Introductory methods of numerical analysis / S.S. Sastry.

Includes index.

1. Errors in Numerical Calculations
1.1 Introduction 1
1.1.1 Computer and Nmnerical Software 3
1.1.2 Computer Languages 3
1.1.3 So^are Packages 4
1.2 Mathematical Preliminaries 5
1.3 Errors and Their Con:q)utations 7
1.4 A General Error Formula 11
1.5 Error in a Series Approximation 12
Exercises 17
2. Solution of Algebraic and Transcendental Equations
2.1 Introduction 20
2.2 The Bisection Method 21
2.3 The Method of False Position 24
2.4 The Iteration Method 26
2.5 Newton-Raphson Method 33
2.6 Ramanujan's Method 38
2.7 The Secant Method 43
2.8 Muller's Method 44
2.9 Graeffe's Root-squaring Method 46
2.10 Lin-Bairstow's Method 48
2.11 The Quotient-difference Method 51
2.12 Solution to Systems of Nonlinear Equations 54
2.12.1 The Method of Iteration 54
2.12.2 Newton-Raphson Method 57
Exercises 59
3. Interpolation
3.1 Introduction 63
3.2 Errors in Polynomial Interpolation 64
3.3 Finite Differences 65
3.3.1 Forward Differences 65
3.3.2 Backward Differences 66
3.3.3 Central Differences 67
3.3.4 Symbolic Relations and Separation of
Symbols 68
3.4 Detection of Errors by Use of Difference Tables 71
3.5 Differences of a Polynomial 72
3.6 Newton's Formulae for Interpolation 73
3.7 Central Difference Interpolation Formulae 7P
3.7.1 Gauss' Central Difference Formulae 79
3.7.2 Stirling's Formula 83
3.7.3 Bessel's Formula 83
3.7A Everett's Formula 85
3.7.5 Relation between Bessel's and Everett's
Formulae 85
3.8 Practical Interpolation 86
3.9 Interpolation with Unevenly Spaced Points 90
3.9.1 Lagrange's Interpolation Formula 91
3.9.2 Error in Lagrange's Interpolation Formula 96
3.9.3 Hermite's Interpolation Formula 97
3.10 Divided Differences and Their Properties 100
3.10.1 Newton's General Interpolation Formula 102
3.10.2 Interpolation by Iteration 104
3.11 Inverse Interpolation 105
3.12 Double Interpolation 107
3.13 Spline Interpolation 108
3.13.1 Linear Splines 109
3.13.2 Quadratic Splines 110
3.14 Cubic Splines 112
3.14.1 Minimizing Property of Cubic Splines 117
3.14.2 Error in the Cubic Spline and Its Derivatives 119
3.15 Surface Fitting by Cubic Splines 122
Exercises 125
4. Least Squares, B-splines and Fourier Transforms 137—186
4.1 Introduction 137
4.2 Least-squares Curve Fitting Procedures 138
4.2.1 Fitting a Straight Line 138
4.2.2 Nonlinear Curve Fitting 140
4.2.3 Curve Fitting by a Sum of Exponentials 143
4.3 Weighted Least Squares Approximation 146
4.3.1 Linear Weighted Least Squares Approximation 146
4.3.2 Nonlinear Weighted Least Squares
Approximation 148
4.4 Method of Least Squares for Continuous Functions 149
4.4.1 Orthogonal Polynomials 151
4.4.2 Gram-Schmidt Orthogonalization Process 154
4.5 Cubic B-splines 157
4.5.1 Least-squares Solution 159
4.5.2 Representations of B-splines 159
4.5.3 Computation of B-splines 162
4.6 Fourier Approximation 164
4.6.1 The Fourier Transform 167
4.6.2 The Fast Fourier Transform 169
4.6.3 Cooley-Tukey Algorithm 170
4.6.4 Sande-Tukey Algorithm 176
4.6.5 Computation of the Inverse DFT 177
4.7 Approximation of Fimctions 178
4.7.1 Chebyshev Polynomials 178
4.7.2 Economization of Power Series 181
Exercises 182
5. Numerical Dinerentiatlou and Integration
5.1 Introduction 187
5.2 Numerical Differentiation 187
5.2.1 Errors in Numerical Differentiation 192
5.2.2 The Cubic Spline Method 194
5.3 Maximum and Minimum Values of a Tabulated
Function 196
5.4 Numerical Integration 197
5.4.1 Trapezoidal Rule 198
5.4.2 Simpson's 1/3-Rule 200
5.4.3 Simpson's 3/8-Rule 201
