Editing Condition number (section)

=== Several variables ===

Condition numbers can be defined for any function <math>f</math> mapping its data from some [[domain of a function|domain]] (e.g. an <math>m</math>-tuple of [[real number]]s <math>x</math>) into some [[codomain]] (e.g. an <math>n</math>-tuple of real numbers <math>f(x)</math>), 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, [[Root-finding algorithms#Roots of polynomials|polynomial root finding]] or computing [[eigenvalue]]s.

The condition number of <math>f</math> at a point <math>x</math> (specifically, its '''relative condition number'''<ref name=TrefethenBau />) is then defined to be the maximum ratio of the fractional change in <math>f(x)</math> to any fractional change in <math>x</math>, in the limit where the change <math>\delta x</math> in <math>x</math> becomes infinitesimally small:<ref name=TrefethenBau>{{cite book |isbn= 978-0-89871-361-9 |first1=L. N. |last1= Trefethen |first2= D. |last2=Bau |title=Numerical Linear Algebra |publisher=SIAM |year=1997 |url=https://books.google.com/books?id=JaPtxOytY7kC&q=978-0898713619}}</ref>

: <math>\lim_{\varepsilon \to 0^+} \sup_{\|\delta x\| \leq \varepsilon} \left[ \left. \frac{\left\|f(x + \delta x) - f(x)\right\|}{\|f(x)\|} \right/ \frac{\|\delta x\|}{\|x\|} \right],</math>

where <math>\|\cdot\|</math> is a [[Norm (mathematics)|norm]] on the domain/codomain of <math>f</math>.

If <math>f</math> is differentiable, this is equivalent to:<ref name=TrefethenBau />

: <math>\frac{\|J(x)\|}{\|f(x) \| / \|x\|},</math>

where {{tmath|J(x)}} denotes the [[Jacobian matrix]] of [[partial derivative]]s of <math>f</math> at <math>x</math>, and <math>\|J(x)\|</math> is the [[induced norm]] on the matrix.