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StudyMath# Fixed-point iteration method

##### This online calculator computes fixed points of iterated functions using fixed-point iteration method (method of successive approximations)

In numerical analysis, fixed-point iteration is a method of computing fixed points of iterated functions.

More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of f, the fixed point iteration is

x_{n+1}=f(x_n), \, n=0, 1, 2, \dots

which gives rise to the sequence x_0, x_1, x_2, \dots which is hoped to converge to a point x. If f is continuous, then one can prove that the obtained x is a fixed point of f, i.e., f(x)=x.

This method actually is sort of successive approximations method, the method of solving mathematical problems by means of a sequence of approximations that converges to the solution and is constructed recursively— that is, each new approximation is calculated on the basis of the preceding approximation; the choice of the initial approximation is, to some extent, arbitrary. The method is used to approximate the roots of algebraic and transcendental equations. It is also used to prove the existence of a solution and to approximate the solutions of differential, integral, and integro-differential equations.

Usage of this method is quite simple:

- assume an approximate value for the variable (initial value)
- solve for the variable
- use the answer as the second approximate value and solve the equation again
- repeat this process until a desired precision for the variable is obtained

This is exactly what calculator below does. It makes iterative calculations of x by given formula and stops when two successive values differ less than given precision.

It is also worth to mention that function used as example, i.e.

x=\frac{1}{2}(\frac{a}{x}+x),

is the iterated function for calculating square root of a. This is perhaps the first algorithm used for approximating square root and it is known as the "Babylonian method", named after the Babylonians, or "Hero's method", named after the first-century Greek mathematician Hero of Alexandria who gave the first explicit description of the method.

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