Root finding

In mathematics and computing, a root-finding algorithm is an algorithm, for finding values x such that f(x) = 0, for a given continuous function f from the real numbers to real numbers or from the complex numbers to the complex numbers. Such an x is called a root or zero of the function f. As, generally, the roots may not be described exactly, they are approximated as floating point numbers, or isolated in small 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.

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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