**Graduate Studies in Mathematics**

Volume: 57;
2003;
453 pp;
Hardcover

MSC: Primary 65; 49;

Print ISBN: 978-0-8218-2953-0

Product Code: GSM/57

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

**Electronic ISBN: 978-1-4704-2103-8
Product Code: GSM/57.E**

List Price: $93.00

AMS Member Price: $74.40

MAA Member Price: $83.70

#### Supplemental Materials

# Concise Numerical Mathematics

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*Robert Plato*

This book succinctly covers the key 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 by applying them to problems in physics or engineering.

The author manages to cover the essentials 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.

#### Readership

Advanced undergraduates, graduate students, and research mathematicians interested in numerical methods; students in neighboring fields such as engineering, physics, and computer science.

#### Reviews & Endorsements

Appealing result of [the author's] endeavours … The presentation is concise … avoiding unnecessary redundancies, but nevertheless is self-contained … even instructors are offered new views and insights … the author offers many well-chosen exercises … The book really is a valuable contribution to the literature on its subject.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Concise Numerical Mathematics

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface to the English Edition xi12 free
- Preface to the German Edition xiii14 free
- Chapter 1. Interpolation by Polynomials 116 free
- §1.1. General prerequisites and Landau symbols 116
- §1.2. Existence and uniqueness of an interpolating polynomial 318
- §1.3. Neville's algorithm 621
- §1.4. Newton's interpolation formula, divided differences 823
- §1.5. The interpolation error 1126
- §1.6. Chebyshev polynomials 1429
- Additional topics and literature 1833
- Exercises 1934

- Chapter 2. Spline Functions 2338
- §2.1. Introductory remarks 2338
- §2.2. Interpolating linear spline functions 2439
- §2.3. Minimality properties of cubic spline functions 2540
- §2.4. The calculation of interpolating cubic spline functions 2742
- §2.5. Error estimates for interpolating cubic splines 3348
- Additional topics and literature 3853
- Exercises 3853

- Chapter 3. The Discrete Fourier Transform and Its Applications 4156
- Chapter 4. Solution of Linear Systems of Equations 5974
- §4.1. Triangular systems 5974
- §4.2. Gaussian elimination 6176
- §4.3. The factorization PA = LR 6681
- §4.4. LR factorization 7489
- §4.5. Cholesky factorization for positive definite matrices 7691
- §4.6. Banded matrices 7994
- §4.7. Norms and error estimates 8196
- §4.8. The factorization A = QS 91106
- Additional topics and literature 100115
- Exercises 100115

- Chapter 5. Nonlinear Systems of Equations 105120
- Chapter 6. The Numerical Integration of Functions 123138
- §6.1. Quadrature by interpolation formulas 124139
- §6.2. Special quadrature by interpolation formulas 125140
- §6.3. The error due to quadrature by interpolation 129144
- §6.4. Degree of exactness for the closed Newton–Cotes formulas, n even 132147
- §6.5. Composite Newton–Cotes formulas 137152
- §6.6. Asymptotic form of the composite trapezoidal rule 141156
- §6.7. Extrapolation methods 142157
- §6.8. Gaussian quadrature 146161
- §6.9. Appendix: Proof of the asymptotic form for the composite trapezoidal rule 155170
- Additional topics and literature 159174
- Exercises 159174

- Chapter 7. Explicit One–Step Methods for Initial Value Problems in Ordinary Differential Equations 161176
- §7.1. An existence and uniqueness theorem 162177
- §7.2. Theory of one–step methods 163178
- §7.3. One–step methods 166181
- §7.4. Analysis of round-off error 170185
- §7.5. Asymptotic expansion of the approximations 172187
- §7.6. Extrapolation methods for one–step methods 178193
- §7.7. Step size control 182197
- Additional topics and literature 186201
- Exercises 186201

- Chapter 8. Multistep Methods for Initial Value Problems of Ordinary Differential Equations 189204
- §8.1. Fundamental terms 189204
- §8.2. The global discretization error for multistep methods 192207
- §8.3. Specific linear multistep methods – preparations 201216
- §8.4. Adams method 204219
- §8.5. Nyström and Milne–Simpson methods 210225
- §8.6. BDF method 214229
- §8.7. Predictor–corrector methods 216231
- §8.8. Linear homogeneous difference equations 222237
- §8.9. Stiff differential equations 232247
- Additional topics and literature 241256
- Exercises 242257

- Chapter 9. Boundary Value Problems for Ordinary Differential Equations 247262
- Chapter 10. Jacobi, Gauss-Seidel and Relaxation Methods for the Solution of Linear Systems of Equations 281296
- §10.1. Iteration methods for the solution of linear systems of equations 281296
- §10.2. Linear fixed point iteration 282297
- §10.3. Some special classes of matrices and their properties 287302
- §10.4. The Jacobi method 289304
- §10.5. The Gauss–Seidel method 292307
- §10.6. The relaxation method and first convergence results 295310
- §10.7. The relaxation method for consistently ordered matrices 300315
- Additional topics and literature 305320
- Exercises 305320

- Chapter 11. The Conjugate Gradient and GMRES Methods 311326
- §11.1. Prerequisites 311326
- §11.2. The orthogonal residual approach (11.2) for positive definite matrices 313328
- §11.3. The CG method for positive definite matrices 316331
- §11.4. The convergence rate of the CG method 319334
- §11.5. The CG method for the normal equations 323338
- §11.6. Arnoldi process 324339
- §11.7. Realization of GMRES on the basis of the Arnoldi process 328343
- §11.8. Convergence rate of the GMRES method 333348
- §11.9. Appendix 1: Krylov subspaces 334349
- §11.10. Appendix 2: Interactive program systems with multifunctionality 335350
- Additional topics and literature 336351
- Exercises 337352

- Chapter 12. Eigenvalue Problems 339354
- §12.1. Introduction 339354
- §12.2. Perturbation theory for eigenvalue problems 339354
- §12.3. Localization of eigenvalues 343358
- §12.4. Variational formulation for eigenvalues of symmetric matrices 346361
- §12.5. Perturbation results for the eigenvalues of symmetric matrices 349364
- §12.6. Appendix: Factorization of matrices 350365
- Additional topics and literature 351366
- Exercises 351366

- Chapter 13. Numerical Methods for Eigenvalue Problems 355370
- §13.1. Introductory remarks 355370
- §13.2. Transformation to Hessenberg form 357372
- §13.3. Newton's method for the calculation of the eigenvalues of Hessenberg matrices 362377
- §13.4. The Jacobi method for the off–diagonal element reduction for symmetric matrices 366381
- §13.5. The QR algorithm 373388
- §13.6. The LR algorithm 386401
- §13.7. The vector iteration 387402
- Additional topics and literature 389404
- Exercises 390405

- Chapter 14. Peano's Error Representation 393408
- Chapter 15. Approximation Theory 401416
- §15.1. Introductory remarks 401416
- §15.2. Existence of a best approximation 402417
- §15.3. Uniqueness of a best approximation 404419
- §15.4. Approximation theory in spaces with a scalar product 408423
- §15.5. Uniform approximation of continuous functions by polynomials of maximum degree n – 1 411426
- §15.6. Applications of the alternation theorem 415430
- §15.7. Haar spaces, Chebyshev systems 417432
- Additional topics and literature 420435
- Exercises 420435

- Chapter 16. Computer Arithmetic 423438
- Bibliography 443458
- Index 449464
- Back Cover Back Cover1472