Dickson polynomial - Revision history

en>Yobot: /* References */WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T09:53:34Z

References: WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 11:53, 5 May 2014
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	A '~~''nonlinear eigenproblem''' is a generalization of an ordinary [[Eigenvalue~~, ~~eigenvector and eigenspace\|eigenproblem]] to equations that depend [[nonlinearly]] on~~ the ~~eigenvalue~~. ~~Specifically, it refers to equations of the form:~~		I'm Clara and I live with my husband and our 3 children in Schneeberg, in the LOWER AUSTRIA south area. My hobbies are Seashell Collecting, Microscopy and College football.<br><br>Look into my web-site - [http://www.sg-ac-eintracht-berlin.de/index/users/view/id/96120 backup plugin]

	~~:<math>A(\lambda) \mathbf{x} = 0 , \~~,~~</math>~~

	~~where '''x''' is a [[vector (mathematics)\|vector]] (the nonlinear "eigenvector")~~ and ~~''A'' is a [[matrix (mathematics)\|matrix]]-valued [[function (mathematics)\|function]] of the number <math>\lambda</math> (the nonlinear "eigenvalue")~~. ~~(More generally,~~ <~~math~~>~~A(\lambda)~~<~~/math~~> ~~could be a [[linear map]], but most commonly it is a finite~~-~~dimensional, usually square, matrix.) ''A'' is usually required to be a [[holomorphic]] function of <math>\lambda</math> (in some [[domain (mathematics)\|domain]]).~~

	~~For example, an ordinary linear eigenproblem <math>B\mathbf{v} = \lambda \mathbf{v}</math>, where ''B'' is a square matrix, corresponds to <math>A(\lambda) = B~~ - ~~\lambda I</math>, where ''I'' is the [[identity matrix]].~~

	~~One common case is where ''A'' is a [[polynomial matrix]], which is called a '''polynomial eigenvalue problem'''. In particular, the specific case where the polynomial has~~ [~~[degree of a polynomial\|degree]] two is called a [[quadratic eigenvalue problem]], and can be written in the form~~:

	~~:<math>A(\lambda) \mathbf{x} = ( A_2 \lambda^2 + A_1 \lambda + A_0) \mathbf{x} = 0 , \,<~~/~~math>~~

	~~in terms of the constant square matrices ''A''<sub>0,1,2<~~/~~sub>~~. This can be converted into an ordinary linear generalized eigenproblem of twice the size by defining a new vector <math>\mathbf{y} = \lambda \mathbf{x}</math>. In terms of '''x''' and '''y''', the quadratic eigenvalue problem becomes:

	~~:<math>\begin{pmatrix}~~ -~~A_0 & 0 \\ 0 & I \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix} = \lambda~~
	~~\begin{pmatrix} A_1 & A_2 \\ I & 0 \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix}~~
	~~, </math>~~

	where ''I'' is the identity matrix. More generally, if ''A'' is a matrix polynomial of degree ''d'', then one can convert the nonlinear eigenproblem into a linear (generalized) eigenproblem of ''d'' times the size.

	~~Besides converting them to ordinary eigenproblems, which only works if ''A'' is polynomial, there are other methods of solving nonlinear eigenproblems based on the [[Jacobi~~-~~Davidson algorithm]] or based on [[Newton's method]] (related to [[inverse iteration]]).~~

	~~==References==~~
	* [[Françoise Tisseur]] and Karl Meerbergen, "The quadratic eigenvalue problem," ''SIAM Review'' '''43''' (2), 235-~~286 (2001)~~.
	* Gene H. Golub and Henk A. van der Vorst, "Eigenvalue computation in the 20th century," ''Journal of Computational and Applied Mathematics'' '''123''', 35-65 (2000).
	* Philippe Guillaume, "Nonlinear eigenproblems," ''SIAM J. Matrix. Anal. Appl.'' '''20''' (3), 575-595 (1999).
	* Axel Ruhe, "Algorithms for the nonlinear eigenvalue problem," ''SIAM Journal on Numerical Analysis'' '''10''' (4), 674-689 (1973).

