Mathematical background

Ridder method is an iterative scheme used to obtain one root of the equation $F(x)=0$ within an interval $[x_1,x_2]$. It assumes that $F(x_1)F(x_2)<0$. Let $x_3$ be the mid-point of the interval $[x_1,x_2]$. A new estimate of the root is $x_4$ with:

$$x_4=x_3+(x_3-x_1)\frac{{\rm sign(F(x_1)-F(x_2))F(x_3)}}{\sqrt{F(x_3)^2-F(x_1)F(x_2)}}$$

under the assumption $F(x_1)F(x_2)<0$ it may be seen that $x_4$ is guaranteed to lie within the interval $[x_1,x_2]$. As soon as $x_4$ as been determined we choose as new $[x_1,x_2]$ the interval $[x_1,x_4]$ if $F(x_1)F(x_4)<0$ or $[x_4,x_2]$ if $F(x_2)F(x_4)<0$. The convergence of this algorithm is quadratic.