<sod>  
 ==Other contexts==
 
Condition numbers for [[singular-value decomposition]]s, [[polynomial]] [[root finding]], [[eigenvalue]] and many other problems may be defined.
 
Generally, if a numerical problem is well-posed, it can be expressed as
a function ''&fnof;'' mapping its data, which is an ''m''-tuple of real numbers ''x'', into its solution, an ''n''-tuple of real numbers ''&fnof;''(''x'').
 
Its condition number is then defined to be the maximum value of the ratio
of the [[Approximation error|relative errors]] in the solution to the [[relative error]] in the data, over the problem domain:
 
:<math>\max \left\{ \left| \frac{f(x) - f(x^*)}{f(x)} \right| \left/ \left| \frac{x - x^*}{x} \right| \right. : |x - x^*| < \varepsilon \right\}</math>
 
where ''&epsilon;'' is some reasonably small value in the variation of data for the problem.
 
If ''&fnof;'' is also differentiable, this is approximately
 
:<math>\left| \frac{ f'(x) }{ f(x) } \right| \cdot \left| x \right|. </math>

And the condition number of the inverse of ''&fnof;'' at ''&fnof;''(''x'') is approximately

:<math>\left| \frac{ f(x) }{ f'(x) } \right| \cdot \left| \frac{1}{x} \right|. </math>
 
 ==External links==
 
 <eod> <sod> 
 ==Other contexts==
 
Condition numbers can be defined for any function ''&fnof;'' mapping its data from some [[function domain|domain]] (e.g. an ''m''-tuple of real numbers ''x'') into some [[codomain]] [e.g. an ''n''-tuple of real numbers ''&fnof;''(''x'')], where both the domain and codomain are [[Banach space]]s.  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 [[eigenvalue]]s.  
 
The condition number of ''&fnof;'' at a point ''x'' (specifically, its '''relative condition number'''<ref name=TrefethenBau/>) is then defined to be the maximum ratio of the fractional change in ''&fnof;''(''x'') to any fractional change in ''x'', in the limit where the change &delta;''x'' in ''x'' becomes infinitesimally small:<ref name=TrefethenBau>L. N. Trefethen and D. Bau, ''Numerical Linear Algebra'' (SIAM, 1997).</ref>
 
:<math>\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}  
               \left/  \frac{\Vert \delta x \Vert }{\Vert x \Vert}  \right .
       \right]</math>
 
where <math>\Vert \cdots \Vert</math> is a [[Norm (mathematics)|norm]] on the domain/codomain of ''&fnof;''(''x'').
 
If ''&fnof;'' is differentiable, this is equivalent to:<ref name=TrefethenBau/>
 
:<math>\frac{\Vert J \Vert}{ \Vert f(x) \Vert / \Vert x \Vert}, </math>
 
where ''J'' denotes the [[Jacobian matrix]] of [[partial derivative]]s of ''&fnof;'' and <math>\Vert J \Vert</math> is the [[induced norm]] on the matrix.
 
 ==External links==
 
 <eod>