Page 6 - Applied Numerical Methods Using MATLAB

viii CONTENTS
2 System of Linear Equations 71

2.1 Solution for a System of Linear Equations / 72
2.1.1 The Nonsingular Case (M = N)/ 72
2.1.2 The Underdetermined Case (M< N): Minimum-Norm
Solution / 72
2.1.3 The Overdetermined Case (M> N): Least-Squares Error
Solution / 75
2.1.4 RLSE (Recursive Least-Squares Estimation) / 76
2.2 Solving a System of Linear Equations / 79
2.2.1 Gauss Elimination / 79
2.2.2 Partial Pivoting / 81
2.2.3 Gauss–Jordan Elimination / 89
2.3 Inverse Matrix / 92
2.4 Decomposition (Factorization) / 92
2.4.1 LU Decomposition (Factorization):
Triangularization / 92
2.4.2 Other Decomposition (Factorization): Cholesky, QR,
and SVD / 97
2.5 Iterative Methods to Solve Equations / 98
2.5.1 Jacobi Iteration / 98
2.5.2 Gauss–Seidel Iteration / 100
2.5.3 The Convergence of Jacobi and Gauss–Seidel
Iterations / 103
Problems / 104

3 Interpolation and Curve Fitting 117

3.1 Interpolation by Lagrange Polynomial / 117
3.2 Interpolation by Newton Polynomial / 119
3.3 Approximation by Chebyshev Polynomial / 124
3.4 Pade Approximation by Rational Function / 129
3.5 Interpolation by Cubic Spline / 133
3.6 Hermite Interpolating Polynomial / 139
3.7 Two-dimensional Interpolation / 141
3.8 Curve Fitting / 143
3.8.1 Straight Line Fit: A Polynomial Function of First
Degree / 144
3.8.2 Polynomial Curve Fit: A Polynomial Function of Higher
Degree / 145
3.8.3 Exponential Curve Fit and Other Functions / 149
3.9 Fourier Transform / 150
3.9.1 FFT Versus DFT / 151
3.9.2 Physical Meaning of DFT / 152
3.9.3 Interpolation by Using DFS / 155
Problems / 157