Condition number: Difference between revisions

Latest revision as of 15:13, 1 May 2014

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:

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

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:

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

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:

$lim_{x \to x_{0}} | \frac{x f^{'} (x)}{f (x)} |$

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:

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

Some example functions

Function $f$ (in terms of input variable $x$ )	derivative $f^{'}$	relative logarithmic derivative $x f^{'} (x) / f (x)$	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)	$r x^{r - 1}$	$r$	$\| r \|$ (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

@@ Line 1: / Line 1: @@
-==Definition==
+==Definition for a function of one variable==
 ===For an arbitrary function of one variable===
-The condition number for a function <math>f</math> at a point <math>x_0</math> in the interior of its domain can be defined formally as:
+The condition number for a function <math>f</math> at a point <math>x_0</math> in the interior of its domain, and where <math>f</math> is a continuous function, can be defined formally as:
-<math>\lim \sup_{x \to x_0} \left| \frac{\frac{f(x) - f(x_0)}{f(x_0)}}{\frac{(x - x_0}{x_0}}\right|</math>
+<math>\lim \sup_{x \to x_0} \left| \frac{\frac{f(x) - f(x_0)}{f(x)}}{\frac{x - x_0}{x}} \right|</math>
+where <math>| \cdot |</math> denotes the [[absolute value]].
 ===For a differentiable function of one variable===
-Consider a function <math>f</math> of one variable. The condition number of <math>f</math> is defined as the [[absolute value]] of the [[relative logarithmic derivative]] of <math>f</math>. Explicitly, the condition number of <math>f</math> at a point <math>x_0</math> in the domain of <math>f</math> satisfying the conditions that the [[derivative]] <math>f'(x_0)</math> exists, <math>x_0 \ne 0</math>, and <math>f(x_0) \ne 0</math>, is defined as:
+Consider a function <math>f</math> of one variable. The condition number of <math>f</math> is defined as the [[absolute value]] of the [[relative logarithmic derivative]] of <math>f</math>. Explicitly, the condition number of <math>f</math> at a point <math>x_0</math> in the domain of <math>f</math> satisfying the conditions that the [[derivative]] <math>f'(x_0)</math> exists, <math>x_0 \ne 0</math>, and <math>f(x_0) \ne 0</math>, simplifies to:
 <math>\left|\frac{x_0 f'(x_0)}{f(x_0)}\right|</math>
+In cases where <math>f'</math> is continuous at and around <math>x_0</math>, we may be able to compute the ''limit'' of this expression to obtain that condition number in cases where <math>f(x_0) = 0</math>. Explicitly:
+<math>\lim_{x \to x_0} \left|\frac{xf'(x)}{f(x)}\right|</math>
+===For a function with one-sided derivatives===
+For a function that is not differentiable but has one-sided derivatives <math>f'_-(x_0)</math> and <math>f'_+(x_0)</math> at a point <math>x_0</math>, the condition number can be defined as:
+<math>\left|\frac{x_0 \max \{ |f'_-(x_0)|,|f'_+(x_0)| \}}{f(x_0)}\right|</math>
+==Some example functions==
+{| class="sortable" border="1"
+! Function <math>f</math> (in terms of input variable <math>x</math>) !! [[derivative]] <math>f'</math> !! [[relative logarithmic derivative]] <math>xf'(x)/f(x)</math> !! condition number (itself a function of <math>x</math>) !! limiting value as <math>x \to \infty</math>
+|-
+| [[power function]] <math>x^r</math> for some real number <math>r</math> (domain conditions apply) || <math>rx^{r-1}</math> || <math>r</math> || <math>|r|</math> (note that the condition number is in this case a ''constant'' function) || <math>|r|</math>
+|-
+| [[exponential function]] <math>e^x</math> || <math>e^x</math> || <math>x</math> || <math>|x|</math> || <math>\infty</math>
+|-
+| [[logarithm function]] <math>\ln x</math> (<math>x > 0</math>) || <math>1/x</math> || <math>1/ \ln x</math> || <math>1/|\ln x|</math> || 0
+|-
+| [[sine function]] <math>\sin x</math> || <math>\cos x</math> || <math>\frac{x \cos x}{\sin x}</math> || <math>\frac{|x \cos x|}{|\sin x|}</math> || undefined, fluctuates wildly
+|}