# Condition number

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## 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, $x_0 \ne 0$, and $f(x_0) \ne 0$, 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