A first course in numerical methods

A First Course on Numerical Methods is designed for students and researchers who seek practical knowledge of modern techniques in scientific computing. Avoiding encyclopaedic and heavily theoretical exposition, the book provides an in-depth treatment of fundamental issues and methods, the reasons behind the success and failure of numerical software, and fresh and easy-to-follow approaches and techniques. The authors focus on current methods, issues and software while providing a comprehensive theoretical foundation, enabling those who need to apply the techniques to successfully design solutions to nonstandard problems. The book also illustrates algorithms using the programming environment of MATLAB®, with the expectation that the reader will gradually become proficient in it while learning the material covered in the book. The book takes an algorithmic approach, focusing on techniques that have a high level of applicability to engineering, computer science and industrial mathematics.

Author Notes

Uri M. Ascher is Professor of Computer Science at the University of British Columbia in Vancouver, Canada. He has previously co-authored three other SIAM books as well as many research papers in the general area of numerical methods and their applications. He is a SIAM Fellow and a recipient of the CAIMS Research Prize.
Chen Greif is Associate Professor of Computer Science at the University of British Columbia in Vancouver, Canada. His research interests are in the field of scientific computing, with specialization in numerical linear algebra. He is currently an associate editor for the SIAM Journal on Scientific Computing.

List of Figures	p. xi
List of Tables	p. xix
Preface	p. xxi
1 Numerical Algorithms	p. 1
1.1 Scientific computing	p. 1
1.2 Numerical algorithms and errors	p. 3
1.3 Algorithm properties	p. 9
1.4 Exercises	p. 14
1.5 Additional notes	p. 15
2 Roundoff Errors	p. 17
2.1 The essentials	p. 17
2.2 Floating point systems	p. 21
2.3 Roundoff error accumulation	p. 26
2.4 The IEEE standard	p. 29
2.5 Exercises	p. 32
2.6 Additional notes	p. 36
3 Nonlinear Equations in One Variable	p. 39
3.1 Solving nonlinear equations	p. 39
3.2 Bisection method	p. 43
3.3 Fixed point iteration	p. 45
3.4 Newton's method and variants	p. 50
3.5 Minimizing a function in one variable	p. 55
3.6 Exercises	p. 58
3.7 Additional notes	p. 64
4 Linear Algebra Background	p. 65
4.1 Review of basic concepts	p. 65
4.2 Vector and matrix norms	p. 73
4.3 Special classes of matrices	p. 78
4.4 Singular values	p. 80
4.5 Examples	p. 83
4.6 Exercises	p. 89
4.7 Additional notes	p. 92
5 Linear Systems: Direct Methods	p. 93
5.1 Gaussian elimination and backward substitution	p. 94
5.2 LU decomposition	p. 100
5.3 Pivoting strategies	p. 105
5.4 Efficient implementation	p. 110
5.5 The Cholesky decomposition	p. 114
5.6 Sparse matrices	p. 117
5.7 Permutations and ordering strategies	p. 122
5.8 Estimating errors and the condition number	p. 127
5.9 Exercises	p. 133
5.10 Additional notes	p. 139
6 Linear Least Squares Problems	p. 141
6.1 Least squares and the normal equations	p. 141
6.2 Orthogonal transformations and QR	p. 151
6.3 Householder transformations and Gram-Schmidt orthogonalization	p. 157
6.4 Exercises	p. 163
6.5 Additional notes	p. 166
7 Linear Systems: Iterative Methods	p. 167
7.1 The need for iterative methods	p. 167
7.2 Stationary iteration and relaxation methods	p. 173
7.3 Convergence of stationary methods	p. 179
7.4 Conjugate gradient method	p. 182
7.5 *Krylov subspace methods	p. 191
7.6 *Multigrid methods	p. 204
7.7 Exercises	p. 210
7.8 Additional notes	p. 218
8 Eigenvalues and Singular Values	p. 219
8.1 The power method and variants	p. 219
8.2 Singular value decomposition	p. 229
8.3 General methods for computing eigenvalues and singular values	p. 236
8.4 Exercises	p. 245
8.5 Additional notes	p. 249
9 Nonlinear Systems and Optimization	p. 251
9.1 Newton's method for nonlinear systems	p. 251
9.2 Unconstrained optimization	p. 258
9.3 *Constrained optimization	p. 271
9.4 Exercises	p. 286
9.5 Additional notes	p. 293
10 Polynomial Interpolation	p. 295
10.1 General approximation and interpolation	p. 295
10.2 Monomial interpolation	p. 298
10.3 Lagrange interpolation	p. 302
10.4 Divided differences and Newton's form	p. 306
10.5 The error in polynomial interpolation	p. 313
10.6 Chebyshev interpolation	p. 316
10.7 Interpolating also derivative values	p. 319
10.8 Exercises	p. 323
10.9 Additional notes	p. 330
11 Piecewise Polynomial Interpolation	p. 331
11.1 The case for piecewise polynomial interpolation	p. 331
11.2 Broken line and piecewise Hermite interpolation	p. 333
11.3 Cubic spline interpolation	p. 337
11.4 Hat functions and B-splines	p. 344
11.5 Parametric curves	p. 349
11.6 *Multidimensional interpolation	p. 353
11.7 Exercises	p. 359
11.8 Additional notes	p. 363
12 Best Approximation	p. 365
12.1 Continuous least squares approximation	p. 366
12.2 Orthogonal basis functions	p. 370
12.3 Weighted least squares	p. 373
12.4 Chebyshev polynomials	p. 377
12.5 Exercises	p. 379
12.6 Additional notes	p. 382
13 Fourier Transform	p. 383
13.1 The Fourier transform	p. 383
13.2 Discrete Fourier transform and trigonometric interpolation	p. 388
13.3 Fast Fourier transform	p. 396
13.4 Exercises	p. 405
13.5 Additional notes	p. 406
14 Numerical Differentiation	p. 409
14.1 Deriving formulas using Taylor series	p. 409
14.2 Richardson extrapolation	p. 413
14.3 Deriving formulas using Lagrange polynomial interpolation	p. 415
14.4 Roundoff and data errors in numerical differentiation	p. 420
14.5 *Differentiation matrices and global derivative approximation	p. 426
14.6 Exercises	p. 434
14.7 Additional notes	p. 438
15 Numerical Integration	p. 441
15.1 Basic quadrature algorithms	p. 442
15.2 Composite numerical integration	p. 446
15.3 Gaussian quadrature	p. 454
15.4 Adaptive quadrature	p. 462
15.5 Romberg integration	p. 469
15.6 *Multidimensional integration	p. 472
15.7 Exercises	p. 475
15.8 Additional notes	p. 479
16 Differential Equations	p. 481
16.1 Initial value ordinary differential equations	p. 481
16.2 Euler's method	p. 485
16.3 Runge-Kutta methods	p. 493
16.4 Multistep methods	p. 500
16.5 Absolute stability and stiffness	p. 507
16.6 Error control and estimation	p. 515
16.7 *Boundary value ODEs	p. 520
16.8 *Partial differential equations	p. 524
16.9 Exercises	p. 531
16.10 Additional notes	p. 537
Bibliography	p. 539
Index	p. 543

Available:*

On Order

Summary

Summary

Author Notes

Table of Contents