Introduction to Numerical Methods and Optimization Techniques

Title Page......Page 1
Copyright......Page 2
Contents......Page 3
Preface......Page 7
1.2 Numerical Methods......Page 10
1.4 Computer Philosophy Used in the Text......Page 13
1.5 Error Analysis......Page 15
Problems......Page 20
2.1 Introduction......Page 24
2.2 Cramer\'s Rule......Page 25
2.3 The Matrix Solution......Page 26
2.4 Gauss Elimination......Page 28
2.5 Crout Reduction......Page 38
2.6 Suggested Reading in Related Topics......Page 45
Problems......Page 47
3.1 Introduction......Page 52
3.2 Iterative Procedures......Page 53
3.3 Newton\'s Method......Page 54
3.4 Quadratic Interpolation and Muller\'s Method......Page 59
3.5 Bairstow\'s Method......Page 66
3.6 Suggested Reading in Related Topics......Page 73
Problems......Page 76
4.1 Introduction......Page 80
4.2 A Unique Solution......Page 81
4.3 The Normalized Variable......Page 82
4.4 Some Useful Operators, A and E......Page 83
4.5 Difference Tables......Page 86
4.6 The Newton-Gregory Polynomial......Page 87
4.7 The Lagrange Polynomial......Page 91
4.8 Inverse Interpolation......Page 94
4.9 Introduction to Least-Squares Data Fitting......Page 95
4.10 Spline Functions......Page 98
4.11 Fourier Series Applied to Interpolation......Page 100
4.12 Suggested Reading in Related Topics......Page 106
Problems......Page 109
5.1 Introduction......Page 112
5.2 Method of Interpolating Polynomials......Page 113
5.3 Method of Undetermined Coefficients......Page 116
5.4 Application of Interpolating Programs......Page 120
Problems......Page 122
6.1 Introduction......Page 124
6.2 Trapezoidal Rule......Page 125
6.3 Simpson\'s Rules......Page 130
6.4 Examples......Page 134
6.5 Romberg Prediction......Page 136
6.6 Method of Undetermined Coefficients......Page 138
6.7 Predictor and Corrector Equations......Page 140
6.8 Gaussian Quadrature......Page 143
6.9 Suggested Reading in Related Topics......Page 146
Problems......Page 148
7.1 Introduction......Page 152
7.3 Euler Method......Page 153
7.4 Stability Analysis......Page 156
7.5 Modified Euler Method......Page 160
7.6 Runge-Kutta Method......Page 162
7.7 Adams Method and Automatic Error Control......Page 165
7.8 Solution of Higher-Order Differential Equations......Page 172
7.9 Boundary-Value Problems......Page 178
7.10 Suggested Reading in Related Topics......Page 181
Problems......Page 183
8.1 Preliminary Remarks......Page 186
8.2 Formulation of Optimization Problems......Page 188
8.3 Overview of Various Optimization Techniques......Page 190
8.4 The Simplex Optimization Technique......Page 192
8.5 Applications of Simplex......Page 199
8.6 Test Functions......Page 204
Problems......Page 211
9.1 Introduction......Page 214
9.2 Quadratic Interpolation for a Specific Direction......Page 215
9.3 The Gradient......Page 217
9.4 The Steepest-Descent Optimization Technique......Page 220
9.5 Applications of Steepest Descent......Page 229
9.6 The Fletcher-Powell Optimization Technique......Page 235
Problems......Page 242
10.1 Introduction......Page 246
10.2 The Least-Squares Algorithm......Page 247
10.3 The Least-pth Algorithm......Page 251
10.4 A Least-pth Program......Page 253
10.5 Application to Least-Squares Data Fitting......Page 261
10.6 Chebyshev Approximations......Page 265
Problems......Page 272
11.1 Introduction......Page 274
11.2 Active Constraints versus Inactive Constraints......Page 275
11.3 Transformations......Page 278
11.4 Penalty Functions......Page 285
11.5 Concluding Comments about Optimization Techniques......Page 289
Problems......Page 291
References......Page 294
Answers to Selected Problems......Page 296
Index......Page 300