An Introduction to Numerical Analysis

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Paperback | 27 November 2003

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Numerical analysis provides the theoretical foundation for the numerical algorithms we rely on to solve a multitude of computational problems in science. Based on a successful course at Oxford University, this book covers a wide range of such problems ranging from the approximation of functions and integrals to the approximate solution of algebraic, transcendental, differential and integral equations. Throughout the book, particular attention is paid to the essential qualities of a numerical algorithm - stability, accuracy, reliability and efficiency. The authors go further than simply providing recipes for solving computational problems. They carefully analyse the reasons why methods might fail to give accurate answers, or why one method might return an answer in seconds while another would take billions of years. This book is ideal as a text for students in the second year of a university mathematics course. It combines practicality regarding applications with consistently high standards of rigour.

Industry Reviews

'The book is easy to read. It flows smoothly and I would consider it to be well worth the money and a useful update to the literature.' Mathematics Today 'The book is well written and offers a good level of rigor for the intended readership ... good book for mathematics students who come armed with some background in analysis.' Eugene L. Allgower, Colorado State University 'This is an excellent book for introducing undergraduates to the fascinating discipline of numerical analysis. It is the result of many years of lectures held by the authors at the University of Oxford, a fact that explains the evident ability of the book of addressing questions frequently asked by students.' Internationale Mathematische Nachrichten 'This is a very good textbook to introduce mathematics students to numerical analysis ... The authors have made a good choice of introductory topics in numerical analysis ... The book is very carefully written and can indeed be recommended as a textbook for a course for second or third year mathematics students ... It is a pleasure to browse through this book; it is written in a pleasant style and contains many historic references to ancient and modern mathematicians with some details about their lives.' Zentralblatt MATH 'The book is well written and offers a good level of rigor for the intended readership ... This is a good book for mathematics students who come armed with some background in analysis. It is carefully written with a good level of rigor.' Eugene L. Allgower, Colorado State University, Society for Industrial and Applied Mathematics

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Non-Fiction Mathematics Calculus & Mathematical Analysis Numerical Analysis

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Non-Fiction Mathematics Calculus & Mathematical Analysis Numerical Analysis

Preface	p. vii
Solution of equations by iteration	p. 1
Introduction	p. 1
Simple iteration	p. 2
Iterative solution of equations	p. 17
Relaxation and Newton's method	p. 19
The secant method	p. 25
The bisection method	p. 28
Global behaviour	p. 29
Notes	p. 32
Exercises	p. 35
Solution of systems of linear equations	p. 39
Introduction	p. 39
Gaussian elimination	p. 44
LU factorisation	p. 48
Pivoting	p. 52
Solution of systems of equations	p. 55
Computational work	p. 56
Norms and condition numbers	p. 58
Hilbert matrix	p. 72
Least squares method	p. 74
Notes	p. 79
Exercises	p. 82
Special matrices	p. 87
Introduction	p. 87
Symmetric positive definite matrices	p. 87
Tridiagonal and band matrices	p. 93
Monotone matrices	p. 98
Notes	p. 101
Exercises	p. 102
Simultaneous nonlinear equations	p. 104
Introduction	p. 104
Simultaneous iteration	p. 106
Relaxation and Newton's method	p. 116
Global convergence	p. 123
Notes	p. 124
Exercises	p. 126
Eigenvalues and eigenvectors of a symmetric matrix	p. 133
Introduction	p. 133
The characteristic polynomial	p. 137
Jacobi's method	p. 137
The Gerschgorin theorems	p. 145
Householder's method	p. 150
Eigenvalues of a tridiagonal matrix	p. 156
The QR algorithm	p. 162
The QR factorisation revisited	p. 162
The definition of the QR algorithm	p. 164
Inverse iteration for the eigenvectors	p. 166
The Rayleigh quotient	p. 170
Perturbation analysis	p. 172
Notes	p. 174
Exercises	p. 175
Polynomial interpolation	p. 179
Introduction	p. 179
Lagrange interpolation	p. 180
Convergence	p. 185
Hermite interpolation	p. 187
Differentiation	p. 191
Notes	p. 194
Exercises	p. 195
Numerical integration-I	p. 200
Introduction	p. 200
Newton-Cotes formulae	p. 201
Error estimates	p. 204
The Runge phenomenon revisited	p. 208
Composite formulae	p. 209
The Euler-Maclaurin expansion	p. 211
Extrapolation methods	p. 215
Notes	p. 219
Exercises	p. 220
Polynomial approximation in the [infinity]-norm	p. 224
Introduction	p. 224
Normed linear spaces	p. 224
Best approximation in the [infinity]-norm	p. 228
Chebyshev polynomials	p. 241
Interpolation	p. 244
Notes	p. 247
Exercises	p. 248
Approximation in the 2-norm	p. 252
Introduction	p. 252
Inner product spaces	p. 253
Best approximation in the 2-norm	p. 256
Orthogonal polynomials	p. 259
Comparisons	p. 270
Notes	p. 272
Exercises	p. 273
Numerical integration - II	p. 277
Introduction	p. 277
Construction of Gauss quadrature rules	p. 277
Direct construction	p. 280
Error estimation for Gauss quadrature	p. 282
Composite Gauss formulae	p. 285
Radau and Lobatto quadrature	p. 287
Note	p. 288
Exercises	p. 288
Piecewise polynomial approximation	p. 292
Introduction	p. 292
Linear interpolating splines	p. 293
Basis functions for the linear spline	p. 297
Cubic splines	p. 298
Hermite cubic splines	p. 300
Basis functions for cubic splines	p. 302
Notes	p. 306
Exercises	p. 307
Initial value problems for ODEs	p. 310
Introduction	p. 310
One-step methods	p. 317
Consistency and convergence	p. 321
An implicit one-step method	p. 324
Runge-Kutta methods	p. 325
Linear multistep methods	p. 329
Zero-stability	p. 331
Consistency	p. 337
Dahlquist's theorems	p. 340
Systems of equations	p. 341
Stiff systems	p. 343
Implicit Runge-Kutta methods	p. 349
Notes	p. 353
Exercises	p. 355
Boundary value problems for ODEs	p. 361
Introduction	p. 361
A model problem	p. 361
Error analysis	p. 364
Boundary conditions involving a derivative	p. 367
The general self-adjoint problem	p. 370
The Sturm-Liouville eigenvalue problem	p. 373
The shooting method	p. 375
Notes	p. 380
Exercises	p. 381
The finite element method	p. 385
Introduction: the model problem	p. 385
Rayleigh-Ritz and Galerkin principles	p. 388
Formulation of the finite element method	p. 391
Error analysis of the finite element method	p. 397
A posteriori error analysis by duality	p. 403
Notes	p. 412
Exercises	p. 414
An overview of results from real analysis	p. 419
WWW-resources	p. 423
Bibliography	p. 424
Index	p. 429
Table of Contents provided by Ingram. All Rights Reserved.

An Introduction to Numerical Analysis

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