We consider some of Jon Borwein's contributions to the high-precision
computation of $\pi$ and the elementary functions, with particular
reference to the fascinating book Pi and the AGM
(Wiley, 1987) by
Jon and his brother Peter Borwein.
Here "AGM" is the arithmetic-geometric mean
, first studied by Euler, Gauss and Legendre.
Because the AGM has second-order convergence, it can be combined with
the fast multiplication algorithms of Schönhage and Strassen (1971),
Fürer (2007), or Harvey, van der Hoeven and Lecerf (2014),
to give fast algorithms for the $n$-bit computation of
$\pi$ and more generally the elementary functions $\exp$, $\log$, $\sin$, $\tan$ etc.
These algorithms run in "almost linear" time
$O(M(n)\log n)$, where $M(n)$
is the time for $n$-bit multiplication.
The talk will survey a few of the results and algorithms, from the time of
Archimedes to the present day, that were of interest to Jon. In several
cases they were either discovered or improved by him.