This lecture will consider the computation of elementary functions such as exp, ln, sin, cos, and constants such as pi, using quadratically convergent algorithms based on the arithmetic-geometric mean (AGM) or Landen transformations. These are asymptotically the fastest known algorithms for such problems, and were discovered independently by Brent and Salamin. One of the most elegant algorithms is called the Gauss-Legendre algorithm because it is based on old results of Gauss and Legendre, combined with modern algorithms for multiplication and square roots.
We shall also describe the best known practical algorithm, discovered by Brent and McMillan, for the computation of Euler's constant 0.57721566... High-precision computation of Euler's constant and its continued fraction expansion is of interest because it is not yet known whether Euler's constant is rational or irrational. The algorithms are related to an (incorrect) result of Ramanujan.
Compressed postscript file of the transparencies used in the lecture