Other Contexts
Condition numbers can be defined for any function ƒ mapping its data from some domain (e.g. an m-tuple of real numbers x) into some codomain, where both the domain and codomain are Banach spaces. They express how sensitive that function is to small changes (or small errors) in its arguments. This is crucial in assessing the sensitivity and potential accuracy difficulties of numerous computational problems, for example polynomial root finding or computing eigenvalues.
The condition number of ƒ at a point x (specifically, its relative condition number) is then defined to be the maximum ratio of the fractional change in ƒ(x) to any fractional change in x, in the limit where the change δx in x becomes infinitesimally small:
![\lim_{ \varepsilon \to 0^+ } \sup_{ \Vert \delta x \Vert \leq \varepsilon } \left[ \frac{ \left\Vert f(x + \delta x) - f(x)\right\Vert }{ \Vert f(x) \Vert } / \frac{ \Vert \delta x \Vert }{ \Vert x \Vert } \right],](http://upload.wikimedia.org/math/f/b/e/fbe98d08a1671533b857bacb2091cd90.png)
where is a norm on the domain/codomain of ƒ(x).
If ƒ is differentiable, this is equivalent to:
where J denotes the Jacobian matrix of partial derivatives of ƒ and is the induced norm on the matrix.
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