Interactive Educational Modules in
Scientific Computing

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) = x2x − 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