Finite Difference methods. Vol.11. Special Volume. Foundations putational Mathematics (Elsevier, 2003) (473s).pdf

Contents of Volume XI
SPECIAL VOLUME:FOUNDATIONS PUTATIONAL MATHEMATICS
GENERAL PREFACE v
On the Foundations putational Mathematics, . Baxter,
A. Iserles 3
Geometric Integration and its Applications, . Budd, . Piggott 35
Linear Programming and Condition Numbers under the Real Number
Computation Model, D. Cheung, F. Cucker, Y. Ye 141
Numerical Solution of Polynomial Systems by Homotopy Continuation
Methods, . Li 209
Chaos in Finite Difference Scheme, M. Yamaguti, Y. Maeda 305
Introduction to Partial Differential Equations and Variational Formulations
in Image Processing, G. Sapiro 383
SUBJECT INDEX 463
vii
On the Foundations putational
Mathematics
. Baxter
School of Economics, Mathematics and Statistics, Birkbeck College,
University of London, Malet Street, London WC1E 7HX, UK
A. Iserles
Department of Applied Mathematics and Theoretical Physics, University of Cambridge,
Silver Street, Cambridge CB3 9EW, UK
“...thereisnopermanentplaceintheworldforuglymathematics.”
. HARDY [1992]
1. On the place of numerical analysis in the mathematical universe
A long time ago, when younger and rasher mathematicians, we both momentarily har-
boured the ambition that one day, older and wiser, we might write a multivolume treatise
titled “On the Mathematical Foundations of Numerical Analysis”. And then it dawned
that such a creation already exists: it is called ‘a mathematics library’. Indeed, it is al-
most impossible to identify a mathematical theme, no matter how ‘pure’, that has never
influenced numerical reasoning: analysis of all kinds, of course, but also algebra, graph
theory, number theory, probability theory, differential geometry, combinatorial topol-
ogy, category theory . . . The mainspring of numerical analysis is, indeed, the entire
aquifer of mathematics, pure and applied, and this is an enduring attraction of the dis-
cipline. Good luck to those content to spend their entire mathematical existen