Numerical Methods for Scientists and Engineers

884 Pages·2012·87.134 MB·English

by  H. M. Antia (auth.)

TEXTS AND READINGS IN PHYSICAL SCIENCES - 2
Numerical Methods
for Scientists and Engineers
Third Edition
Texts and Readings in Physical Sciences
Managing Editors
H. S. Mani, Chennai Mathematical Institute, Chennai.
[email protected]
Ram Ramaswamy, Vice Chancellor, University of Hyderabad, Hyderabad.
[email protected]
Editors
Kedar Damle (fIFR, Mumbai) [email protected]
Debashis Ghoshal GNU, New Delhi) [email protected]
Rajaram Nityananda (NCRA, Pune) [email protected]
Gautam Menon (IMSc, Chennai) [email protected]
Tarun Souradeep (IUCAA, Pune) [email protected]
Volumes published so far
1. Field Theories in Condensed Matter Physics, Sumathi Roo (Ed.)
2. Numerical Methods for Scientists and Engineers (3/E), H. M. Antia
3. Lectures on Quantum Mechanics (2/E), Ashok Oas
4. Lectures on Electromagnetism, Ashok Oas
5. Current Perspectives in High Energy Physics, Oebashis Ghoshal (Ed.)
6. Linear Algebra and Group Theory for Physicists (2/E), K. N. Srinivasa Roo
7. Nonlinear Dynamics: Near and Far from Equilibrium,] K. Bhattacharjee
and S. Bhattacharyya
8. Spacetime, Geometry and Gravitation, Pankcg Sharan
9. Lectures on Advanced Mathematical Methods for Physicists, Sunil Mukhi
and N. Mukunda
10. Computational Statistical Physics, Sitangshu Bikas Santra and
Purusattam Roy (Eds.)
11. The Physics of Disordered Systems, Gautam 1. Menon and
Purusattam Roy (Eds.)
Numerical Methods
for Scientists and Engineers
Third Edition
H. M. Antia
~HINDUSTAN
U LQJ UB OOK AGENCY
Published by
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Copyright © 1991, First Edition, Tata Mcgraw-Hill Publishing Company
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Copyright © 2002, Second Edition, Hindustan Book Agency (India)
Copyright © 2012, Hindustan Book Agency (India)
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ISBN 978-93-80250-40-3 ISBN 978-93-86279-52-1 (eBook)
DOI 10.1007/978-93-86279-52-1
Texts and Readings in the Physical Sciences
The Texts and Readings in the Physical Sciences (TRiPS) series of books
aims to provide a forum for physical scientists to describe their fields of
research interest. Each book is intended to cover a subject in a detailed
manner and from a personal viewpoint, so to give students in the field a
unique exposure to the subject that is at once accessible, pedagogic, and
contemporary. The monographs and texts that have appeared so far, as
well as the volumes that we plan to bring out in the coming years, cover a
range of areas at the core as well as at the frontiers of physics.
In addition to texts on a specific topic, the TRiPS series includes lecture
notes from a thematic School or Workshop and topical volumes of
contributed articles focussing on a currently active area of research.
Through these various forms of exposition, we hope that the series will
be valuable both for the beginning graduate student as well as the
experienced researcher.
H.S. Mani R. Ramaswamy
Chennai Hyderabad
Contents
Preface xiii
Preface to the second edition xv
Preface to the First Edition xix
Notation xxiii
List of Computer Programs xxv
1 Introduction 1
1.1 Errors in Numerical Computation 2
1.2 Truncation Error 4
1.3 Programming 8
Bibliography 11
Exercises 11
2 Roundoff Error 13
2.1 Number Representation 13
2.2 Roundoff Error ..... 22
2.3 Error Analysis . . . . . 32
2.4 Condition and Stability 42
Bibliography 50
Exercises ..... . 51
3 Linear Algebraic Equations 59
3.1 Introduction ..... 60
3.2 Gaussian Elimination . . . . . . . 63
3.3 Direct Triangular Decomposition 71
3.4 Error Analysis . . . . . . . . . 77
3.5 Matrix Inversion ...... . 87
3.6 Singular Value Decomposition 88
3.7 Iterative Methods 98
Bibliography 102
Exercises .... 103
viii Contents
4 Interpolation 111
4.1 Polynomial Interpolation . . . . . . . . . . 112
4.2 Divided Difference Interpolation Formula. 117
4.3 Hermite Interpolation. . . 128
