Collins G. W. - Fundamental Numerical Methods and Data Analysis.pdf

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George W. Collins, II
Fundamental Numerical
Methods and Data Analysis
by
George W. Collins, II
© George W. Collins, II 2003
Table of Contents
List of Figures .....................................................................................................................................vi
List of Tables .......................................................................................................................................ix
Preface ............................................................................................................................. xi
Notes to the Internet Edition ................................................................................... xiv
1.
Introduction and Fundamental Concepts .......................................................................... 1
1.1
Basic Properties of Sets and Groups.......................................................................... 3
1.2 Scalars, Vectors, and Matrices................................................................................... 5
1.3
Coordinate Systems and Coordinate Transformations.............................................. 8
1.4 Tensors and Transformations.................................................................................... 13
1.5 Operators ................................................................................................................... 18
Chapter 1 Exercises ............................................................................................................... 22
Chapter 1 References and Additional Reading..................................................................... 23
2.
The Numerical Methods for Linear Equations and Matrices ........................................ 25
2.1
Errors and Their Propagation.................................................................................... 26
2.2
Direct Methods for the Solution of Linear Algebraic Equations............................. 28
a.
Solution by Gaussian Elimination................................................................ 30
c.
Solution by Gauss Jordan Elimination......................................................... 31
d.
Solution by Matrix Factorization: The Crout Method................................. 34
e.
The Solution of Tri-diagonal Systems of Linear Equations ........................ 38
2.3
Solution of Linear Equations by Iterative Methods ................................................. 39
a.
Solution by The Gauss and Gauss-Seidel Iteration Methods ...................... 39
b.
The Method of Hotelling and Bodewig ..................................................... 41
c.
Relaxation Methods for the Solution of Linear Equations .......................... 44
d.
Convergence and Fixed-point Iteration Theory........................................... 46
2.4
The Similarity Transformations and the Eigenvalues and Vectors of a
Matrix ........................................................................................................................ 48
i
b.
Solution by Cramer's Rule............................................................................ 28
Chapter 2 Exercises ............................................................................................................... 53
Chapter 2 References and Supplemental Reading................................................................ 54
3.
Polynomial Approximation, Interpolation, and Orthogonal Polynomials ................... 55
3.1
Polynomials and Their Roots.................................................................................... 56
a.
Some Constraints on the Roots of Polynomials........................................... 57
b.
Synthetic Division......................................................................................... 58
c.
The Graffe Root-Squaring Process .............................................................. 60
d.
Iterative Methods .......................................................................................... 61
3.2
Curve Fitting and Interpolation................................................................................. 64
a.
Lagrange Interpolation ................................................................................. 65
b.
Hermite Interpolation.................................................................................... 72
c.
Splines ........................................................................................................... 75
3.3
Orthogonal Polynomials ........................................................................................... 85
a.
The Legendre Polynomials........................................................................... 87
b.
The Laguerre Polynomials ........................................................................... 88
d.
Additional Orthogonal Polynomials ............................................................ 90
e.
The Orthogonality of the Trigonometric Functions..................................... 92
Chapter 3 Exercises ................................................................................................................ 93
Chapter 3 References and Supplemental Reading................................................................. 95
4.
Numerical Evaluation of Derivatives and Integrals ......................................................... 97
4.1
Numerical Differentiation .......................................................................................... 98
a.
Classical Difference Formulae ...................................................................... 98
b.
Richardson Extrapolation for Derivatives................................................... 100
4.2
Numerical Evaluation of Integrals: Quadrature ...................................................... 102
The Trapezoid Rule ..................................................................................... 102
b.
Simpson's Rule............................................................................................. 103
c.
Quadrature Schemes for Arbitrarily Spaced Functions .............................. 105
d.
Gaussian Quadrature Schemes .................................................................... 107
e.
Romberg Quadrature and Richardson Extrapolation.................................. 111
f.
Multiple Integrals......................................................................................... 113
ii
d.
Extrapolation and Interpolation Criteria ...................................................... 79
c.
The Hermite Polynomials............................................................................. 89
a.
