Numerical Analysis. A Mathematical Introduction

Looks are important and, sadly for numerical analysis, its burden of notation does not make it obviously attractive. Furthermore, it appears to be between two camps: that of the physicist and engineer who want methods and are satisfied with experimental numerics; and that of the mathematicians who seek beautiful problems. However, numerical analysis can offer the best of both worlds: motivated problems where getting a solution fast is the primary concern, and tools which are both elementary and powerful. This book is divided into four parts. Part 1 starts with a guided tour of floating number systems and machine arithmetic. Exponentials and logarithms are constructed from scratch to prescrit a new point of view on well-known questions, and the linear algebra needed to do this is also summarized. Part II starts with polynomial approximation (polynomial interpolation, mean-square approximations, splines). It then deals with Fourier series, providing the trigonometric version of least-square approximations, 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 about numerical linear algebra, while Part IV treats a selection of non-linear 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 does not assume any previous knowledge of numerical methods, and is written for advanced undergraduate students in mathematics who are interested in the spice and spirit of numerical analysis. It will also be useful to scientists and engineers wishing to learn what mathematics has to say about why their numerical methods work - or fail.