Figure: Fixed point iteration for the very simple case where g(x) is a linear function

of x. In this figure the liney = g(x) has been chosen to have a positive slope less than one

and its iteration started from the value x0. Similarly, the line has been chosen

to have a positive slope greater than one and its iteration started from the value . The

convergent behavior of the g(x) fixed point iteration is quite different from that

for which diverges away from the fixed point at xp. The divergence for is

caused solely by the fact that the slope of is greater than the slope of the

line y= x.

The conditions for which the fixed point iteration scheme is convergent can be

understood by inspection on the figure 1. In this case we consider the

determination of the zero of the simple function:

A straightforward choice for the function g(x) with which to do fixed point

iteration is:

This example is particularly simple since we can solve f(x) = 0 analytically

and find the fixed point of g(x), xp = ma/(m-1). It is easy to verify that g(xp)

= xp, confirming that xp is indeed a fixed point. The fixed point iteration

sequence is shown for two choices of the slope, m, both positive. The

curve y = g(x) has m < 1 and the curve has m > 1. It is clear that

the m < 1 case results in monotonic convergence to the fixed point xp, so that

the fixed point is strongly attractive in this case. The m > 1 case

illustrates monotonic divergence away from the fixed point xp, so that the

fixed point is strongly repellent in this case.

While this simple linear case may seem special, it displays the behaviour

which applies in general to a continuous mapping function, g(x). In order to

understand the reasons for the difference in behaviour for the two cases m < 1

and m > 1, we need to follow the iteration sequence in some detail. Once

given the starting value x0, we compute g(x0), the corresponding point on

the y = g(x) curve. We then move along the line to intersect

the y = x line, and there read the value ofx, and use this as the next iteration

value for x. Examination of the m < 1 iteration sequence in the figure shows

that each motion along the arrows of the iteration sequence leads towards the

intersection point of y = x and y = g(x), thus assuring convergence. A similar

examination of the m > 1 case shows that each motion along the arrows of the

iteration sequence leads away from the intersection point at xp, thus assuring

divergence. While the point xp remains a fixed point of , it is an unstable

fixed point in the sense that starting arbitrarily close to the fixed point still

results in an iterative path that leads away from the fixed point. The

termsattractor and repeller then naturally describe the fixed point xp for the

maps associated with m < 1 and m > 1 respectively.

Figure 6.3: Fixed point iteration for a general function g(x) for the four cases of

interest. Generalizations of the two cases of positive slope shown in the figure 1 are

shown on the left, and illustrate monotonic convergence and divergence. The cases

where g(x) has negative slope are shown on the right, and illustrate oscillating

convergence and divergence. The top pair of panels illustrate strong and

weak attractors, while the bottom pair of panels illustrate strong and weak repellers.

We have considered iteration functions, like g(x), which have positive slopes

in the neighborhood of the fixed point, and shown that these lead to either

monotonic convergence or monotonic divergence. When g(x) has negative

slope in the neighborhood of the fixed point, the result is oscillating

convergence or divergence, with convergence requiring |m| < 1. The iteration

sequences for all four cases are shown in the figure 1 for more general g(x).

The conditions leading to convergence are unchanged from those derived for

the linear case as long as the neighborhood of the fixed point considered is

small enough.