Numerical Methods

Preface......Page 8
Acknowledgement......Page 12
Contents......Page 14
1.1 Taylor\'s Theorem......Page 18
1.2 Number Representation......Page 21
1.3 Error Considerations......Page 25
1.3.1 Absolute and Relative Errors......Page 26
1.3.3 Round-off Errors......Page 27
1.3.4 Truncation Errors......Page 31
1.3.6 Error Propagation......Page 33
1.4 Error Estimation......Page 34
1.5.1 Function Approximation......Page 35
1.5.2 Stability and Condition......Page 36
1.6.1 Linear Convergence......Page 37
1.6.2 Quadratic Convergence......Page 38
1.6.3 Aitken’s Acceleration Formula......Page 40
Problems......Page 41
2.1 Introduction......Page 46
2.3 The Inverse of a Matrix......Page 47
2.4 Matrix Inversion Method......Page 49
2.4.1 Augmented Matrix......Page 52
2.5 Gauss Elimination Method......Page 53
2.6 Gauss-Jordan Method......Page 57
2.7 Cholesky\'s Triangularisation Method......Page 61
2.8 Crout\'s Method......Page 67
2.9 Thomas Algorithm for Tridiagonal System\n......Page 72
2.10 Jacobi\'s Iteration Method......Page 76
2.11 Gauss-Seidal Iteration Method......Page 81
Problems......Page 86
3.1 Introduction......Page 92
3.2 Bisection Method......Page 93
3.2.1 Error Bounds\n......Page 94
3.3 Method of False Position......Page 97
3.4 Newton-Raphson Method......Page 100
3.4.2 Rate of Convergence of Newton-Raphson Method......Page 102
3.4.3 Modified Newton-Raphson Method......Page 105
3.4.4 Rate of Convergence of Modified Newton-Raphson Method......Page 106
3.5 Successive Approximation Method......Page 107
3.5.1 Error Estimate in the Successive Approximation Method......Page 108
3.6.1 Convergence of the Secant Method......Page 111
3.7 Muller\'s Method......Page 114
3.9 Aitken\'s Δ2 Method......Page 117
3.10 Comparison of Iterative Methods......Page 119
Problems......Page 120
4.2 Derivatives Based on Newton\'s Forward Interpolation Formula......Page 124
4.3 Derivatives based on Newton\'s Backward Interpolation Formula......Page 128
4.4 Derivatives based on Stirling\'s Interpolation Formula......Page 129
4.5 Maxima and Minima of a Tabulated Function......Page 132
4.6 Cubic Spline Method......Page 134
4.7 Summary......Page 135
5.1 Introduction......Page 138
5.2.1 Forward Differences......Page 139
5.2.2 Backward Differences......Page 140
5.2.3 Central Differences......Page 141
5.2.4 Error Propagation in a Difference Table......Page 144
5.2.6 Difference Operators......Page 147
5.2.7 Relation between the Operators......Page 148
5.2.8 Representation of a Polynomial using Factorial Notation......Page 153
5.3.2 Newton’s Binomial Expansion Formula......Page 157
5.3.3 Newton’s Forward Interpolation Formula......Page 159
5.3.4 Newton’s Backward Interpolation Formula......Page 165
5.3.5 Error in the Interpolation Formula......Page 169
5.4.1 Lagrange’s Formula for Unequal Intervals......Page 171
5.4.2 Hermite’s Interpolation Formula......Page 173
5.4.4 Lagrange’s Formula for Inverse Interpolation......Page 175
5.5 Central Difference Interpolation Formulae......Page 176
5.5.1 Gauss’s Forward Interpolation Formula......Page 177
5.5.2 Gauss’s Backward Interpolation Formula......Page 179
5.5.3 Bessel’s Formula......Page 181
5.5.4 Stirling’s Formula......Page 183
5.5.5 Laplace-Everett’s Formula......Page 184
5.6 Divided Differences......Page 186
5.6.1 Newton’s Divided Difference Interpolation Formula......Page 188
5.7 Cubic Spline Interpolation......Page 190
5.8 Summary......Page 195
Problems......Page 196
6.1 Introduction......Page 206
6.2 Linear Equation......Page 207
6.3 Curve Fitting with a Linear Equation......Page 208
6.4 Criteria for a \"Best\" Fit......Page 210
6.5 Linear Least-Squares Regression......Page 211
6.6 Linear Regression Analysis......Page 213
6.6.1 Matlab Functions: Polyfit and Polyval......Page 215
6.7 Interpretation of a and b......Page 216
6.9 Coefficient of Determination......Page 218
6.10 Linear Correlation......Page 220
6.11 Linearisation of Non-Linear Relationships......Page 224
6.12 Polynomial Regression......Page 227
6.13 Quantification of Error of Linear Regression......Page 230
6.14 Multiple Linear Regression......Page 232
6.15 Weighted Least Squares Method......Page 234
6.17 Least Squares Method for Continuous Data......Page 235
6.18 Approximation using Orthogonal Polynomials......Page 237
6.19 Gram-Schmidt Orthogonalisation Process......Page 238
6.20 Additional Example Problems and Solutions......Page 240
Problems......Page 244
7.1 Introduction......Page 254
7.2 Newton-Cotes Closed Quadrature Formula......Page 255
7.3 Trapezoidal Rule......Page 256
7.3.1 Error Estimate in Trapezoidal Rule......Page 258
7.4 Simpson\'s 1/3 Rule......Page 261
7.4.1 Error Estimate in Simpson’s 1/3 Rule......Page 262
7.5 Simpson\'s 3/8 Rule......Page 265
7.6.1 Boole’s Rule......Page 267
7.6.2 Weddle’s Rule......Page 268
7.7.1 Richardson’s Extrapolation......Page 271
7.7.2 Romberg Integration Formula......Page 272
Problems......Page 278
8.1 Introduction......Page 282
8.2.1 Picard’s Method of Successive Approximation......Page 284
8.2.2 Taylor’s Series Method......Page 288
8.3.1 Euler’s Method......Page 292
8.3.2 Modified Euler’s Method......Page 298
8.3.3.1 Runge-Kutta Method of Order Two......Page 303
8.3.3.2 Runge-Kutta Method of Order Four......Page 306
8.3.4 Predictor-Corrector Methods......Page 312
8.3.4.1 Adams-Moulton Predictor-Corrector Method......Page 313
8.3.4.2 Milne’s Predictor-Corrector Method......Page 318
8.4 Summary......Page 320
Problems......Page 321
Bibliography......Page 326
Appendix-A — Partial Fraction Expansions......Page 332
Appendix-B — Basic Engineering Mathematics......Page 337
Appendix-C — Cramer’s Rule......Page 351
Answers to Selected Problems......Page 356