Numerical Analysis: A Mathematical Introduction

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Clarendon Press, 2002 - Mathematics - 496 pages
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Numerical analysis explains why numerical computations work, or fail. This book is divided into four parts. Part I starts Part I starts with a guided tour of floating number systems and machine arithmetic. The exponential and the logarithm are constructed from scratch to present a new point ofview on questions well-known to the reader, and the needed knowledge of linear algebra is summarized. Part II starts with polynomial approximation (polynomial interpolation, mean-square approximation, splines). It then deals with Fourier series, providing the trigonometric version of least squareapproximations, and one of the most important numerical algorithms, the fast Fourier transform. Any scientific computation program spends most of its time solving linear systems or approximating the solution of linear systems, even when trying to solve non-linear systems. Part III is therefore aboutnumerical linear algebra, while Part IV treats a selection of non-linear or complex problems: resolution of linear equations and systems, ordinary differential equations, single step and multi-step schemes, and an introduction to partial differential equations. The book has been written having inmind the advanced undergraduate students in mathematics who are interested in the spice and spirit of numerical analysis. The book does not assume previous knowledge of numerical methods. It will also be useful to scientists and engineers wishing to learn what mathematics has to say about thereason why their numerical methods work - or fail.
 

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Contents

The entrance fee
1
A flavour of numerical analysis
11
Algebraic preliminaries
25
Polynomial and trigonometric approximation
47
Leastsquares approximation for polynomials
77
Splines
106
Fouriers world
133
Quadrature
165
Pythagoras world
290
Nonlinear problems
305
Nonlinear equations and systems
331
Solving differential systems
362
Singlestep schemes
385
Linear multistep schemes
414
Towards partial differential equations
439
References
479

Numerical linear algebra
205
Theoretical interlude
240
Iterations and recurrence
257

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About the author (2002)

Professor Michelle Schatzman, MAPLY, 21 Avenue Claude Bernard, UCBL, F-69622 Villeurbanne Cedex, Tel: +33 4 72 44 85 26, Fax: +33 4 74 22 80 53, Email: schatz@maply.univ-lyon1.fr

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