Editing Legendre form (section)

== Numerical evaluation ==

The classic method of evaluation is by means of '''[[Landen's transformation|Landen's transformations]]'''. Descending Landen transformation decreases the '''modulus''' <math>\scriptstyle{k}</math> towards zero, while increasing the amplitude <math>\scriptstyle{\phi}</math>. Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude. In either limit of <math>\scriptstyle{k}</math> approaching zero or one, the integral is readily evaluated.

Most modern authors recommend evaluation in terms of the [[Carlson symmetric form]]s, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by [[Boost C++ Libraries]], [[GNU Scientific Library]]<!--see gsl/specfunc/ellint.c--> and [[Numerical Recipes]].<ref name=numerical_recipes/>