Numerical Mathematics

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri (auth.)9783540346586, 3-540-34658-9

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 performance on examples and counterexamples which outline their pros and cons. This is done using the MATLABTM 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 MATLABTM 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.

In this second edition, the readability of pictures, tables and program headings has been improved. Several changes in the chapters on iterative methods and on polynomial approximation have also been added.

From the reviews of the first edition:

“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 to understand the theory and to apply it. MATLAB-software satisfies the need of user-friendliness. [….] In the reviewers opinion, the presented book is the best textbook in numerical mathematics edited in the last ten years.”

Zentralblatt für Mathematik 2001, 991.38387

Table of contents :
Front Matter….Pages 1-1
Foundations of Matrix Analysis….Pages 3-31
Principles of Numerical Mathematics….Pages 33-56
Front Matter….Pages 58-58
Direct Methods for the Solution of Linear Systems….Pages 59-124
Iterative Methods for Solving Linear Systems….Pages 125-182
Approximation of Eigenvalues and Eigenvectors….Pages 183-244
Front Matter….Pages 246-246
Rootfinding for Nonlinear Equations….Pages 247-284
Nonlinear Systems and Numerical Optimization….Pages 285-331
Polynomial Interpolation….Pages 333-377
Numerical Integration….Pages 379-422
Front Matter….Pages 424-424
Orthogonal Polynomials in Approximation Theory….Pages 425-478
Numerical Solution of Ordinary Differential Equations….Pages 479-538
Two-Point Boundary Value Problems….Pages 539-587
Parabolic and Hyperbolic Initial Boundary Value Problems….Pages 589-633