Implicit Function

In mathematics, an implicit function is a function that is defined implicitly by a relation between its argument and its value.

The implicit function theorem provides a condition under which a relation defines an implicit function. It states that if the left hand side of the equation R(x, y) = 0 is differentiable and satisfies some mild condition on its partial derivatives at some point (a, b) such that R(a, b) = 0, then it defines a function y = f(x) over some interval containing a. Geometrically, the graph defined by R(x,y) = 0 will overlap locally with the graph of some equation y = f(x).

For most implicit functions, there is no formula which define them explicitly. However various numerical methods exist for computing approximatively the value of y corresponding to any fixed value of x; this allows to find an explicit approximation to the implicit function. Most of these methods can achieve any prescribed accuracy but only a few can certify the accuracy of the result. Most are iterative and many are based on some form of Newton's method, which has the advantage of using the value computed for some x for a faster computation in a neighborhood of x.