Quarteroni / Sacco / Saleri Numerical Mathematics

2007
ISBN: 978-0-387-22750-4
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 37, 720 Seiten, eBook

Reihe: Texts in Applied Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Quarteroni, Alfio

Sacco, Riccardo

Saleri, Fausto

Inhaltsverzeichnis

Getting Started.- Foundations of Matrix Analysis.- Principles of Numerical Mathematics.- Numerical Linear Algebra.- Direct Methods for the Solution of Linear Systems.- Iterative Methods for Solving Linear Systems.- Approximation of Eigenvalues and Eigenvectors.- Around Functions and Functionals.- Rootfinding for Nonlinear Equations.- Nonlinear Systems and Numerical Optimization.- Polynomial Interpolation.- Numerical Integration.- Transforms, Differentiation and Problem Discretization.- Orthogonal Polynomials in Approximation Theory.- Numerical Solution of Ordinary Differential Equations.- Two-Point Boundary Value Problems.- Parabolic and Hyperbolic Initial Boundary Value Problems.

Leseproben

5 Approximation of Eigenvalues and Eigenvectors (p. 183)

In this chapter we deal with approximations of the eigenvalues and eigenvectors of a matrix A ¸ Cn×n. Two main classes of numerical methods exist to this purpose, partial methods, which compute the extremal eigenvalues of A (that is, those having maximum and minimum module), or global methods, which approximate the whole spectrum of A.

It is worth noting that methods which are introduced to solve the matrix eigenvalue problem are not necessarily suitable for calculating the matrix eigenvectors. For example, the power method (a partial method, see Section 5.3) provides an approximation to a particular eigenvalue/eigenvector pair. The QR method (a global method, see Section 5.5) instead computes the real Schur form of A, a canonical form that displays all the eigenvalues of A but not its eigenvectors. These eigenvectors can be computed, starting from the real Schur form of A, with an extra amount of work, as described in Section 5.8.2.

Finally, some ad hoc methods for dealing e.ectively with the special case where A is a symmetric (n × n) matrix are considered in Section 5.10.

5.1 Geometrical Location of the Eigenvalues

Since the eigenvalues of A are the roots of the characteristic polynomial pA(?) (see Section 1.7), iterative methods must be used for their approximation when n ¡Ý 5. Knowledge of eigenvalue location in the complex plane can thus be helpful in accelerating the convergence of the process.