Banach fixed-point theorem: Difference between revisions

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==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

Banach theorem allows for example fast and accurate calculation of the {{pi}} number using the trigonometric

functions which numerically are the power [[Taylor series]].

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

Because <math>\sin(\pi)=0</math> and the {{pi}} is the fixed point of for example the function

<math>f(x)=\sin(x)+x</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.

i.e.

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>f(\pi)=\pi</math>

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

and also the function <math>f</math> is around {{pi}} the contraction mapping from the obvious reasons because its derivative in {{pi}} vanishes therefore {{pi}} can be obtained from the infinite superposition for example for the argument value 3:

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

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

Already the triple superposition of this function at <math>3</math> gives {{pi}} with accuracy to 33 digits:

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