Condition number

From Calculus
Jump to: navigation, search

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