Interactive Educational Modules in
Scientific Computing

Linear Fractional Interpolation

This module demonstrates the linear fractional 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 as the zero of a rational function (linear fraction) interpolating the function values at the three approximate solution values. 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 starting points x or accept default values. The successive steps of the linear fractional interpolation method are then carried out sequentially by repeatedly clicking on NEXT or on the currently highlighted step. The current values of x and f(x) are indicated by bullets on the plot and are also shown numerically in the table below. At each iteration of the linear fractional interpolation method, a rational function is fit to the three current values of f(x), the next approximate solution is taken to be the zero of the rational function, and the process is then repeated. If the starting guesses are close enough to the true solution, then the linear fractional 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 5.5.6, especially Example 5.14.

Developers: Jeffrey Naisbitt and Michael Heath