Practical Scientific Computing

197 Pages·2011·7.45 MB·English

by  Ali Muhammad, Victor Zalizniak

Practical Scientific Computing
0 Woodhead Publishing Limited, 201 1
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0 Woodhead Publishing Limited, 201 1
Practical Scientific
Computing
Ali Muhammad and Victor Zalizniak
WOODHEAD
PUBLISHING
Oxford Cambridge Philadelphia New Delhi
0 Woodhead Publishing Limited, 201 1
Published by Woodhead Publishing Limited,
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First published 201 1, Woodhead Publishing Limited
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Table of Contents
Preface ........................................................................................................... ix
Acknowledgements ..........................................................................................
Part I ............................................................................................................... 1
Chapter 1 Introduction ................................................................................. 3
1.1 Getting Started ........................................................................................... 4
1.2 Interpreter .................................................................................................. 7
1.3 Prog ram .................................................................................................... 10
Chapter 2 Expressions ................................................................................. 12
2.1 Matrix ...................................................................................................... 12
2.2 Real Number ............................................................................................ 13
2.3 Complex Number ..................................................................................... 14
2.4 Boolean .................................................................................................... 15
2.5 String. ....................................................................................................... 16
2.6 Structure. .................................................................................................. 17
2.7 Cell ........................................................................................................... 19
2.8 Range Expression .................................................................................... 20
2.9 Boolean Expression ................................................................................. 21
2.10 Relational Expression ............................................................................ 22
2.1 1 Numerical Expression ............................................................................ 23
Chapter 3 Statements .................................................................................. 25
3.1 Assignment Statement ............................................................................. 25
3.2 Loop Statements ...................................................................................... 27
3.2.1 “for” Statement ..................................................................................... 27
3.2.2 “while” Statement ................................................................................. 28
3.3 Conditional Statements ............................................................................ 28
3.3.1 “if” Statement ....................................................................................... 29
3.3.2 ‘‘switch” Statement .............................................................................. 30
3.4 Continue and Break Statements ............................................................... 31
Chapter 4 Programming ............................................................................. 32
4.1 Program. ................................................................................................... 32
4.2 Function ................................................................................................... 34
4.3 Procedure ................................................................................................. 42
4.4 Java Programming ................................................................................... 43
4.5 C Programming ........................................................................................ 49
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vi Table of Contents
Chapter 5 Architecture ............................................................................... 53
5.1 Front-end .................................................................................................. 53
5.2 Back-end .................................................................................................. 54
5.3 User Interface ........................................................................................... 55
5.4 Gnuplot Interface ..................................................................................... 56
5.5 Execution Engine ..................................................................................... 57
Chapter 6 Plotting ........................................................................................ 61
6.1 Simple Function Plot (fplot) .................................................................... 61
6.2 Two-Dimensional Plots ........................................................................... 64
6.2.1 Basic plot (plot) .................................................................................... 64
6.2.2 Logarithmic plot (plot) .......................................................................... 66
6.2.3 Polar plot (polar) ................................................................................... 67
6.2.4 Bar chart (bar) ....................................................................................... 67
6.2.5 Histogram plot (hist) ............................................................................. 67
6.2.6 Stairs plot (stair) .................................................................................... 68
6.3 Three-Dimensional Plots ......................................................................... 68
6.3.1 Line plot (plot3) .................................................................................... 68
6.3.2 Mesh surface plot (mesh, meshc, contour) ........................................... 69
Part I1 ............................................................................................................ 71
Chapter 7 Solving Nonlinear Equations .................................................... 73
7.1 Calculation of Roots with the use of Iterative Functions ......................... 73
7.1.1 One-point iterative processes ................................................................ 76
7.1.2 Multi-point iterative processes .............................................................. 82
7.1.3 One-point iterative processes with memory ......................................... 85
7.2 Exercises .................................................................................................. 86
Chapter 8 Solving Systems of Linear Equations ....................................... 89
8.1 Linear Algebra Background ..................................................................... 89
8.2 Systems of Linear Equations ................................................................... 92
