# 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*) = *x*^{2} − *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