Lecture Notes in Numerical Analysis with Mathematica


by

Krystyna STYŚ, Tadeusz STYŚ

DOI: 10.2174/97816080594231140101
eISBN: 978-1-60805-942-3, 2014
ISBN: 978-1-60805-943-0



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Table of Contents

Foreword

- Pp. i

O. A. Daman

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Preface

- Pp. iii-iv (2)

Krystyna STYš and Tadeusz STYš

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The List of Mathematica Functions and Modulae

- Pp. v

Krystyna STYš and Tadeusz STYš

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Floating Point Computer Arythmetic

- Pp. 1-26 (26)

Krystyna STYš and Tadeusz STYš

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Natural and Generalized Interpolating Polynomials

- Pp. 27-62 (36)

Krystyna STYš and Tadeusz STYš

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Polynomial Splines

- Pp. 63-102 (40)

Krystyna STYš and Tadeusz STYš

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Uniform Approximation

- Pp. 103-132 (30)

Krystyna STYš and Tadeusz STYš

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Introduction to the Least Squares Analysis

- Pp. 133-156 (24)

Krystyna STYš and Tadeusz STYš

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Selected Methods for Numerical Integration

- Pp. 157-198 (42)

Krystyna STYš and Tadeusz STYš

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Solving Nonlinear Equations by Iterative Methods

- Pp. 199-229 (31)

Krystyna STYš and Tadeusz STYš

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References

- Pp. 230-231 (2)

Krystyna STYš and Tadeusz STYš

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Index

- Pp. 233-235 (3)

Krystyna STYš and Tadeusz STYš

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Foreword

The lecture notes have been developed during over thirty years of teaching experience by the authors in the area of Numerical Analysis. They have taught the course on Numerical Methods designed for Science and Engineering students, at the University of Warsaw, University of Jos, Nigeria and the University of Botswana. The content of the notes covers the following topics: Computer Numbers and Roundoff Errors Estimates, Interpolation and Approximation of Functions, Polynomial Splines and Applications, Numerical Integration and Solu- tion of Non-linear Equations. The authors have presented the subjects in exact and comprehensive way with emphasis put on formulation of fundamental theorems with proofs supported by well selected examples. They used Mathematica, System for doing Mathematics, in solving problems specific to the subjects. In the notes, the reader will find interesting algorithms and their implementation in the Mathematica System. The lecture notes are well written and recommended as reading material for a basic course in Numerical Methods for science and engineering students.

O. A. Daman
University of Botswana,
Botswana


Preface

This text is intended for science and engineering students. It covers most of the topics taught in a first course on numerical analysis and requires some basic knowledge in calculus, linear algebra and computing. The text has been used as recommended handbook for courses taught on numerical analysis at undergraduate level. Each chapter ends with a number of questions. It is taken for granted if the reader has access to computer facilities for solving some of these questions using Mathematica. There is extensive literature published on numerical analysis including books on undergraduate and postgraduate levels. As the text for a first course in numerical analysis, this handbook contains classical content of the subject with emphases put on error analysis, optimal algorithms and their implementation in computer arithmetic. There is also a desire that the reader will find interesting theorems with proofs and verity of examples with programs in Mathematica which help reading the text. The first chapter is designed for floating point computer arithmetic and round-off error analysis of simple algorithms. It also includes the notion of well conditioned problems and concept of complexity and stability of algorithms.

Within chapter 2, interpolation of functions is discussed. The problem of interpolation first is stated in its simplest form for polynomials, and then is extended to generalized polynomials. Different Chebyshev’s systems for generalized interpolating polynomials are considered.

In chapter 3, polynomial splines are considered for uniform approximation of an one variable function.

Fundamental theorems on uniform approximation (Taylor’s theorem,Weierstrass theorem, Equi-Oscillation Chebeshev’s theorem) are stated and proved in chapter 4.

Chapter 5 is an introduction to the least squares method and contains approximation of functions in the norm of L2(a, b) space. Also, it contains approximation of discrete data and an analysis of experimental data.

In the chapter 6, two techniques of numerical integration are developed, the Newton-Cotes methods and Gauss methods. In both methods an error analysis is carried out.

For solution of non-linear algebraic equations, the most popular methods, such as Fixed Point Iterations, NewtonMethod, SecantMethod and BisectionMethod, are described in chapter 7.

Krystyna STYŠ
University of Warsaw, Poland
Tadeusz STYŠ
University of Warsaw, Poland

List of Contributors

Author(s):
Krystyna STYŚ
University of Warsaw
Poland


Tadeusz STYŚ
University of Warsaw
Poland




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