Condition number

Definition

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 can be defined formally as:

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

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 $f(x_{0})\neq 0$, is defined as:

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