Editing Nonlinear system (section)

==Nonlinear systems of equations==
A nonlinear system of equations consists of a set of equations in several variables such that at least one of them is not a [[linear equation]].

For a single equation of the form <math>f(x)=0,</math> many methods have been designed; see [[Root-finding algorithm]]. In the case where {{mvar|f}} is a [[polynomial]], one has a ''[[polynomial equation]]'' such as
<math>x^2 + x - 1 = 0.</math> The general root-finding algorithms apply to polynomial roots, but, generally they do not find all the roots, and when they fail to find a root, this does not imply that there is no roots. Specific methods for polynomials allow finding all roots or the [[real number|real]] roots; see [[real-root isolation]].

Solving [[systems of polynomial equations]], that is finding the common zeros of a set of several polynomials in several variables is a difficult problem for which elaborate algorithms have been designed, such as [[Gröbner base]] algorithms.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |doi-access= free }}</ref>

For the general case of system of equations formed by equating to zero several [[differentiable function]]s, the main method is [[Newton's method#Systems of equations|Newton's method]] and its variants. Generally they may provide a solution, but do not provide any information on the number of solutions.