Numerical Methods Iii Approximation Of Functions
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Numerical Methods III - Approximation of Functions
Author: Boris Obsieger
language: en
Publisher: university-books.eu
Release Date: 2013-10-25
The book is written primarily for the students on technical universities, but also as a useful handbook for engineers and PhD students. It introduces reader into various types of approximations of functions, which are defined either explicitly or by their values in the distinct set of points, as well as into economisation of existing approximation formulas. Why the approximation of functions is so important? Simply because various functions cannot be calculated without approximation. Approximation formulas for some of these functions (such as trigonometric functions and logarithms) are already implemented in the calculators and standard computer libraries, providing the precision to all bits of memory in which a value is stored. So high precision is not usually required in the engineering practice, and use more numerical operations that is really necessary. Economised approximation formulas can provide required precision with less numerical operation, and can made numerical algorithms faster, especially when such formulas are used in nested loops. The other important use of approximation is in calculating functions that are defined by values in the chosen set of points, such as in solving integral equations (usually obtained from differential equations). The book is divided into five chapters. In the first chapter are briefly explained basic principles of approximations, i.e. approximations near the chosen point (by Maclaurin, Taylor or Padé expansion), principles of approximations with orthogonal series and principles of least squares approximations. In the second chapter, various types of least squares polynomial approximations, particularly those by using orthogonal polynomials such as Legendre, Jacobi, Laguerre, Hermite, Zernike and Gram polynomials are explained. Third chapter explains approximations with Fourier series, which are the base for developing approximations with Chebyshev polynomials (fourth chapter). Uniform approximation and further usage of Chebyshev polynomials in the almost uniform approximation, as well as in economisation of existing approximation formulas, are described in fifth chapter. Practical applications of described approximation procedures are supported by 35 algorithms and 40 examples. Besides its practical usage, the given text with 36 figures and 11 tables, partially in colour, represents a valuable background for understanding, developing and applying various numerical methods, such as interpolation, numerical integration and solving partial differential equations, which are topics in the further volumes of the series Numerical Methods.
Numerical Approximation Methods
Author: Harold Cohen
language: en
Publisher: Springer Science & Business Media
Release Date: 2011-09-28
This book presents numerical and other approximation techniques for solving various types of mathematical problems that cannot be solved analytically. In addition to well known methods, it contains some non-standard approximation techniques that are now formally collected as well as original methods developed by the author that do not appear in the literature. This book contains an extensive treatment of approximate solutions to various types of integral equations, a topic that is not often discussed in detail. There are detailed analyses of ordinary and partial differential equations and descriptions of methods for estimating the values of integrals that are presented in a level of detail that will suggest techniques that will be useful for developing methods for approximating solutions to problems outside of this text. The book is intended for researchers who must approximate solutions to problems that cannot be solved analytically. It is also appropriate for students taking courses in numerical approximation techniques.
Analysis of Approximation Methods for Differential and Integral Equations
Author: Hans-Jürgen Reinhardt
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-12-06
This book is primarily based on the research done by the Numerical Analysis Group at the Goethe-Universitat in Frankfurt/Main, and on material presented in several graduate courses by the author between 1977 and 1981. It is hoped that the text will be useful for graduate students and for scientists interested in studying a fundamental theoretical analysis of numerical methods along with its application to the most diverse classes of differential and integral equations. The text treats numerous methods for approximating solutions of three classes of problems: (elliptic) boundary-value problems, (hyperbolic and parabolic) initial value problems in partial differential equations, and integral equations of the second kind. The aim is to develop a unifying convergence theory, and thereby prove the convergence of, as well as provide error estimates for, the approximations generated by specific numerical methods. The schemes for numerically solving boundary-value problems are additionally divided into the two categories of finite difference methods and of projection methods for approximating their variational formulations.