	~~[[Category:Linear algebra]~~]

en>R. J. Mathar: /* References */ one more reference

2013-12-13T17:29:14Z

References: one more reference

← Older revision		Revision as of 19:29, 13 December 2013
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	~~Hi there, my name~~ is ~~Leah. Participating in region audio is anything I will hardly ever give up. My~~ [~~http://www.alexa.com/search?q=position&r=topsites_index&p=bigtop position~~] ~~is a travel agent.~~ [~~http://www.Britannica.com/search?query=Michigan Michigan~~] ~~is wherever we have been dwelling for years~~. ~~If you want~~ to ~~find out additional check out out my web page: http~~:~~//www.deboeck-incoming.com/aspnet_client/goedkoop/nike-air-max-one-143285250.htm<br><br>~~		A '''nonlinear eigenproblem''' is a generalization of an ordinary [[Eigenvalue, eigenvector and eigenspace\|eigenproblem]] to equations that depend [[nonlinearly]] on the eigenvalue. Specifically, it refers to equations of the form:

	~~Here~~ is ~~my blog post~~ - [~~http:~~//~~www~~.~~deboeck~~-~~incoming~~.~~com~~/~~aspnet_client~~/~~goedkoop~~/~~nike~~-~~air~~-~~max~~-~~one~~-~~143285250~~.~~htm nike air max one~~]		:<math>A(\lambda) \mathbf{x} = 0 , \,</math>

			where '''x''' is a [[vector (mathematics)\|vector]] (the nonlinear "eigenvector") and ''A'' is a [[matrix (mathematics)\|matrix]]-valued [[function (mathematics)\|function]] of the number <math>\lambda</math> (the nonlinear "eigenvalue"). (More generally, <math>A(\lambda)</math> could be a [[linear map]], but most commonly it is a finite-dimensional, usually square, matrix.) ''A'' is usually required to be a [[holomorphic]] function of <math>\lambda</math> (in some [[domain (mathematics)\|domain]]).

			For example, an ordinary linear eigenproblem <math>B\mathbf{v} = \lambda \mathbf{v}</math>, where ''B'' is a square matrix, corresponds to <math>A(\lambda) = B - \lambda I</math>, where ''I'' is the [[identity matrix]].

			One common case is where ''A'' is a [[polynomial matrix]], which is called a '''polynomial eigenvalue problem'''. In particular, the specific case where the polynomial has [[degree of a polynomial\|degree]] two is called a [[quadratic eigenvalue problem]], and can be written in the form:

			:<math>A(\lambda) \mathbf{x} = ( A_2 \lambda^2 + A_1 \lambda + A_0) \mathbf{x} = 0 , \,</math>

			in terms of the constant square matrices ''A''<sub>0,1,2</sub>. This can be converted into an ordinary linear generalized eigenproblem of twice the size by defining a new vector <math>\mathbf{y} = \lambda \mathbf{x}</math>. In terms of '''x''' and '''y''', the quadratic eigenvalue problem becomes:

			:<math>\begin{pmatrix} -A_0 & 0 \\ 0 & I \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix} = \lambda
			\begin{pmatrix} A_1 & A_2 \\ I & 0 \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix}
			, </math>

			where ''I'' is the identity matrix. More generally, if ''A'' is a matrix polynomial of degree ''d'', then one can convert the nonlinear eigenproblem into a linear (generalized) eigenproblem of ''d'' times the size.

			Besides converting them to ordinary eigenproblems, which only works if ''A'' is polynomial, there are other methods of solving nonlinear eigenproblems based on the [[Jacobi-Davidson algorithm]] or based on [[Newton's method]] (related to [[inverse iteration]]).

			==References==
			* [[Françoise Tisseur]] and Karl Meerbergen, "The quadratic eigenvalue problem," ''SIAM Review'' '''43''' (2), 235-286 (2001).
			* Gene H. Golub and Henk A. van der Vorst, "Eigenvalue computation in the 20th century," ''Journal of Computational and Applied Mathematics'' '''123''', 35-65 (2000).
			* Philippe Guillaume, "Nonlinear eigenproblems," ''SIAM J. Matrix. Anal. Appl.'' '''20''' (3), 575-595 (1999).
			* Axel Ruhe, "Algorithms for the nonlinear eigenvalue problem," ''SIAM Journal on Numerical Analysis'' '''10''' (4), 674-689 (1973).

			[[Category:Linear algebra]]

en>Xnn: removed merge tag

2012-07-23T15:18:10Z

removed merge tag

New page

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