5.4.4 Boole's and Weddle's Rules 201
5.4.5 Use of Cubic Splines 202
5.4.6 Romberg Integration 202
5.4.7 Newton-Cotes Integration Formulae 204
5.5 Euler-Maclaurin Formula 211
5.6 Adaptive Quadratxire Methods 213
5.7 Gaussian Integration 216
5.8 Numerical Evaluation of Singular Integrals 220
5.8.1 Evaluation of Principal Value Integrals 220
5.8.2 Generalized Quadrature 222
5.9 Numerical Calculation of Fourier Integrals 224
5.9.1 Trapezoidal Rule 224
5.9.2 Filon's Formula 225
5.9.3 The Cubic Spline Method 227
5.10 Numerical Double Integration 230
Exercises 232
6. Matrices and Linear Systems of Equations
6.1 Introduction 240
6.2 Basic Definitions 240
6.2.1 Matrix Operations 243
6.2.2 Transpose of a Matrix 245
6.2.3 The hiverse of a Matrix 248
6.2.4 Rank of a Matrix 249
6.2.5 Consistency of a Linear System of
Equations 250
6.2.6 Vector and Matrix Norms 252
6.3 Solution of Linear Systems—^Direct Methods 255
6.3.1 Matrix Inversion Method 255
6.3.2 Gauss Elimination 257
6.3.3 Gauss-Jordan Method 260
6.3.4 Modification of the Gauss Method to Compute
the Inverse 261
6.3.5 Number of Arithmetic Operations 264
6.3.6 LU Decomposition 265
6.3.7 LU Decomposition firom Gauss Elimination 269
6.3.8 Solution of Tridiagonal Systems 270
6.3.9 Solution of Centro-symmetric Equations 271
6.3.10 Ill-conditioned Linear Systems 272
6.3.11 Method for Ill-conditioned Matrices 274
6.4 Solution of Linear Systems—Alterative Methods 275
6.5 The Eigenvalue Problem 278
6.5.1 Eigenvalues of a Symmetric Tridiagonal
Matrix 282
6.5.2 Householder's Method 283
6.5.3 The QR Method 287
6.6 Singular Value Decompoisition 288
Exercises 290
7. Numerical Solution of Ordinary Differential Equations 295-332
7.1 Introduction 295
7.2 Solution by Taylor's Series 296
7.3 Picard's Method of Successive Approximations 298
7.4 Euler's Method 300
7.4.1 Error Estimates for the Euler Method 301
7.4.2 Modified Euler's Method 303
7.5 Runge-Kutta Methods 304
7.6 Predictor-Corrector Methods 309
7.6.1 Adams-Moulton Method 309
7.6.2 Milne's Method 311
7.7 The Cubic Spline Method 314
7.8 Simultaneous and Higher-order Equations 316
7.9 Some General Remarks 317
7.10 Boundary-value Problems 318
7.10.1 Finite-difference Method 318
7.10.2 The Shooting Method 323
7.10.3 The Cubic Spline Method 325
Exercises. 328
8. Numerical Solution of Partial Differential Equations
8.1 Introduction 333
8.2 Finite-Difference Approximations to Derivatives 335
8.3 Laplace's Equation 338
8.3.1 Jacobi's Method 339
8.3.2 Gauss-Seidel Method 339
8.3.3 Successive Over-relaxation (or SOR
Method) 339
8.3.4 The ADI Method 345
8.4 Parabolic Equations 349
8.5 Iterative Methods for the Solution of Equations 355
8.6 Hyperbolic Equations 358
8.7 Software for Partial Differential Equations 362
Exercises 362
9. Numerical Solution of Integral Equations
9.1 Introduction 365
9.2 Numerical Methods for Fredholm Equations 367
9.2.1 Method of Degenerate Kernels 367
9.2.2 Quadrature Methods 370
9.2.3 Use of Chebyshev Series 372
9.2.4 The Cubic Spline Method 376
9.3 Singular Kernels 378
9.4 Method of Invariant Imbedding 382
Exercises 385
10. The Finite Element Method
10.1 Introduction 387
10.1.1 Functionals 388
10.1.2 Base Fimctions 392
10.2 Methods of Approximation 392
10.2.1 The Rayleigh-Ritz Method 393
10.2.2 The Galerkin Method 399
10.3 Application to Two-dimensional Problems 401
10.4 The Finite Element Method 402
10.4.1 Finite Element Method for One-dimensional
Problems 404
10.4.2 Application to Two-dimensional Problems 411
10.5 Concluding Remarks 419
Exercises 419