This article is concerned with finding scalar, real or complex roots, approximated as floating point numbers. Finding integer roots or exact algebraic roots are separate problems, whose algorithms have little in common with those discussed here. (See: Diophantine equation as for integer roots)
Finding a root of f(x) − g(x) = 0 is the same as solving the equation f(x) = g(x). Here, x is called the unknown in the equation. Conversely, any equation can take the canonical form f(x) = 0, so equation solving is the same thing as computing (or finding) a root of a function.
Numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converge towards a limit (the so-called "fixed point") which is a root. The first values of this series are initial guesses. The method computes subsequent values based on the old ones and the function f.
The behaviour of root-finding algorithms is studied in numerical analysis. Algorithms perform best when they take advantage of known characteristics of the given function. Thus an algorithm to find isolated real roots of a low-degree polynomial in one variable may bear little resemblance to an algorithm for complex roots of a "black-box" function which is not even known to be differentiable. Questions include ability to separate close roots, robustness in achieving reliable answers despite inevitable numerical errors, and rate of convergence.
Read more about Root-finding Algorithm: Finding Roots of Polynomials
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