Successive Parabolic Interpolation
This module demonstrates successive parabolic interpolation for
minimizing a nonlinear function in one dimension. Given three
approximate solution values, a new approximate solution value is given
by the minimum of a quadratic polynomial interpolating the three given
approximate solution values. The new approximate solution value
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 objective function f(x)
and initial guesses x0, x1,
x2. The steps of successive parabolic interpolation
are then carried out sequentially by repeatedly clicking on NEXT or on
the currently highlighted step. The current points x and
corresponding function values f(x) are
indicated by bullets on the plot and are also shown numerically in the
table below. At each iteration, a quadratic polynomial
p(x) is fit to the three current function
values, the next approximate solution is taken to be the minimum of the
quadratic function, and the process is then repeated. If the starting
guesses are close enough to the true minimum, then the successive
parabolic 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 6.4.2, especially Example 6.9.
Developers: Jeffrey Naisbitt and Michael Heath