Editing Banach fixed-point theorem (section)

==Example==

An application of the Banach fixed-point theorem and fixed-point iteration can be used to quickly obtain an approximation of {{pi}} with high accuracy. Consider the function <math>f(x)=\sin(x)+x</math>. It can be verified that {{pi}} is a fixed point of ''f'', and that ''f'' maps the interval <math>\left[3\pi/4,5\pi/4\right]</math> to itself. Moreover, <math>f'(x)=1+\cos(x)</math>, and it can be verified that

:<math>0\leq1+\cos(x)\leq1-\frac{1}{\sqrt{2}}<1</math>

on this interval. Therefore, by an application of the [[mean value theorem]], ''f'' has a Lipschitz constant less than 1 (namely <math>1-1/\sqrt{2}</math>). Applying the Banach fixed-point theorem shows that the fixed point {{pi}} is the unique fixed point on the interval, allowing for fixed-point iteration to be used.

For example, the value 3 may be chosen to start the fixed-point iteration, as <math>3\pi/4\leq3\leq5\pi/4</math>. The Banach fixed-point theorem may be used to conclude that

: <math>\pi=f(f(f(\cdots f(3)\cdots)))).</math>

Applying ''f'' to 3 only three times already yields an expansion of {{pi}} accurate to 33 digits:

: <math>f(f(f(3)))=3.141592653589793238462643383279502\ldots\,.</math>