Author | Kress, Rainer. author |
---|---|

Title | Numerical Analysis [electronic resource] / by Rainer Kress |

Imprint | New York, NY : Springer New York : Imprint: Springer, 1998 |

Connect to | http://dx.doi.org/10.1007/978-1-4612-0599-9 |

Descript | XII, 326 p. online resource |

SUMMARY

No applied mathematician can be properly trained without some basic unยญ derstanding ofnumerical methods, Le., numerical analysis. And no scientist and engineer should be using a package program for numerical computaยญ tions without understanding the program's purpose and its limitations. This book is an attempt to provide some of the required knowledge and understanding. It is written in a spirit that considers numerical analysis not merely as a tool for solving applied problems but also as a challenging and rewarding part of mathematics. The main goal is to provide insight into numerical analysis rather than merely to provide numerical recipes. The book evolved from the courses on numerical analysis I have taught since 1971 at the University ofGottingen and may be viewed as a successor of an earlier version jointly written with Bruno Brosowski [10] in 1974. It aims at presenting the basic ideas of numerical analysis in a style as concise as possible. Its volume is scaled to a one-year course, i.e., a two-semester course, addressing second-yearstudents at a German university or advanced undergraduate or first-year graduate students at an American university

CONTENT

1 Introduction -- 2 Linear Systems -- 2.1 Examples for Systems of Equations -- 2.2 Gaussian Elimination -- 2.3 LR Decomposition -- 2.4 QR Decomposition -- Problems -- 3 Basic Functional Analysis -- 3.1 Normed Spaces -- 3.2 Scalar Products -- 3.3 Bounded Linear Operators -- 3.4 Matrix Norms -- 3.5 Completeness -- 3.6 The Banach Fixed Point Theorem -- 3.7 Best Approximation -- Problems -- 4 Iterative Methods for Linear Systems -- 4.1 Jacobi and Gaussโ{128}{148}Seidel Iterations -- 4.2 Relaxation Methods -- 4.3 Two-Grid Methods -- Problems -- 5 Ill-Conditioned Linear Systems -- 5.1 Condition Number -- 5.2 Singular Value Decomposition -- 5.3 Tikhonov Regularization -- Problems -- 6 Iterative Methods for Nonlinear Systems -- 6.1 Successive Approximations -- 6.2 Newtonโ{128}{153}s Method -- 6.3 Zeros of Polynomials -- 6.4 Least Squares Problems -- Problems -- 7 Matrix Eigenvalue Problems -- 7.1 Examples -- 7.2 Estimates for the Eigenvalues -- 7.3 The Jacobi Method -- 7.4 The QR Algorithm -- 7.5 Hessenberg Matrices -- Problems -- 8 Interpolation -- 8.1 Polynomial Interpolation -- 8.2 Trigonometric Interpolation -- 8.3 Spline Interpolation -- 8.4 Bรฉzier Polynomials -- Problems -- 9 Numerical Integration -- 9.1 Interpolatory Quadratures -- 9.2 Convergence of Quadrature Formulae -- 9.3 Gaussian Quadrature Formulae -- 9.4 Quadrature of Periodic Functions -- 9.5 Romberg Integration -- 9.6 Improper Integrals -- Problems -- 10 Initial Value Problems -- 10.1 The Picardโ{128}{148}Lindelรถf Theorem -- 10.2 Eulerโ{128}{153}s Method -- 10.3 Single-Step Methods -- 10.4 Multistep Methods -- Problems -- 11 Boundary Value Problems -- 11.1 Shooting Methods -- 11.2 Finite Difference Methods -- 11.3 The Riesz and Lax-Milgram Theorems -- 11.4 Weak Solutions -- 11.5 The Finite Element Method -- Problems -- 12 Integral Equations -- 12.1 The Riesz Theory -- 12.2 Operator Approximations -- 12.3 Nystrรถmโ{128}{153}s Method -- 12.4 The Collocation Method -- 12.5 Stability -- Problems -- References

Mathematics
Numerical analysis
Applied mathematics
Engineering mathematics
Mathematics
Numerical Analysis
Appl.Mathematics/Computational Methods of Engineering