Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-25T22:42:09.081Z Has data issue: false hasContentIssue false
  • English
  • Français

An approximation to the arithmetic-geometric mean

Published online by Cambridge University Press:  23 January 2015

G. J. O. Jameson
G. J. O. Jameson*
Affiliation:
Dept. of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF e-mail: g.jameson@lancaster.ac.uk
Get access Rights & Permissions [Opens in a new window]

Extract

Given positive numbers a > b, consider the ‘agm iteration’ given by a0 = a, b0 = b and

At each stage, the two new numbers are the arithmetic and geometric means of the previous two. It is easily seen that bn < an, (an) is decreasing, (bn) is increasing and an + 1 − bn + 1 < ½(an − bn), and hence that (an) and (bn) converge to a common limit, which is called the arithmetic-geometric mean of a and b. We will denote it by M (a, b).

Type
Articles
Information
The Mathematical Gazette , Volume 98 , Issue 541 , March 2014 , pp. 85 - 95
Copyright
Copyright © The Mathematical Association 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Lord, Nick, Recent calculations of π: the Gauss-Salamin algorithm, Math. Gaz. 76 (July 1992) pp. 231–242.CrossRefGoogle Scholar
2. Young, Robert M., On the area enclosed by the curve x4 + y4 = 1, Math. Gaz. 93 (July 2009) pp. 295–299.Google Scholar
3. Lord, Nick, Evaluating integrals using polar areas, Math. Gaz. 96 (July 2012) pp. 289–296.CrossRefGoogle Scholar
4. Newman, D. J., Rational approximation versus fast computer methods, in Lectures on approximation and value distribution, Sém, Math. Sup. 79 Presses University Montreal (1982) pp. 149–174.Google Scholar
5. Newman, D. J., A simplified version of the fast algorithm of Brent and Salamin, Math. Comp. 44 (1985) pp. 207–210, reprinted in Pi: A Source Book, Springer (1999) pp. 553-556.CrossRefGoogle Scholar
6. Whittaker, E. T. and Watson, G. N., A course of modem analysis, Cambridge University Press (1927).Google Scholar
7. Bowman, F., Introduction to elliptic junctions, Dover Publications (1961).Google Scholar
8. Borwein, J. M. and Borwein, P. B., The arithmetic-geometric mean and fast computation of elementary functions, SIAM Review 26 (1984) pp. 351–365, reprinted in Pi: A Source Book, Springer (1999) pp. 537-552.CrossRefGoogle Scholar
9. Borwein, J. M. and Borwein, P. B., Pi and the AGM, Wiley (1987).Google Scholar