Graphical illustration of a fixed point iteration

External JARs

For this program to work, following libraries must be included:

• exp4j to evaluate function value
• jfreechart to draw a chart

Both JARs are included.

What is a fixed point?

In mathematics, a fixed point of a function is an element, that is mapped on itself by the function; thus $f(x)=x$ for some $x \in D(f)$. This can be illustrated by an iteration starting in a starting point $x_0$ and ending in a fixed point $x $. If the fixed point does not exist or the starting point is not chosen suitably, then the fixed point is not found.

Inputs

To illustrate fixed point iteration, these inputs must be defined:

• function $f(x)$

• starting point $x_0$

• precision $e$ representing condition $f(x)=x$ in a numerical sense $\lvert f(x)-x \rvert \leq e$

• $x_{min}$ ... minimal x-axis value

• $x_{max}$ ... maximal x-axis value

• $y_{min}$ ... minimal y-axis value

• $y_{max}$ ... maximal y-axis value

Examples

For every example, function $f(x)$, precision $e=10^{-3}$ and starting point $x_0$ is defined. Points of scale can be seen in every image.

Example 1

$$f(x)=sqrt(1-x)=\sqrt{1-x} \hspace{4em} x_0=0.2 $$

Example 2

$$f(x)=sqrt(x)=\sqrt{x} \hspace{4em} x_0=0.2 $$

Example 3

$$f(x)=(-1/4) * x \ \hat \ \ 3 +(3/4) * x+1/4=-\frac{1}{4} x^3+\frac{3}{4}x+\frac{1}{4} \hspace{4em} x_0=-1.3 $$

Example 4

$$f(x)=(-1/4) * x \ \hat \ \ 3 +(3/4) * x+1/4=-\frac{1}{4} x^3+\frac{3}{4}x+\frac{1}{4} \hspace{4em} x_0=-2.45 $$