en>Christopher.D.Tucker: typo "pole" should read "poles"

2013-06-30T16:10:11Z

typo "pole" should read "poles"

← Older revision		Revision as of 18:10, 30 June 2013
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	[~~http~~://~~apcbook~~.~~com~~.ng/~~index.php?do~~=/~~profile~~-~~446~~/~~info~~/ over the ~~counter std test] author~~ is ~~called Irwin~~. ~~Years ago we moved~~ to ~~North Dakota~~. ~~To collect cash~~ is a ~~factor that I~~'~~m totally addicted~~ to. ~~For years I've been working as a payroll clerk~~.		{{dablink\|For other senses of the word "residual" in mathematics, see [[residual (mathematics)]].}}

			Loosely speaking, a '''residual''' is the [[Error#Experimental_science\|error]] in a result. To be precise, suppose we want to find ''x'' such that

			: <math>f(x)=b.\,</math>

			Given an approximation ''x''<sub>0</sub> of ''x'', the residual is

			: <math>b - f(x_0)\,</math>

			whereas the error is

			: <math>x_0 - x.\,</math>

			If we do not know ''x'', we cannot compute the error but we can compute the residual.

			==Residual of the approximation of a function==
			Similar terminology is used dealing with
			[[differential equation\|differential]],
			[[integral equation\|integral]] and
			[[functional equation]]s.
			For the approximation
			<math>~f_{\rm a}~</math> of the solution
			<math>~f~</math> of the equation
			:<math> T(f)(x)=g(x) </math> ,
			the residual can either be the function
			: <math>~g(x)~ - ~T(f_{\rm a})(x)</math>
			or can be said to be the maximum of the norm of this difference
			: <math>\max_{x\in \mathcal X} \|g(x)-T(f_{\rm a})(x)\| </math>
			over the domain <math>\mathcal X</math>, where the function
			<math>~f_{\rm a}~</math>
			is expected to approximate the solution <math>~f~</math>,
			or some integral of a function of the difference, for example:

			: <math>~\int_{\mathcal X} \|g(x)-T(f_{\rm a})(x)\|^2~{\rm d} x.</math>

			In many cases, the smallness of the residual means that the approximation is close to the solution, i.e.,

			: <math>~\left\|\frac{f_{\rm a}(x) - f(x)}{f(x)}\right\| \ll 1.~</math>

			In these cases, the initial equation is considered as [[well-posed]]; and the residual can be considered as a measure of deviation of the approximation from the exact solution.

			==Use of residuals==

			While one does not know the exact solution, one may look for the approximation with small residual.

			Residuals appear in many areas in mathematics, from [[iterative solver]]s such as the [[generalized minimal residual method]], which seeks solutions to equations by systematically minimizing the residual.

			==External links==
			* Jonathan Richard Shewchuk. ''[http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf An Introduction to the Conjugate Gradient Method Without the Agonizing Pain]'', p. 6.

			[[Category:Numerical analysis]]

en>Tkuvho: /* The Hessian */

2012-02-07T15:01:38Z

The Hessian

New page

[http://apcbook.com.ng/index.php?do=/profile-446/info/ over the counter std test] author is called Irwin. Years ago we moved to North Dakota. To collect cash is a factor that I'm totally addicted to. For years I've been working as a payroll clerk.

Polar curve - Revision history

en>Christopher.D.Tucker: typo "pole" should read "poles"

en>Tkuvho: /* The Hessian */