Root-finding

In mathematics and computing, a root-finding algorithm is an algorithm for finding roots of continuous functions. A root of a function f, from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number x such that f(x) = 0. As, generally, the roots of a function cannot be computed exactly, nor expressed in closed form, root-finding algorithms provide approximations to roots, expressed either as floating point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound).

Solving an equation f(x) = g(x) is the same as finding the roots of the function f – g. Thus root-finding algorithms allows solving any equation defined by continuous functions. However, generally, root-finding algorithm find only some roots without any indication that all roots are found; in particularly, if no root is found, it is possible that there are roots that have not been found.

Most numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converge towards the root as a limit. They require one or more initial guesses of the root as starting values, then each iteration of the algorithm produces a successively more accurate approximation to the root. Since the iteration must be stopped at some point these methods produce an approximation to the root, not an exact solution. Many methods compute subsequent values by evaluating an auxiliary function on the preceding values. The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points, and for converging rapidly to these fixed points.

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