Numerical mathematics is the branch of mathematics that proposes, develops, analyzes and applies methods from scientific computing to several fields including analysis, linear algebra, geometry, approximation theory, functional equations, optimization and differential equations. Other disciplines, such as physics, the natural and biological sciences, engineering, and economics and the financial sciences frequently give rise to problems that need scientific computing for their solutions.As such, numerical mathematics is the crossroad of several disciplines of great relevance in modern applied sciences, and can become a crucial tool for their qualitative and quantitative analysis.One of the purposes of this book is to provide the mathematical foundations of numerical methods, to analyze their basic theoretical properties (stability, accuracy, computational complexity) and demonstrate their performances on examples and counterexamples which outline their pros and cons. This is done using the MATLAB software environment which is user-friendly and widely adopted. Within any specific class of problems, the most appropriate scientific computing algorithms are reviewed, their theoretical analyses are carried out and the expected results are verified on a MATLAB computer implementation. Every chapter is supplied with examples, exercises and applications of the discussed theory to the solution of real-life problems.This book is addressed to senior undergraduate and graduate students with particular focus on degree courses in Engineering, Mathematics, Physics and Computer Sciences. The attention which is paid to the applications and the related development of software makes it valuable also for researchers and users of scientific computing in a large variety of professional fields.
Author(s): Alfio Quarteroni, Riccardo Sacco, Fausto Saleri
Publisher: Springer
Year: 2000
Language: English
Pages: 675
Tags: Математика;Вычислительная математика;
Preface......Page 5
Preface to the Second Edition......Page 7
Contents......Page 9
Part I Getting Started......Page 19
1.1 Vector Spaces......Page 21
1.2 Matrices......Page 23
1.3 Operations with Matrices......Page 24
1.4 Trace and Determinant of a Matrix......Page 28
1.5 Rank and Kernel of a Matrix......Page 29
1.6 Special Matrices......Page 30
1.7 Eigenvalues and Eigenvectors......Page 31
1.8 Similarity Transformations......Page 33
1.9 The Singular Value Decomposition (SVD)......Page 35
1.10 Scalar Product and Norms in Vector Spaces......Page 36
1.11 Matrix Norms......Page 40
1.12 Positive De.nite, Diagonally Dominant and M-matrices......Page 45
1.13 Exercises......Page 48
2.1 Well-posedness and Condition Number of a Problem......Page 51
2.2 Stability of Numerical Methods......Page 55
2.3 A priori and a posteriori Analysis......Page 60
2.4 Sources of Error in Computational Models......Page 61
2.5 Machine Representation of Numbers......Page 63
2.6 Exercises......Page 72
Part II Numerical Linear Algebra......Page 75
3 Direct Methods for the Solution of Linear Systems......Page 77
3.1 Stability Analysis of Linear Systems......Page 78
3.2 Solution of Triangular Systems......Page 84
3.3 The Gaussian Elimination Method (GEM) and LU Factorization......Page 88
3.4 Other Types of Factorization......Page 99
3.5 Pivoting......Page 105
3.6 Computing the Inverse of a Matrix......Page 109
3.7 Banded Systems......Page 110
3.8 Block Systems......Page 114
3.9 Sparse Matrices......Page 117
3.10 Accuracy of the Solution Achieved Using GEM......Page 124
3.11 An Approximate Computation of K(A)......Page 126
3.12 Improving the Accuracy of GEM......Page 130
3.13 Undetermined Systems......Page 132
3.14 Applications......Page 135
3.15 Exercises......Page 141
4.1 On the Convergence of Iterative Methods......Page 143
4.2 Linear Iterative Methods......Page 146
4.3 Stationary and Nonstationary Iterative Methods......Page 156
4.4 Methods Based on Krylov Subspace Iterations......Page 178
4.5 The Lanczos Method for Unsymmetric Systems......Page 188
4.6 Stopping Criteria......Page 191
4.7 Applications......Page 193
4.8 Exercises......Page 198
5.1 Geometrical Location of the Eigenvalues......Page 201
5.2 Stability and Conditioning Analysis......Page 204
5.3 The Power Method......Page 210
5.4 The QR Iteration......Page 217
5.5 The Basic QR Iteration......Page 219
5.6 The QR Method for Matrices in Hessenberg Form......Page 221
5.7 The QR Iteration with Shifting Techniques......Page 232
