[[Equation solving|Solving an equation]] {{math|1=''f''(''x'') = ''g''(''x'')}} is the same as finding the roots of the function {{math|1=''h''(''x'') = ''f''(''x'') – ''g''(''x'')}}. Thus root-finding algorithms allow solving any [[equation (mathematics)|equation]] defined by continuous functions. However, most root-finding algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists.
Most numerical root-finding methods use [[iteration]], producing a [[sequence]] of numbers that hopefully convergeconverges towards the root as aits [[Limit of a sequence|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 (mathematics)|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.
The behaviourbehavior of general root-finding algorithms is studied in [[numerical analysis]]. However, for polynomials, root-finding study belongs generally to [[computer algebra]], since algebraic properties of polynomials are fundamental for the most efficient algorithms. The efficiency of an algorithm may depend dramatically on the characteristics of the given functions. For example, many algorithms use the [[derivative]] of the input function, while others work on every [[continuous function]]. In general, numerical algorithms are not guaranteed to find all the roots of a function, so failing to find a root does not prove that there is no root. However, for [[polynomial]]s, there are specific algorithms that use algebraic properties for certifying that no root is missed, and locating the roots in separate intervals (or [[disk (mathematics)|disks]] for complex roots) that are small enough to ensure the convergence of numerical methods (typically [[Newton's method]]) to the unique root so located.