8.3 Types of Matrices that arise from Applications and Analysis ................. 94
8.3.1 Sparse matrices ..................................................................................... 94
8.3.2 Band matrices ....................................................................................... 94
8.3.3 Symmetric positive definite matrices .................................................... 94
8.3.4 Triangular matrices ............................................................................... 94
8.3.5 Orthogonal matrices .............................................................................. 98
8.3.6 Reducible matrices ................................................................................ 98
8.3.7 Matrices with diagonal dominance ....................................................... 98
8.4 Error Sources ........................................................................................... 99
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Table of Contents vii
8.5 Condition Number ................................................................................... 99
8.6 Direct Methods ....................................................................................... 100
8.6.1 Basic principles of direct methods ...................................................... 101
8.6.2 Error estimation for linear systems ..................................................... 103
8.7 Iterative Methods ................................................................................... 104
8.7.1 Basic principles of iterative methods .................................................. 104
8.7.2 Jacobi method ..................................................................................... 106
8.7.3 Gauss-Seidel method .......................................................................... 108
8.7.4 Method of relaxation ........................................................................... 110
8.7.5 Variational iterative methods .............................................................. 113
8.8 Exercises ................................................................................................ 115
Chapter 9 Computational Eigenvalue Problems .................................... 122
9.1 Basic Facts concerning Eigenvalue Problems ....................................... 122
9.2 Localization of Eigenvalues. .................................................................. 123
9.3 Power Method ........................................................................................ 124
9.4 Inverse Iteration ..................................................................................... 126
9.5 Iteration with a Shift of Origin .............................................................. 128
9.6 The QR Method ..................................................................................... 130
9.7 Exercises ................................................................................................ 133
Chapter 10 Introduction to Finite Difference Schemes for Ordinary
Differential Equations ............................................................................... 138
10.1 Elementary Example ofa Finite Difference Scheme ........................... 138
10.2 Approximation and Stability ................................................................ 141
10.3 Numerical Solution of Initial Value Problems. .................................... 143
10.3.1 Stability of difference schemes for linear problems ......................... 144
10.3.2 Runge-Kutta methods ....................................................................... 146
10.3.3 Adams type methods ......................................................................... 150
10.3.4 Systems of differential equations ...................................................... 153
10.4 Numerical Solution of Boundary Value Problems .............................. 156
10.4.1 Conversion of difference schemes to systems of equations. .............1 56
10.4.2 Method of time development ............................................................ 158
10.4.3 Accurate approximation of boundary conditions when derivatives are
specified at boundary points ........................................................................ 160
10.5 Error Estimation and Control ............................................................... 161
10.6 Exercises .............................................................................................. 164
Chapter 11 Interpolation and Approximation ....................................... 168
1 1. 1 Interpolation ........................................................................................ 168
11.1.1 Polynomial interpolation. .................................................................. 168
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Vlll Table of Contents
1 1.1.2 Trigonometric interpolation .............................................................. 171
1 1.1.3 Interpolation by splines ..................................................................... 175
1 1.2 Approximation of Functions and Data Representation ........................ 176
11.2.1 Least-squares approximation ............................................................ 177
11.2.2 Approximation by orthogonal functions ........................................... 180
1 1.2.3 Approximation by interpolating polynomials ................................... 185
1 1.3 Exercises .............................................................................................. 186
Bibliography ................................................................................................ 189
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Scientific computing is about developing mathematical models, numerical
methods and computer implementations to study and solve real problems in science,
engineering, business and even social sciences. Mathematical modeling requires
deep understanding of classical numerical methods. There are a number of
commercial and open source tools available which provide a rich collection of built-
in mathematical functions. These tools are widely used by researchers to model,
solve and simulate mathematical problems. These tools also offer easy to use
programming languages for users to develop more functions.
This book is divided into two parts. In the first part, we present an open source
tool, numEclipse. It is a Java based tool modeled after MATLAB@ and it is
implemented as a plug-in for Eclipse, which is a leading integrated development
environment (IDE) for Java programming. In the second part, we study the classical
methods of numerical analysis. We present the numerical algorithms and their
implementations using numEclipse.
Although we have tried to make this book as self-contained as possible,
knowledge of calculus, linear algebra and differential equations is a prerequisite.
Some exposure to a programming language will be helpful to follow the computer
programs.
Such a small volume as this book cannot do justice to the vast area of classical
numerical methods. The choice of some of the topics is based on our own
preferences and teaching experience. This book provides enough foundations for a
beginner to venture into more advanced texts in the subject area.
Ali Muhammad
Victor Zalizniak
0 Woodhead Publishing Limited, 201 1