Numerical Methods

NUMERICAL METHODS
Problems and Solutions

M.K. JAIN
S.R..K. IYENGAR
R.K. JAIN
Department of Mathematics
Indian Institute of fichnology Delhi
India

Contents

Preface (v)

1 TRANSCENDENTAL AND POLYNOMIAL EQUATIONS 1
1.1 Introduction 1
1.2 Iterative methods for simple roots. 2
1.3 Iterative methods for multiple roots. 6
1.4 Iterative methods for a system of nonlinear equations . 7
1.5 Complex roots. 8
1.6 Iterative methods for polynomial equations. 9
1.7 Problems and solutions. 13

2 LINEAR ALGEBRAIC EQUATIONS AND EIGENVALUE PROBLEMS 71
2.1 Introduction . 71
2.2 Direct methods . 74
2.3 Iteration methods . 78
2.4 Eigenvalue problems . 80
2.5 Special system of equations. 84
2.6 Problems and solutions. 86

3 INTERPOLATION AND APPROXIMATION 144
3.1 Introduction . 144
3.2 Lagrange and Newton interpolations . 145
3.3 Gregory-Newton interpolations . 147
3.4 Hermite interpolation . 150
3.5 Piecewise and Spline interpolation. 150
3.6 Bivariate interpolation . 153
3.7 Approximation. 154
3.8 Problems and solutions. 158

4 DIFFERENTIATION AND INTEGRATION 212
4.1 Introduction . 212
4.2 Numerical differentiation . 212
4.3 Extrapolation methods . 216
4.4 Partial differentiation . 217
4.5 Optimum choice of step-length . 218
4.6 Numerical integration . 219
4.7 Newton-Cotes integration methods . 220
4.8 Gaussian integration methods . 222
4.9 Composite integration methods . 228
4.10 Romberg integration . 229
4.11 Double integration . 229
4.12 Problems and solutions. 231

5 NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS 272
5.1 Introduction . 272
5.2 Singlestep methods . 275
5.3 Multistep methods . 279
5.4 Predictor Corrector methods . 282
5.5 Stability analysis. 284
5.6 System of differential equations . 286
5.7 Shooting methods. 288
5.8 Finite difference methods . 292
5.9 Problems and solutions. 296

Appendix
Bibliography
Index