4.4 Cubic Spline Interpolation . . . 130
4.5 B-splines . .. . . . . . . . . . . 135
4.6 Rational Function Interpolation 141
4.7 Interpolation in Two or More Dimensions 147
4.8 Spline Interpolation in Two or More Dimensions 150
Bibliography 152
Exercises 153
5 Differentiation 159
5.1 Differentiation of Interpolating Polynomials 159
5.2 Method of Undetermined Coefficients 164
5.3 Extrapolation Method 168
Bibliography 172
Exercises 172
6 Integration 175
6.1 Newton-Cotes Quadrature Formulae 177
6.2 Extrapolation Methods . 187
6.3 Gaussian Quadrature 194
6.4 Roundoff Error . . 198
6.5 Weight Function 201
6.6 Improper Integrals 209
6.7 Automatic Integration 220
6.8 Summation. . .... 229
6.9 Multiple Integrals . .. 236
6.10 Rules Exact for Monomials. 242
6.11 Monte Carlo Method .. . 247
6.12 Equidistributed Sequences 254
Bibliography 259
Exercises .. . . . .. . 260
7 Nonlinear Algebraic Equations 271
7.1 Real Roots ... . . . . 273
7.2 Fixed-Point Iteration .. 276
7.3 Method of False Position 278
7.4 Secant Method ... . . 281
7.5 Newton-Raphson Method 285
7.6 Brent's Method 289
7.7 Complex Roots . ... . . 294
7.8 Muller's Method ... . . 297
7.9 Quadrature Based Method 301
Contents ix
7.10 Real Roots of Polynomials 304
7.11 Laguerre's Method .... 308
7.12 Roundoff Error . . . . . . 313
7.13 Criterion for Acceptance of a Root 315
7.14 Ill-conditioning . . . . . . . . . 319
7.15 System of Nonlinear Equations 323
7.16 Newton's Method . 328
7.17 Broyden's Method 333
Bibliography 335
Exercises 336
8 Optimisation 345
8.1 Golden Section Search 347
8.2 Brent's Method .... 353
8.3 Methods Using Derivative 356
8.4 Minimisation in Several Dimensions. 359
8.5 Quasi-Newton Methods. 363
8.6 Direction Set Methods 371
8.7 Linear Programming 379
8.8 Simulated Annealing 390
Bibliography 395
Exercises .. 396
9 Statistical Inferences 401
9.1 Elementary statistics ............ . 401
9.2 Monte Carlo Methods ........... . 416
9.3 Experimental Errors and their Propagation 418
Bibliography 422
Exercises ..... . 423
10 Functional Approximations 425
10.1 Choice of Norm and Model. 426
10.2 Linear Least Squares . . . . 432
10.3 Nonlinear Least Squares .. 451
10.4 Least Squares Approximation in Two Dimensions 457
10.5 Discrete Fourier Transform. . . . 459
10.6 Fast Fourier Transform . . . . . . 464
10.7 FFT in Two or More Dimensions 473
10.8 Inversion of Laplace Transform 475
10.9 Pade Approximations . . . 479
10.10 Chebyshev Expansions . . . . . 485
10.11 Minimax Approximations ... 494
10.12 Discrete Minimax Approximations 503
10.13 L1-approximations 508
Bibliography . . . . . . . . . . . . 510
x Contents
Exercises ....... . 511
11 Algebraic Eigenvalue Problem 523
11.1 Introduction .. . 524
11.2 Power Method ..... . 529
11.3 Inverse Iteration. . . . . 534
11.4 Eigenvalues of a Real Symmetric Matrix 541
11.5 The QL Algorithm ........... . 549
11.6 Reduction of a Matri~ to Hessenberg Form. 553
11.7 Lanczos Method. . . . . . . . . . . . . . . . 557
11.8 QR Algorithm for a Real Hessenberg Matrix . 558
11.9 Roundoff Errors. 564
Bibliography 566
Exercises .... 567
12 Ordinary Differential Equations 573
12.1 Initial Value Problem ... 574
12.2 Stability of Numerical Integration Methods 578
12.3 Predictor-Corrector Methods 587
12.4 Runge-Kutta Methods .. . 600
12.5 Extrapolation Methods .. . 607
12.6 Stiff Differential Equations . 610
12.7 Boundary Value Problem. 618
12.8 Finite Difference Methods 626
12.9 Eigenvalue Problem .. . 633
12.10 Expansion Methods .. . 640
12.11 Some Special Techniques 642
Bibliography 647
Exercises 648
13 Integral Equations 657
13.1 Introduction ............... . 658
13.2 Fredholm Equations of the Second Kind 661
13.3 Expansion Methods .......... . 669
13.4 Eigenvalue Problem .......... . 672
13.5 Fredholm Equations of the First Kind 676
13.6 Inverse Problems ........... . 680
13.7 Volterra Equations of the Second Kind 685
13.8 Volterra Equations of the First Kind 692
Bibliography 695
Exercises .............. . 696