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 $f^{\prime }(x_0)$ exists, $x_0\ne 0$, and $f(x_0)\ne 0$, simplifies to:

$\left|\frac{x_0{f}^{\prime }(x_0)}{f(x_0)}\right|$

In cases where $f^{\prime }$ 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:

$\underset{x\to x_0}{lim}\left|\frac{x{f}^{\prime }(x)}{f(x)}\right|$

For a function with one-sided derivatives

For a function that is not differentiable but has one-sided derivatives $f{\text{'}}_{-}(x_0)$ and $f{\text{'}}_{+}(x_0)$ at a point $x_0$, the condition number can be defined as:

$\left|\frac{x_{0}max\{|f{\text{'}}_{-}(x_0)|,|f{\text{'}}_{+}(x_0)|\}}{f(x_0)}\right|$

Some example functions

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