Ridders' Method

Ridders' method is a root bracketing algorithm for finding a root of a real-valued continuous function. It is a variation of the method of false position that uses an exponential function to obtain a faster root estimate without requiring the derivative of the function.

Suppose and . The intermediate value theorem then ensures that a root is bracketed in . Let be the midpoint. If , then is the root. Otherwise, Ridders' method multiplies by an exponential factor chosen so that the transformed values at the three equally spaced points , , and are collinear. Applying the method of false position to the transformed function and eliminating the exponential factor gives the new estimate

where is the sign function. The expression under the square root is positive, and lies in the original interval (Ridders 1979).

After evaluating , the sign-changing bracket is updated. If , take and as the new endpoints. Otherwise, if , take and . In the remaining case, take and . Exact zeros at any sampled point terminate the iteration. Thus the root remains bracketed as in bisection, while the exponential interpolation generally gives a more accurate interior point. Each iteration requires two new function evaluations, at and . Near a sufficiently smooth simple root, the local rate of convergence is quadratic or better (Ridders 1979).

For example, applying one iteration to on gives , , and , already close to the root (Ridders 1979).