Page 7 - Applied Numerical Methods Using MATLAB

CONTENTS ix
4 Nonlinear Equations 179

4.1 Iterative Method Toward Fixed Point / 179
4.2 Bisection Method / 183
4.3 False Position or Regula Falsi Method / 185
4.4 Newton(–Raphson) Method / 186
4.5 Secant Method / 189
4.6 Newton Method for a System of Nonlinear Equations / 191
4.7 Symbolic Solution for Equations / 193
4.8 A Real-World Problem / 194
Problems / 197

5 Numerical Differentiation/Integration 209

5.1 Difference Approximation for First Derivative / 209
5.2 Approximation Error of First Derivative / 211
5.3 Difference Approximation for Second and Higher
Derivative / 216
5.4 Interpolating Polynomial and Numerical Differential / 220
5.5 Numerical Integration and Quadrature / 222
5.6 Trapezoidal Method and Simpson Method / 226
5.7 Recursive Rule and Romberg Integration / 228
5.8 Adaptive Quadrature / 231
5.9 Gauss Quadrature / 234
5.9.1 Gauss–Legendre Integration / 235
5.9.2 Gauss–Hermite Integration / 238
5.9.3 Gauss–Laguerre Integration / 239
5.9.4 Gauss–Chebyshev Integration / 240
5.10 Double Integral / 241
Problems / 244

6 Ordinary Differential Equations 263

6.1 Euler’s Method / 263
6.2 Heun’s Method: Trapezoidal Method / 266
6.3 Runge–Kutta Method / 267
6.4 Predictor–Corrector Method / 269
6.4.1 Adams–Bashforth–Moulton Method / 269
6.4.2 Hamming Method / 273
6.4.3 Comparison of Methods / 274
6.5 Vector Differential Equations / 277
6.5.1 State Equation / 277
6.5.2 Discretization of LTI State Equation / 281
6.5.3 High-Order Differential Equation to State Equation / 283
6.5.4 Stiff Equation / 284