Classification of Fatou components - Revision history

en>Adam majewski: /* Transcendental case */ link int

2014-06-15T11:49:52Z

Transcendental case: link int

← Older revision		Revision as of 13:49, 15 June 2014
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	~~{{Context\|date=October 2009}}~~		Ed is what individuals call me and my spouse doesn't like it at all. I am really fond of handwriting but I can't make it my occupation truly. Her family members lives in Ohio but her husband desires them to move. My working day occupation is a journey agent.<br><br>Have a look at my web page [http://mybrandcp.com/xe/board_XmDx25/107997 clairvoyants]
	In [[numerical analysis]], the '''Nyström method'''<ref>{{cite journal\|last=Nyström\|first=Evert Johannes\|title=Über die praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben\|journal=Acta Mathematica\|year=1930\|volume=54\|issue=1\|pages=185–204\|doi=10.1007/BF02547521}}</ref> or '''quadrature method''' seeks the numerical solution of an [[integral equation]] by replacing the integral with a representative weighted sum. The continuous problem is broken into <math>n</math> discrete intervals, quadrature or [[numerical integration]] determines the weights and locations of representative points for the integral. The discrete problem to be solved is ~~now a [[system of linear equations]] with n equations~~ and ~~n unknowns.~~

	~~From the n solved points the function value~~ at ~~other points is interpolated consistent with the chosen quadrature method~~.
	~~Depending on the original problem and the choice~~ of ~~quadrature the problem may be ill-conditioned~~.

	~~Since the linear equations require <math>O(n^3)</math> operations~~ to ~~solve, hence high-order quadrature rules perform better because low-order quadrature rules require large <math>n</math> for a given accuracy~~. ~~[[Gaussian quadrature]]~~ is ~~normally~~ a ~~good choice for smooth, non-singular problems~~.

	~~== Discretization of the integral ==~~

	:<~~math~~>~~\int_a^b h (x) \;\mathrm d x \approx \sum_{k=1}^n w_k h (x_k)~~<~~/math~~>

	:~~where <math>w_k<~~/~~math> are the weights of the quadrature rule, and points <math>x_k<~~/~~math> are the abscissas~~.

	~~== Example ==~~

	~~Applying this to the inhomogeneous [[Fredholm equation]] of the second kind~~

	~~:<math>f (x) = \lambda u (x) - \int_a^b K (x, x') f (x') \;\mathrm d x'<~~/~~math>,~~

	~~results in~~

	~~:<math>f (x) \approx \lambda u (x) - \sum_{k=1}^n w_k K (x, x_k) f (x_k)<~~/~~math>.~~

	~~== References ==~~

	~~{{reflist}}~~
	* Leonard M. Delves & Joan Walsh (eds): ''Numerical Solution of Integral Equations'', Clarendon, Oxford, 1974.
	* Hans-Jürgen Reinhardt: ''Analysis of Approximation Methods for Differential and Integral Equations'', Springer, New York, 1985.

	~~{{DEFAULTSORT:Nystrom method}}~~
	~~[[Category:Integral equations]]~~
	~~[[Category:Numerical analysis]]~~
	~~[[Category:Numerical integration (quadrature)]~~]


	~~{{mathapplied-stub}}~~

en>Adam majewski: /* Transcendental case */ link

2014-01-28T18:21:03Z

Transcendental case: link

← Older revision		Revision as of 20:21, 28 January 2014
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	~~Andrew Berryhill~~ is ~~what his spouse enjoys to contact him~~ and ~~he completely digs that title~~. ~~Her family members life in Ohio but her husband desires them~~ to ~~move~~. ~~To perform lacross~~ is the ~~factor I love most~~ of ~~all~~. ~~Distributing production~~ is ~~how he tends to make~~ a ~~living~~.<br><br>~~my web page; [http~~://~~srncomm~~.~~com~~/~~blog~~/~~2014/08/25/relieve~~-~~that~~-~~stress-find-a-new-hobby/ free psychic reading]~~		{{Context\|date=October 2009}}
			In [[numerical analysis]], the '''Nyström method'''<ref>{{cite journal\|last=Nyström\|first=Evert Johannes\|title=Über die praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben\|journal=Acta Mathematica\|year=1930\|volume=54\|issue=1\|pages=185–204\|doi=10.1007/BF02547521}}</ref> or '''quadrature method''' seeks the numerical solution of an [[integral equation]] by replacing the integral with a representative weighted sum. The continuous problem is broken into <math>n</math> discrete intervals, quadrature or [[numerical integration]] determines the weights and locations of representative points for the integral. The discrete problem to be solved is now a [[system of linear equations]] with n equations and n unknowns.

			From the n solved points the function value at other points is interpolated consistent with the chosen quadrature method.
			Depending on the original problem and the choice of quadrature the problem may be ill-conditioned.

			Since the linear equations require <math>O(n^3)</math> operations to solve, hence high-order quadrature rules perform better because low-order quadrature rules require large <math>n</math> for a given accuracy. [[Gaussian quadrature]] is normally a good choice for smooth, non-singular problems.

			== Discretization of the integral ==

			:<math>\int_a^b h (x) \;\mathrm d x \approx \sum_{k=1}^n w_k h (x_k)</math>

			:where <math>w_k</math> are the weights of the quadrature rule, and points <math>x_k</math> are the abscissas.

			== Example ==

			Applying this to the inhomogeneous [[Fredholm equation]] of the second kind

			:<math>f (x) = \lambda u (x) - \int_a^b K (x, x') f (x') \;\mathrm d x'</math>,

			results in

			:<math>f (x) \approx \lambda u (x) - \sum_{k=1}^n w_k K (x, x_k) f (x_k)</math>.

			== References ==

			{{reflist}}
			* Leonard M. Delves & Joan Walsh (eds): ''Numerical Solution of Integral Equations'', Clarendon, Oxford, 1974.
			* Hans-Jürgen Reinhardt: ''Analysis of Approximation Methods for Differential and Integral Equations'', Springer, New York, 1985.

			{{DEFAULTSORT:Nystrom method}}
			[[Category:Integral equations]]
			[[Category:Numerical analysis]]
			[[Category:Numerical integration (quadrature)]]


			{{mathapplied-stub}}

en>Adam majewski: /* Rational case */ description

2012-08-23T20:22:22Z

Rational case: description

New page

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