"Archimedes the Numerical Analyst"
Archimedes the Numerical Analyst
This paper takes as its theme the process used by Archimedes for estimating , with variations on this theme : how Archimedes' approximations can be computed by a "double-mean" process, how the use of repeated extrapolation can give a much more accurate estimate of , and how the same "double-mean" process can be used to compute the logarithm. The paper is reprinted in :
which consists of a collection of papers about , ranging from translations of contributions from antiquity, including the work of Archimedes, to recent work on computing to a billion decimal places.
- Archimedes the Numerical Analyst , G M Phillips,
American Mathematical Monthly 88, 165-169, 1981.
Click here for a list of contents of Pi : A Source Book .
The following books refer to Archimedes the Numerical Analyst .
- Pi and the Arithmetic-Geometric Mean , J M Borwein and P B Borwein, Wiley, 1987.
This book, sub-titled A Study in Analytic Number Theory and Computational Complexity , is concerned with the close relationship between the algebraic-geometric mean iteration and the calculation of . It includes a discussion of the theory of elliptic integrals and functions, theta functions and modular functions.
- Spirals : From Theodorus to Chaos , P J Davis, A K Peters, 1993.
This book grew out of the Hedrick lectures which were presented by Philip Davis on the 75th anniversary of the MAA. It investigates iterative methods which generate spirals, including those which the author has named after the Greek mathematician Theodorus, and contains one chapter by Walter Gautschi and one by Arieh Iserles. In this book, Philip Davis humorously writes :
"I knew that Phillips had shown how, if Archimedes had been really clever in his computations of , he might have squeezed out fifteen more correct decimal digits out of the raw data that he derived from polygons of ninety-six sides."
For information about Archimedes, including a summary of the paper Archimedes the Numerical Analyst , and related work, click on Archimedes.