Concise Numerical Mathematics

By: Robert Plato

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This book succinctly covers many topics of numerical methods. While it is basically a survey of the subject, it has enough depth for the student to walk away with the ability to implement the methods by writing computer programs or applying them to problems in physics or engineering. The author manages to cover the many important topics while avoiding redundancies and using well-chosen examples and exercises. The exposition is supplemented by numerous figures. Work estimates and pseudo codes are provided for many algorithms, which can be easily converted to computer programs. Topics covered include interpolation, the fast Fourier transform, iterative methods for solving systems of linear and nonlinear equations, numerical methods for solving ODEs, numerical methods for matrix eigenvalue problems, approximation theory, and computer arithmetic. In general, the author assumes only a knowledge of calculus and linear algebra. The book is suitable as a text for a first course in numerical methods for mathematics students or students in neighboring fields, such as engineering, physics, and computer science.

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Preface to the English Edition	p. xi
Preface to the German Edition	p. xiii
Interpolation by Polynomials	p. 1
General prerequisites and Landau symbols	p. 1
Existence and uniqueness of an interpolating polynomial	p. 3
Neville's algorithm	p. 6
Newton's interpolation formula, divided differences	p. 8
The interpolation error	p. 11
Chebyshev polynomials	p. 14
Additional topics and literature	p. 18
Exercises	p. 19
Spline Functions	p. 23
Introductory remarks	p. 23
Interpolating linear spline functions	p. 24
Minimality properties of cubic spline functions	p. 25
The calculation of interpolating cubic spline functions	p. 27
Error estimates for interpolating cubic splines	p. 33
Additional topics and literature	p. 38
Exercises	p. 38
The Discrete Fourier Transform and Its Applications	p. 41
Discrete Fourier transform	p. 41
Applications of the discrete Fourier transform	p. 43
Fast Fourier transform (FFT)	p. 49
Additional topics and literature	p. 56
Exercises	p. 57
Solution of Linear Systems of Equations	p. 59
Triangular systems	p. 59
Gaussian elimination	p. 61
The factorization PA = LR	p. 66
LR factorization	p. 74
Cholesky factorization for positive definite matrices	p. 76
Banded matrices	p. 79
Norms and error estimates	p. 81
The factorization A = QS	p. 91
Additional topics and literature	p. 100
Exercises	p. 100
Nonlinear Systems of Equations	p. 105
Preliminary remarks	p. 105
The one-dimensional case (N = 1)	p. 107
Banach's fixed point theorem	p. 109
Newton's method	p. 112
Additional topics and literature	p. 121
Exercises	p. 121
The Numerical Integration of Functions	p. 123
Quadrature by interpolation formulas	p. 124
Special quadrature by interpolation formulas	p. 125
The error due to quadrature by interpolation	p. 129
Degree of exactness for the closed Newton-Cotes formulas, n even	p. 132
Composite Newton-Cotes formulas	p. 137
Asymptotic form of the composite trapezoidal rule	p. 141
Extrapolation methods	p. 142
Gaussian quadrature	p. 146
Appendix: Proof of the asymptotic form for the composite trapezoidal rule	p. 155
Additional topics and literature	p. 159
Exercises	p. 159
Explicit One-Step Methods for Initial Value Problems in Ordinary Differential Equations	p. 161
An existence and uniqueness theorem	p. 162
Theory of one-step methods	p. 163
One-Step methods	p. 166
Analysis of round-off error	p. 170
Asymptotic expansion of the approximations	p. 172
Extrapolation methods for one-step methods	p. 178
Step size control	p. 182
Additional topics and literature	p. 186
Exercises	p. 186
Multistep Methods for Initial Value Problems of Ordinary Differential Equations	p. 189
Fundamental terms	p. 189
The global discretization error for multistep methods	p. 192
Specific linear multistep methods - preparations	p. 201
Adams method	p. 204
Nystrom and Milne-Simpson methods	p. 210
BDF method	p. 214
Predictor-corrector methods	p. 216
Linear homogeneous difference equations	p. 222
Stiff differential equations	p. 232
Additional topics and literature	p. 241
Exercises	p. 242
Boundary Value Problems for Ordinary Differential Equations	p. 247
Problem setting, existence, uniqueness	p. 247
Difference methods	p. 250
Galerkin methods	p. 260
Simple shooting methods	p. 274
Additional topics and literature	p. 276
Exercises	p. 277
Jacobi, Gauss-Seidel and Relaxation Methods for the Solution of Linear Systems of Equations	p. 281
Iteration methods for the solution of linear systems of equations	p. 281
Linear fixed point iteration	p. 282
Some special classes of matrices and their properties	p. 287
The Jacobi method	p. 289
The Gauss-Seidel method	p. 292
The relaxation method and first convergence results	p. 295
The relaxation method for consistently ordered matrices	p. 300
Additional topics and literature	p. 305
Exercises	p. 305
The Conjugate Gradient and GMRES Methods	p. 311
Prerequisites	p. 311
The orthogonal residual approach (11.2) for positive definite matrices	p. 313
The CG method for positive definite matrices	p. 316
The convergence rate of the CG method	p. 319
The CG method for the normal equations	p. 323
Arnoldi process	p. 324
Realization of GMRES on the basis of the Arnoldi process	p. 328
Convergence rate of the GMRES method	p. 333
Appendix 1: Krylov subspaces	p. 334
Appendix 2: Interactive program systems with multifunctionality	p. 335
Additional topics and literature	p. 336
Exercises	p. 337
Eigenvalue Problems	p. 339
Introduction	p. 339
Perturbation theory for eigenvalue problems	p. 339
Localization of eigenvalues	p. 343
Variational formulation for eigenvalues of symmetric matrices	p. 346
Perturbation results for the eigenvalues of symmetric matrices	p. 349
Appendix: Factorization of matrices	p. 350
Additional topics and literature	p. 351
Exercises	p. 351
Numerical Methods for Eigenvalue Problems	p. 355
Introductory remarks	p. 355
Transformation to Hessenberg form	p. 357
Newton's method for the calculation of the eigenvalues of Hessenberg matrices	p. 362
The Jacobi method for the off-diagonal element reduction for symmetric matrices	p. 366
The QR algorithm	p. 373
The LR algorithm	p. 386
The vector iteration	p. 387
Additional topics and literature	p. 389
Exercises	p. 390
Peano's Error Representation	p. 393
Introductory remarks	p. 393
Peano kernels	p. 394
Applications	p. 397
Additional topics and literature	p. 398
Exercises	p. 398
Approximation Theory	p. 401
Introductory remarks	p. 401
Existence of a best approximation	p. 402
Uniqueness of a best approximation	p. 404
Approximation theory in spaces with a scalar product	p. 408
Uniform approximation of continuous functions by polynomials of maximum degree n - 1	p. 411
Applications of the alternation theorem	p. 415
Haar spaces, Chebyshev systems	p. 417
Additional topics and literature	p. 420
Exercises	p. 420
Computer Arithmetic	p. 423
Number representations	p. 423
General floating point number systems	p. 424
Floating point number systems in practical applications	p. 429
Rounding, truncating	p. 432
Arithmetic in floating point number systems	p. 436
Additional topics and literature	p. 441
Bibliography	p. 443
Index	p. 449
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Concise Numerical Mathematics

At a Glance

Paperback

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