5.8 Computing the Eigenvectors and the SVD of a Matrix......Page 238
5.9 The Generalized Eigenvalue Problem......Page 241
5.10 Methods for Eigenvalues of Symmetric Matrices......Page 244
5.11 The Lanczos Method......Page 251
5.12 Applications......Page 254
5.13 Exercises......Page 258
Part III Around Functions and Functionals......Page 263
6 Root.nding for Nonlinear Equations......Page 265
6.1 Conditioning of a Nonlinear Equation......Page 266
6.2 A Geometric Approach to Root.nding......Page 268
6.3 Fixed-point Iterations for Nonlinear Equations......Page 278
6.4 Zeros of Algebraic Equations......Page 282
6.5 Stopping Criteria......Page 291
6.6 Post-processing Techniques for Iterative Methods......Page 293
6.7 Applications......Page 298
6.8 Exercises......Page 301
7 Nonlinear Systems and Numerical Optimization......Page 303
7.1 Solution of Systems of Nonlinear Equations......Page 304
7.2 Unconstrained Optimization......Page 316
7.3 Constrained Optimization......Page 333
7.4 Applications......Page 343
7.5 Exercises......Page 348
8.1 Polynomial Interpolation......Page 351
8.2 Newton Form of the Interpolating Polynomial......Page 357
8.3 Barycentric Lagrange Interpolation......Page 362
8.4 Piecewise Lagrange Interpolation......Page 364
8.5 Hermite-Birko. Interpolation......Page 367
8.6 Extension to the Two-Dimensional Case......Page 369
8.7 Approximation by Splines......Page 373
8.8 Splines in Parametric Form......Page 383
8.9 Applications......Page 388
8.10 Exercises......Page 393
9.1 Quadrature Formulae......Page 397
9.2 Interpolatory Quadratures......Page 399
9.3 Newton-Cotes Formulae......Page 404
9.4 Composite Newton-Cotes Formulae......Page 410
9.5 Hermite Quadrature Formulae......Page 412
9.6 Richardson Extrapolation......Page 414
9.7 Automatic Integration......Page 418
9.8 Singular Integrals......Page 424
9.9 Multidimensional Numerical Integration......Page 429
9.10 Applications......Page 435
9.11 Exercises......Page 439
Part IV Transforms, Di.erentiation and Problem Discretization......Page 441
10.1 Approximation of Functions by Generalized Fourier Series......Page 443
10.2 Gaussian Integration and Interpolation......Page 447
10.3 Chebyshev Integration and Interpolation......Page 451
10.4 Legendre Integration and Interpolation......Page 454
10.5 Gaussian Integration over Unbounded Intervals......Page 456
10.6 Programs for the Implementation of Gaussian Quadratures......Page 457
10.7 Approximation of a Function in the Least-Squares Sense......Page 459
10.8 The Polynomial of Best Approximation......Page 461
10.9 Fourier Trigonometric Polynomials......Page 463
10.10 Approximation of Function Derivatives......Page 470
10.11 Transforms and Their Applications......Page 478
10.12 The Wavelet Transform......Page 486
10.13 Applications......Page 490
10.14 Exercises......Page 494
11.1 The Cauchy Problem......Page 497
11.2 One-Step Numerical Methods......Page 500
11.3 Analysis of One-Step Methods......Page 501
11.4 Di.erence Equations......Page 510
11.5 Multistep Methods......Page 515
11.6 Analysis of Multistep Methods......Page 520
11.7 Predictor-Corrector Methods......Page 529
11.8 Runge-Kutta (RK) Methods......Page 536
11.9 Systems of ODEs......Page 544
11.10 Sti. Problems......Page 546
11.11 Applications......Page 548
11.12 Exercises......Page 554
12.1 A Model Problem......Page 557
12.2 Finite Di.erence Approximation......Page 559
12.3 The Spectral Collocation Method......Page 568
12.4 The Galerkin Method......Page 570
12.5 Advection-Di.usion Equations......Page 586
12.6 A Quick Glance at the Two-Dimensional Case......Page 598
12.7 Applications......Page 601
12.8 Exercises......Page 604
13.1 The Heat Equation......Page 607
13.2 Finite Di.erence Approximation of the Heat Equation......Page 609
13.3 Finite Element Approximation of the Heat Equation......Page 611
13.4 Space-Time Finite Element Methods for the Heat Equation......Page 619
13.5 Hyperbolic Equations: A Scalar Transport Problem......Page 622
13.6 Systems of Linear Hyperbolic Equations......Page 625
13.7 The Finite Di.erence Method for Hyperbolic Equations......Page 627
13.8 Analysis of Finite Di.erence Methods......Page 629
13.9 Dissipation and Dispersion......Page 636
13.10 Finite Element Approximation of Hyperbolic Equations......Page 642
13.11 Applications......Page 648
13.12 Exercises......Page 650
References......Page 653
Index of MATLAB Programs......Page 663
Index......Page 667
