) can be placed in the response, whereby a family of transfer functions between Butterworth and elliptic can be derived, offering a continuous trade-off in group delay and passive sensitivity properties. The approximation algorithms have also been applied to arbitrary group delay correction by all-pass functions. Touch points form a direct link to an iterative passive ladder design method, which bypasses the need for Hurwitz factorisation. The combination of iterative and classical synthesis methods is suggested as the best compromise between accuracy and speed. It is shown that passive ladder prototypes of a minimum-node form can be efficiently simulated by SC networks without additional op-amps. A special technique is introduced for canonic realisation of SC ladder networks from transfer functions with finite transmission at high frequency, solving instability and synthesis difficulties. SC ladder structures are further simplified by synthesising the zeros at +/-2fs which are introduced into the transfer function by bilinear transformation. They cause cancellation of feedthrough branches and yield simplified LDI-type SC filter structures, although based solely on the bilinear transform. Matrix methods are used to design the SC filter simulations. They are shown to be a very convenient and flexible vehicle for computer processing of the linear equations involved in analogue filter design. A wide variety of filter structures can be expressed in a unified form. Scaling and analysis can readily be performed on the system matrices with great efficiency. Finally, the techniques are assembled in a filter compiler for SC filters called PANDDA. The application of the above techniques to practical design problems is then examined. Exact correction of sinc(x), LDI termination error, pre-filter and local loop telephone line weightings are illustrated. An optimisation algorithm is described, which uses the arbitrary passband weighting to predistort the transfer function for response distortions. Compensation of finite amplifier gain-bandwidth and switch resistance effects in SC filters is demonstrated. Two commercial filter specifications which pose major difficulties for traditional design methods are chosen as examples to illustrate PANDDA's full capabilities. Significant reductions in order and total area are achieved. Finally, test results of several SC filters designed using PANDDA for a dual-channel speech-processing ASIC are presented. The speed with which high-quality, standard SC filters can be produced is thus proven

Design, Analysis, and Applications of Iterative Methods for Solving Nonlinear Systems

In this chapter, we present an overview of some multipoint iterative methods for solving nonlinear systems obtained by using different techniques such as composition of known methods, weight function procedure, and pseudo-composition, etc. The dynamical study of these iterative schemes provides us valuable information about their stability and reliability. A numerical test on a specific problem coming from chemistry is performed to compare the described methods with classical ones and to confirm the theoretical results

Guidance, flight mechanics and trajectory optimization. Volume 6 - The N-body problem and special perturbation techniques

Analytical formulations and numerical integration methods for many body problem and special perturbative technique

Finite element solution for elliptic partial differential equations

The contents of this thesis are a detailed study of the implementation of Finite Element method for solving linear and non-linear elliptic partial differential equations. It commences with a description and classification of partial differential equations, the related matrix and eigenvalue theory and the related matrix methods to solve the linear and non-linear systems of equations. In Chapter Three, we discuss the development of the, finite element method and its application with a full description of an orderly step-by-step process. In Chapter Four, we discuss the implementation of developing an efficient easy-to-use finite element program for the general two-dimensional problem along with the capability of handling problems for different domains and boundary conditions and with a fully automated mesh generation and refinement technique along with a description of generalised pre- and post-processors for the Finite Element Method. [Continues.