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