PHYS 2200 Midterm: Midterm 1 Fall 2013
Document Summary
This concise and surprising exponentially converging formula for ir is used by both. As salamin points out, by 1819 gauss was in possession of the agm iteration for computing elliptic integrals of the first kind and also formula (4. 5) for computing elliptic integrals of second kind. 1811, and as watson puts it [20, p. 14] "in the hands of legendre, the transformation. [(3. 23)] became a most powerful method for computing elliptic integrals. " (see also [10], [14] and the footnotes of [ 11 ]. ) King [ 1 1, p. 39] derives (4. 6) which he attributes, in an equivalent form, to gauss. It is perhaps surprising that (4. 6) was not suggested as a practical means of calculating ir to great accuracy until recently. It is worth emphasizing the extraordinary similarity between (1. 1) which leads to linearly convergent algorithms for all the elementary functions, and (3. 1) which leads to exponentially convergent algorithms.
