Interactive Educational Modules in
Scientific Computing

Inverse Interpolation

This module demonstrates the inverse interpolation method for solving a nonlinear equation f(x) = 0 in one dimension. Given three approximate solution values, this method produces a new approximate solution p(0), where p is a quadratic polynomial interpolating the three approximate solution values as a function of their corresponding function values (inverse interpolation). The new approximate solution replaces one of the old ones, and the process is repeated until convergence, which is usually quite rapid.

The user selects a problem either by choosing a preset example or typing in a desired function f(x). The user can also select three starting points x or accept default values. The successive steps of the inverse interpolation method are then carried out sequentially by repeatedly clicking on NEXT or on the currently highlighted step. The current values of x and y = f(x) are indicated by bullets on the plot and are also shown numerically in the table below. At each iteration of the inverse interpolation method, a quadratic polynomial p(y) is fit to the three current values of x as a function of y, the next approximate solution is taken to be p(0), and the process is then repeated. If the starting guesses are close enough to the true solution, then the inverse interpolation method converges to it, typically with a superlinear convergence rate.

Reference: Michael T. Heath, Scientific Computing, An Introductory Survey, 2nd edition, McGraw-Hill, New York, 2002. See Section 5.5.5, especially Example 5.13.

Developers: Jeffrey Naisbitt and Michael Heath