Fixed-Point Iteration
This module demonstrates fixed-point iteration for finding a fixed
point of a nonlinear function g(x) in one
dimension. The function is first evaluated at a chosen starting point,
then the function is repeatedly applied to its own output until the
input and output values differ by as little as desired.
The user selects a problem by choosing one of four preset functions
g(x). The user can also select any desired
starting point x. The successive steps of fixed-point iteration
are then carried out sequentially by repeatedly clicking on NEXT or on
the currently highlighted step. The resulting values of x and
g(x) are shown in the plot by bullets and
are also shown numerically in the table below. The iterates may or may
not converge to the fixed point x =
g(x), which is the intersection of the curve
g(x) and the line y = x,
and the convergence may be relatively slow (linear) or fast
(quadratic). The four examples provided, all of which are fixed-point
problems equivalent to the same nonlinear equation
f(x) = x2 − x
− 2 = 0, demonstrate nonconvergence, monotonic linear
convergence, alternating linear convergence, and quadratic convergence,
respectively.
Reference: Michael T. Heath, Scientific Computing,
An Introductory Survey, 2nd edition, McGraw-Hill, New York,
2002. See Section 5.5.2, especially Examples 5.8 and 5.9 and Figure
5.5.
Developers: Jeffrey Naisbitt and Michael Heath