Cobweb Plots

Suppose we have a one-dimensional Picard sequence which starts at $x^{(0)}$, and then is defined by $$x^{(k+1)}=f(x^{(k)}).$$

The cobweb plot draws the line $y=x$ and the function $y=f(x)$. A fixed point is a point where the line and the function intersect. We visualize the iteration by starting at $(x^{(0)},f(x^{(0)}))=(x^{(0)},x^{(1)})$, then draw a horizontal line to $(f(x^{(0)}),f(x^{(0)}))=(x^{(1)},x^{(1)})$, then draw a vertical line to $(x^{(1)},f(x^{(1)}))=(x^{(1)},x^{(2)})$, and so on. For a convergent sequence we will see zig-zagging parallel to the axes in smaller and smaller steps to a point where the function and the line intersect.

An illustration will make this clear. The Newton iteration for the square root of $a$ is $$x^{(k+1)}=\frac12\left(x^{(k)}+\frac{a}{x^{(k)}}\right).$$ The iterations for $a=.5$ starting with $x^{(0)}=.1$ are in the cobweb plot in figure 1. $$\fbox{Insert Figure 1 here}$$ We also give R code for a general cobweb plotter with a variable number of parameters.

Figure 1: Cobweb plot for Newton Square Root Iteration </center>