Essentials of Scientific Computing

Numerical Methods for Science and Engineering

Victor Zalizniak

• Publisher: Woodhead Publishing
• Year: 2008
• Print ISBN: 9781904275329
• eBook ISBN: 9780857099365
• Topic: Computer Science
• Subtopic: Business Intelligence
• Index: Computer Science

Modern development of science and technology is based to a large degree on computer modelling. To understand the principles and techniques of computer modelling, students should first get a strong background in classical numerical methods, which are the subject of this book. This text is intended for use in a numerical methods course for engineering and science students, but will also be useful as a handbook on numerical techniques for research students.Essentials of Scientific Computing is as self-contained as possible and considers a variety of methods for each type of problem discussed. It covers the basic ideas of numerical techniques, including iterative process, extrapolation and matrix factorization, and practical implementation of the methods shown is explained through numerous examples. An introduction to MATLAB is included, together with a brief overview of modern software widely used in scientific computations. - Outlines classical numerical methods, which is essential for understanding the principles and techniques of computer modelling - Intended for use in a numerical methods course for engineering and science students, but will also be useful as a handbook on numerical techniques for research students - Covers the basic ideas of numerical techniques, including iterative process, extrapolation and matrix factorization

1

Errors in computer arithmetic operations

This chapter briefly considers the machine representation of floating-point numbers and modelling of errors in computer arithmetic operations. The arithmetic performed by a computer is different from the arithmetic that we use in algebra and calculus courses. In the traditional mathematical world, numbers with an infinite number of digits are permitted. In the computational world, however, each representable number has only a finite number of digits. Input data, and also intermediate and final results of arithmetic operations, are placed in the computer memory in some standard way. To represent real numbers in some given number system, the following floating-point representation is chosen:

$\mathit{x}=\pm {\mathit{b}}^{\mathit{c}}{\mathit{m}}_{\mathit{b}}.$

(1.1)

Here, x is a real number, b is a base of the number system (usually 2, 8, or 16), c is an exponent called the characteristic, m_b is a fractional part called the mantissa, and ± denotes the sign of the number x. A mantissa of a number x≠ 0 satisfies the condition:

$\frac{1}{\mathit{b}}\le {\mathit{m}}_{\mathit{b}}<1,$

which makes representation (1.1) unique. One can represent any real number in the floating-point representation and this can be done in a number system with any base b> 0. However, in the case of computers, some real numbers (for instance, irrational numbers) cannot be exactly placed in a limited computer memory. All numbers which can be placed in the memory are represented there in some unique way so that all their exponents are bounded c_m≤ c ≤ c_p, and the mantissa is decomposed into a final b-fraction:

${\mathit{m}}_{\mathit{b}}=\frac{{\mathit{a}}_1}{\mathit{b}}+\frac{{\mathit{a}}_2}{{\mathit{b}}^2}+\cdots +\frac{{\mathit{a}}_{\mathit{k}}}{{\mathit{b}}^{\mathit{k}}},$

where a_n satisfy:

$1\le {\mathit{a}}_1\le \mathit{b}-1,0\le {\mathit{a}}_{\mathit{n}}\le \mathit{b}-1,\mathit{n...}$

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright page
• About the Author
• Dedication
• List of Examples
• Preface
• 1: Errors in computer arithmetic operations
• 2: Solving equations of the form f(x)=0
• 3: Solving systems of linear equations
• 4: Computational eigenvalue problems
• 5: Solving systems of nonlinear equations
• 6: Numerical integration
• 7: Introduction to finite difference schemes for ordinary differential equations
• 8: Interpolation and Approximation
• 9: Programming in MATLAB
• A: Integration formulae of Gaussian type
• B: Transformations of integration domains
• C: Stability regions for Runge-Kutta and Adams schemes
• D: A brief survey of available software
• Bibliography
• Index