Condition number: Difference between revisions

Revision as of 14:56, 1 May 2014

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:

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

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:

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

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:

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

@@ Line 11: / Line 11: @@
 ===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>
+===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>