In numerical analysis, fixed-point iteration is a method of computing fixed points of iterated functions.
More specifically, given a function defined on real numbers with real values, and given a point in the domain of , the fixed point iteration is
This gives rise to the sequence , which it is hoped will converge to a point . If is continuous, then one can prove that the obtained is a fixed point of – i.e., .
This method is actually a sort of successive approximations method – the method of solving mathematical problems by means of a sequence of approximations that converge 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 being, 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 the calculator below does. It makes iterative calculations of x by a given formula, and stops when two successive values differ less than a given precision.
It is also worth mentioning that a function used as an example, i.e.
is the iterated function for calculating the square root of a. This is perhaps the first algorithm used for approximating the square root. 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.