An introduction to numerical analysis / Endre Süli and David F. Mayers.

Format:

Book

Author/Creator:

Süli, Endre, 1956-

Contributor:

Mayers, D. F. (David Francis), 1931-

Language:

English

Subjects (All):

Numerical analysis.

Physical Description:

x, 433 pages : illustrations (some color) ; 24 cm

Place of Publication:

Cambridge ; New York : Cambridge University Press, 2003.

Summary:

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 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.

Contents:

1 Solution of equations by iteration 1

1.2 Simple iteration 2

1.3 Iterative solution of equations 17

1.4 Relaxation and Newton's method 19

1.5 The secant method 25

1.6 The bisection method 28

1.7 Global behaviour 29

2 Solution of systems of linear equations 39

2.2 Gaussian elimination 44

2.3 LU factorisation 48

2.4 Pivoting 52

2.5 Solution of systems of equations 55

2.6 Computational work 56

2.7 Norms and condition numbers 58

2.8 Hilbert matrix 72

2.9 Least squares method 74

3 Special matrices 87

3.2 Symmetric positive definite matrices 87

3.3 Tridiagonal and band matrices 93

3.4 Monotone matrices 98

4 Simultaneous nonlinear equations 104

4.2 Simultaneous iteration 106

4.3 Relaxation and Newton's method 116

4.4 Global convergence 123

5 Eigenvalues and eigenvectors of a symmetric matrix 133

5.2 The characteristic polynomial 137

5.3 Jacobi's method 137

5.4 The Gerschgorin theorems 145

5.5 Householder's method 150

5.6 Eigenvalues of a tridiagonal matrix 156

5.7 The QR algorithm 162

5.7.1 The QR factorisation revisited 162

5.7.2 The definition of the QR algorithm 164

5.8 Inverse iteration for the eigenvectors 166

5.9 The Rayleigh quotient 170

5.10 Perturbation analysis 172

6 Polynomial interpolation 179

6.2 Lagrange interpolation 180

6.3 Convergence 185

6.4 Hermite interpolation 187

6.5 Differentiation 191

7 Numerical integration-I 200

7.2 Newton-Cotes formulae 201

7.3 Error estimates 204

7.4 The Runge phenomenon revisited 208

7.5 Composite formulae 209

7.6 The Euler-Maclaurin expansion 211

7.7 Extrapolation methods 215

8 Polynomial approximation in the [infinity]-norm 224

8.2 Normed linear spaces 224

8.3 Best approximation in the [infinity]-norm 228

8.4 Chebyshev polynomials 241

8.5 Interpolation 244

9 Approximation in the 2-norm 252

9.2 Inner product spaces 253

9.3 Best approximation in the 2-norm 256

9.4 Orthogonal polynomials 259

9.5 Comparisons 270

10 Numerical integration - II 277

10.2 Construction of Gauss quadrature rules 277

10.3 Direct construction 280

10.4 Error estimation for Gauss quadrature 282

10.5 Composite Gauss formulae 285

10.6 Radau and Lobatto quadrature 287

11 Piecewise polynomial approximation 292

11.2 Linear interpolating splines 293

11.3 Basis functions for the linear spline 297

11.4 Cubic splines 298

11.5 Hermite cubic splines 300

11.6 Basis functions for cubic splines 302

12 Initial value problems for ODEs 310

12.2 One-step methods 317

12.3 Consistency and convergence 321

12.4 An implicit one-step method 324

12.5 Runge-Kutta methods 325

12.6 Linear multistep methods 329

12.7 Zero-stability 331

12.8 Consistency 337

12.9 Dahlquist's theorems 340

12.10 Systems of equations 341

12.11 Stiff systems 343

12.12 Implicit Runge-Kutta methods 349

13 Boundary value problems for ODEs 361

13.2 A model problem 361

13.3 Error analysis 364

13.4 Boundary conditions involving a derivative 367

13.5 The general self-adjoint problem 370

13.6 The Sturm-Liouville eigenvalue problem 373

13.7 The shooting method 375

14 The finite element method 385

14.1 Introduction: the model problem 385

14.2 Rayleigh-Ritz and Galerkin principles 388

14.3 Formulation of the finite element method 391

14.4 Error analysis of the finite element method 397

14.5 A posteriori error analysis by duality 403

Appendix A An overview of results from real analysis 419

Appendix B WWW-resources 423.

Notes:

Includes bibliographical references and index.

ISBN:

0521810264

0521007941

OCLC:

50525488