CONTENTS
Preface to the Sixth Edition v
Preface to the First Edition vii
1.HIGH SPEED COMPUTATION 1
1.1 Introduction 1
1.2 Computer Arithmetic 1
1.3 Errors 7
1.4 Machine Computation 12
1.5 Computer Software 15
2. TRANSCENDENTAL AND POLYNOMIAL EQUATIONS 18
2.1 Introduction 18
2.2 Bisection Method 20
2.3 Iteration Methods Based on First Degree Equation 22
2.4 Iteration Methods Based on Second Degree Equation 29
2.5 Rate of Convergence 41
2.6 General Iteration Methods 53
2.7 System of Nonlinear Equations 63
2.8 Methods for Complex Roots 72
Exercise 2.1 74
2.9 Polynomial Equations 83
2.10 Choice of an Iterative Method and Implementation 100
Exercise 2.2 102
3. SYSTEM OF LINEAR ALGEBRAIC EQUATIONS AND
EIGENVALUE PROBLEMS 104
3.1 Introduction 104
3.2 Direct Methods 110
3.3 Error Analysis for Direct •Methods 134
Exercise 3 140
3.4 Iteration Methods 146
Exercise 3.2 165
3.5 Eigenvalues .and Eigenvectors 170
3.6 Bounds on Eigenvalues 174
3.7 Jacobi Method for Symmetric Matrices 179
3.8 Givens Method for Symmetric Matrices 185
3.9 Householder's Method for Symmetric Matrices 189
3.10 Rutishauser Method for Arbitrary Matrices 193
3.11 Power Method 196
3. 12 Inverse Power Method 199
3.13 Choice of a Method 201
Exercise 3.3 203
4. INTERPOLATION AND APPROXIMATION 210
4.1 Introduction 210
4.2 Lagrange and Newton Interpolations 213
4.3 Finite Difference Operators 229
4.4 Interpolating Polynomials Using Finite Differences 235
Exercise 4.1 242
4.5 Hennite Interpolation 247
4.6 Piecewise and Spline Interpolation 251
4.7 Bivariate Interpolation 273
Exercise 4.2 276
4.8 Approxitnation 282
4.9 Least Squares Approximation 284
4.10 Uniform Approximation 302
4.11 Rational Approximation 308
4.12 Choice of the Method 310
Exercise 4.3 313
5. DIFFERENTIATION AND INTEGRATION 320
5.1 Introduction 320
5.2 Numerical Differentiation 320
5.3 Optimum Choice of Step-Length 335
5.4 Extrapolation Methods 339
5.5 Partial Differentiation 343
Exercise 5.1 345
5.6 Numerical Integration 348
5.7 Methods Based on Interpolation 349
5.8 Methods Based on Undetermined Coefficients 356
5.9 Composite Integration Methods 386
5.10 Romberg Integration; 390
5.11 Double Integration 393
Exercise 5.2 396
6. . ORDINARY DIFFERENTIAL EQUATIONS:
INITIAL v;ALUE PROBLEMS 403
6.1 Introduction 403
6.2 Difference Equations 412
Exercise 6.1 419
6.3 Numerical Methods 421
6.4• Sing1estep Methods 434
6.5 Stability Analysis of Singlestep Methods 468
Exercise 6.2 477
6.6 Multistep Methods 485
6.7 Predictor-Corrector Methods 517
6.8 Stability Analysis of Multistep Methods 525
6.9 Stiff System 540
Exercise 6.3 541
7. ORDINARY DIFFERENTIAL EQUATIONS:
BOUNDARY VALUE PROBLEMS 550
7.1 Introduction 550
7.2 Initial Value Problem Method (Shooting Method) 551
Exercise 7.1
7.3 Finite Difference Methods 574
Exercise 7.2 604
7.4 Finite Element Methods 610
Exercise 7.3 630
ANSWERS AND HINTS TO THE PROBLEMS 633
BIBLIORAPHY 717
INDEX 719