4.3
Monte Carlo Integration Schemes and Other Tricks............................................... 115
a.
Monte Carlo Evaluation of Integrals ........................................................... 115
Chapter 4 Exercises ............................................................................................................. 119
Chapter 4 References and Supplemental Reading............................................................... 120
5.
Numerical Solution of Differential and Integral Equations .......................................... 121
5.1
The Numerical Integration of Differential Equations ............................................. 122
One Step Methods of the Numerical Solution of Differential
Equations...................................................................................................... 123
b.
Error Estimate and Step Size Control ......................................................... 131
c.
Multi-Step and Predictor-Corrector Methods ............................................. 134
d.
Systems of Differential Equations and Boundary Value
Problems....................................................................................................... 138
e.
Partial Differential Equations ...................................................................... 146
5.2
The Numerical Solution of Integral Equations........................................................ 147
a.
The Numerical Solution of Fredholm Equations........................................ 148
c.
The Numerical Solution of Volterra Equations .......................................... 150
d.
The Influence of the Kernel on the Solution............................................... 154
Chapter 5 Exercises .............................................................................................................. 156
Chapter 5 References and Supplemental Reading .............................................................. 158
6.
Least Squares, Fourier Analysis, and Related Approximation Norms ....................... 159
6.1
Legendre's Principle of Least Squares..................................................................... 160
a.
The Normal Equations of Least Squares..................................................... 161
b.
Linear Least Squares.................................................................................... 162
c.
The Legendre Approximation ..................................................................... 164
6.2
Least Squares, Fourier Series, and Fourier Transforms .......................................... 165
a.
Least Squares, the Legendre Approximation, and Fourier Series.............. 165
b.
The Fourier Integral ..................................................................................... 166
c.
The Fourier Transform ................................................................................ 167
iii
b.
The General Application of Quadrature Formulae to Integrals ................. 117
a.
b.
Types of Linear Integral Equations ............................................................. 148
d.
The Fast Fourier Transform Algorithm ...................................................... 169
6.3
Error Analysis for Linear Least-Squares ................................................................. 176
a.
The Relation of the Weighted Mean Square Observational Error
to the Weighted Mean Square Residual ...................................................... 178
c.
Determining the Weighted Mean Square Residual .................................... 179
d.
The Effects of Errors in the Independent Variable ..................................... 181
6.4
Non-linear Least Squares ......................................................................................... 182
a.
The Method of Steepest Descent................................................................. 183
b.
Linear approximation of f(a j ,x) ................................................................... 184
c.
Errors of the Least Squares Coefficients..................................................... 186
6.5
Other Approximation Norms ................................................................................... 187
a.
The Chebyschev Norm and Polynomial Approximation ........................... 188
b.
The Chebyschev Norm, Linear Programming, and the Simplex
Method ......................................................................................................... 189
c.
The Chebyschev Norm and Least Squares ................................................. 190
Chapter 6 Exercises .............................................................................................................. 192
Chapter 6 References and Supplementary Reading............................................................. 194
7.
Probability Theory and Statistics ..................................................................................... 197
7.1
Basic Aspects of Probability Theory ....................................................................... 200
a.
The Probability of Combinations of Events................................................ 201
c.
Distributions of Random Variables............................................................. 203
7.2
Common Distribution Functions ............................................................................. 204
a.
Permutations and Combinations.................................................................. 204
c.
The Poisson Distribution ............................................................................. 206
d.
The Normal Curve ....................................................................................... 207
e.
Some Distribution Functions of the Physical World .................................. 210
7.3
Moments of Distribution Functions......................................................................... 211
7.4
The Foundations of Statistical Analysis .................................................................. 217
a.
Moments of the Binomial Distribution ....................................................... 218
b.
Multiple Variables, Variance, and Covariance ........................................... 219
iv
b.
Errors of the Least Square Coefficients ...................................................... 176
b.
Probabilities and Random Variables ........................................................... 202
b.
The Binomial Probability Distribution........................................................ 205
c.
Maximum Likelihood .................................................................................. 221

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