Ridders' method

In numerical analysis, Ridder's method is a root-finding algorithm based on the false position method and the use of an exponential function to successively approximate a root of a function f.

Ridder's method is simpler than Brent's method but usually performs about as well.

Method

Given two values of the independent variable, x₁ and x₂, which are on two different sides of the root being sought, the method begins by evaluating the function at the midpoint x₃ between the two points. One then finds the unique exponential function which, when multiplied by f, transforms the function at the three points into a straight line. The false position method is then applied to the transformed values, leading to a new value x₄, between x₁ and x₂, which can be used as one of the two bracketing values in the next step of the iteration.

The method can be summarized by the formula

$x_{4}=x_{3}+(x_{3}-x_{1}){\frac {sign[f(x_{1})-f(x_{2})]f(x_{3})}{\sqrt {f(x_{3})^{2}-f(x_{1})f(x_{2})}}}.$

References

• Press, W.H. (1992) [1988]. Numerical Recipes in C: The Art of Scientific Computing (2nd ed.). Cambridge UK: Cambridge University Press. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)