Editing Carlson symmetric form (section)

==Numerical evaluation==

The duplication theorem can be used for a fast and robust evaluation of the Carlson symmetric form of elliptic integrals
and therefore also for the evaluation of Legendre-form of elliptic integrals. Let us calculate <math>R_F(x,y,z)</math>:
first, define <math>x_0=x</math>, <math>y_0=y</math> and <math>z_0=z</math>. Then iterate the series

:<math>\lambda_n=\sqrt{x_n}\sqrt{y_n}+\sqrt{y_n}\sqrt{z_n}+\sqrt{z_n}\sqrt{x_n},</math>
:<math>x_{n+1}=\frac{x_n+\lambda_n}{4}, y_{n+1}=\frac{y_n+\lambda_n}{4}, z_{n+1}=\frac{z_n+\lambda_n}{4}</math>
until the desired precision is reached: if <math>x</math>, <math>y</math> and <math>z</math> are non-negative, all of the series will converge quickly to a given value, say, <math>\mu</math>. Therefore,

:<math>R_F\left(x,y,z\right)=R_F\left(\mu,\mu,\mu\right)=\mu^{-1/2}.</math>

Evaluating <math>R_C(x,y)</math> is much the same due to the relation

:<math>R_C\left(x,y\right)=R_F\left(x,y,y\right).</math>