Condition number - Revision history

Vipul at 15:13, 1 May 2014

2014-05-01T15:13:55Z

← Older revision		Revision as of 15:13, 1 May 2014
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	<math>\left\|\frac{x_0 \max \{ \|f'_-(x_0)\|,\|f'_+(x_0)\| \}}{f(x_0)}\right\|</math>		<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
			\|}

Vipul: /* For a differentiable function of one variable */

2014-05-01T15:03:48Z

For a differentiable function of one variable

← Older revision		Revision as of 15:03, 1 May 2014
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	<math>\left\|\frac{x_0 f'(x_0)}{f(x_0)}\right\|</math>		<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 with one-sided derivatives===

Vipul: /* For an arbitrary function of one variable */

2014-05-01T15:01:32Z

For an arbitrary function of one variable

← Older revision		Revision as of 15:01, 1 May 2014
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	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:		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]].		where <math>\| \cdot \|</math> denotes the [[absolute value]].

Vipul: /* For an arbitrary function of one variable */

2014-05-01T15:00:43Z

For an arbitrary function of one variable

← Older revision		Revision as of 15:00, 1 May 2014
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	===For an arbitrary 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_0)}}{\frac{x - x_0}{x_0}} \right\|</math>

Vipul at 14:57, 1 May 2014

2014-05-01T14:57:15Z

← Older revision		Revision as of 14:57, 1 May 2014
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	==Definition==		==Definition for a function of one variable==

	===For an arbitrary function of one variable===		===For an arbitrary function of one variable===

Vipul: /* For a differentiable function of one variable */

2014-05-01T14:56:44Z

For a differentiable function of one variable

← Older revision		Revision as of 14:56, 1 May 2014
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	===For a differentiable function of one variable===		===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>		<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>

Vipul: /* For an arbitrary function of one variable */

2014-05-01T14:50:51Z

For an arbitrary function of one variable

← Older revision		Revision as of 14:50, 1 May 2014
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	<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_0)}}{\frac{x - x_0}{x_0}} \right\|</math>

	where <math>\\| \cdot \\|</math> denotes the [[absolute value]].		where <math>\| \cdot \|</math> denotes the [[absolute value]].

	===For a differentiable function of one variable===		===For a differentiable function of one variable===

Vipul: /* For an arbitrary function of one variable */

2014-05-01T14:50:36Z

For an arbitrary function of one variable

← Older revision		Revision as of 14:50, 1 May 2014
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	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 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_0)}}{\frac{x - x_0}{x_0}} \right\|</math>

	where <math>\\| \cdot \\|</math> denotes the [[absolute value]].		where <math>\\| \cdot \\|</math> denotes the [[absolute value]].

Vipul: /* For an arbitrary function of one variable */

2014-05-01T14:49:56Z

For an arbitrary function of one variable

← Older revision		Revision as of 14:49, 1 May 2014
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	<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_0)}}{\frac{x - x_0}{x_0}} \right\\|</math>

			where <math>\\| \cdot \\|</math> denotes the [[absolute value]].

	===For a differentiable function of one variable===		===For a differentiable function of one variable===

Vipul: /* For an arbitrary function of one variable */

2014-05-01T14:49:32Z

For an arbitrary function of one variable

← Older revision		Revision as of 14:49, 1 May 2014
Line 5:		Line 5:
	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 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\~~mid~~</math>		<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>

	===For a differentiable function of one variable===		===For a differentiable function of one variable===