Numerical Mathematics
Description
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.
Reviews / Votes
From the reviews:
"I found many of the examples to be quite interesting. I find no fault with any of the theoretical portions of the text. The authors are quite thorough in their discussion of the theory underlying each of the topics.This text uses MATLAB for programming the numerical codes. This is a very good choice.It contains a lot of interesting and useful information for experienced users of numerical methods."
John Strikwerda, SIAM Review 2002, Vol. 44, Issue 1, p. 160-162
"This is an excellent and modern textbook in numerical mathematics! It is primarily addressed to undergraduate students in mathematics, physics, computer science and engineering. But you will need a weekly 4 hour lecture for 3 terms lecture to teach all topics treated in this book! Well known methods as well as very new algorithms are given. The methods and their performances are demonstrated by illustrative examples and computer examples. Exercises shall help the reader tounderstand the theory and to apply it. MATLAB-software satisfies the need of user-friendliness. "The spread of numerical software presents an enrichment for the scientific community. However, the user has to make the correct choice of the method which best suits at hand. As a matter of fact, no black-box methods or algorithms exist that can effectively and accurately solve all kinds of problems." All MATLAB-programs are available by internet. ... There are a lot of numerical examples and impressing figures and very useful applications, as for instance: Regularization of a triangular grid, analysis of an electric network and of a nonlinear electrical circuit, finite difference analysis of beam bending, analysis of the buckling of a beam, free dynamic vibration of a bridge, analysis of the state equation for a real gas, solution of a nonlinear system arising from semiconductor device simulation, finite element analysis of a clamped beam, geometric reconstruction based on computer tomographies, computation of the wind action on a sailboat mast, numerical solution of blackbody radiation, compliance of arterial walls, lubrication of a slider, heat conduction in a bar, a hyperbolic model for blood flow interaction with arterial walls. It is a joy to read the book, it is carefully written and well printed. ..... In the reviewers opinion, the presented book is the best textbook in numerical mathematics edited in the last ten years."
W.H.Schmidt, Zentralblatt für Mathematik 2001, 991.38387
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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(λ) (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.
