Condition number

Definition for a function of one variable

For an arbitrary function of one variable

The condition number for a function $f$ at a point $x_{0}$ in the interior of its domain, and where $f$ is a continuous function, can be defined formally as:

$\lim \sup _{x\to x_{0}}\left|{\frac {\frac {f(x)-f(x_{0})}{f(x)}}{\frac {x-x_{0}}{x}}}\right|$

where $|\cdot |$ denotes the absolute value.

For a differentiable function of one variable

Consider a function $f$ of one variable. The condition number of $f$ is defined as the absolute value of the relative logarithmic derivative of $f$. Explicitly, the condition number of $f$ at a point $x_{0}$ in the domain of $f$ satisfying the conditions that the derivative $f'(x_{0})$ exists, $x_{0}\neq 0$, and $f(x_{0})\neq 0$, simplifies to:

$\left|{\frac {x_{0}f'(x_{0})}{f(x_{0})}}\right|$

In cases where $f'$ is continuous at and around $x_{0}$, we may be able to compute the limit of this expression to obtain that condition number in cases where $f(x_{0})=0$. Explicitly:

$\lim _{x\to x_{0}}\left|{\frac {xf'(x)}{f(x)}}\right|$

For a function with one-sided derivatives

For a function that is not differentiable but has one-sided derivatives $f'_{-}(x_{0})$ and $f'_{+}(x_{0})$ at a point $x_{0}$, the condition number can be defined as:

$\left|{\frac {x_{0}\max\{|f'_{-}(x_{0})|,|f'_{+}(x_{0})|\}}{f(x_{0})}}\right|$

Some example functions

Function $f$ (in terms of input variable $x$)	derivative $f'$	relative logarithmic derivative $xf'(x)/f(x)$	condition number (itself a function of $x$)	limiting value as $x\to \infty$
power function $x^{r}$ for some real number $r$ (domain conditions apply)	$rx^{r-1}$	$r$	$|r|$ (note that the condition number is in this case a constant function)	$|r|$
exponential function $e^{x}$	$e^{x}$	$x$	$|x|$	$\infty$
logarithm function $\ln x$ ($x>0$)	$1/x$	$1/\ln x$	$1/|\ln x|$	0
sine function $\sin x$	$\cos x$	${\frac {x\cos x}{\sin x}}$	${\frac {|x\cos x|}{|\sin x|}}$	undefined, fluctuates wildly