Duality (optimization) - Revision history

en>Skyba: /* Duality gap */

2014-12-28T00:44:57Z

Duality gap

en>Davyzhu: /* Articles */

2014-02-24T12:21:02Z

Articles

← Older revision		Revision as of 14:21, 24 February 2014
Line 1:		Line 1:
	In [~~[abstract algebra]], the '''Chien search''', named after R~~. T. ~~Chien, is a fast algorithm for determining [[Root of a function\|root]]s of [[polynomial]]s defined~~ over ~~a [[finite field]~~]. ~~The most typical use of the Chien search is in finding the roots of error-locator polynomials encountered in decoding [[Reed-Solomon code]]~~s ~~and~~ [~~[BCH code]]s.~~		Hello and welcome. My at home std testing name is [http://www.pornextras.info/user/G49Z over the counter std test] Figures Wunder. Since she was 18 she's been operating as [http://www.hooddirectory.com/how-you-can-cure-an-unpleasant-yeast-infection/ home std test] a receptionist but her promotion never comes. California is exactly where her house is but she requirements to [http://Www.Ranker.com/list/famous-people-with-herpes/celebrity-lists transfer simply] because of home std test her family members. at home std test What I love doing is taking part in baseball but I haven't produced a dime with it.<br><br>Also visit my site: [http://www.revleft.com/vb/member.php?u=160656 http://www.revleft.com/vb/member.php?u=160656]

	~~==Algorithm==~~
	~~We denote the polynomial (over the finite field GF(<math>q<~~/~~math>)) whose roots we wish to determine as:~~

	~~:<math>\ \Lambda(x) = \lambda_0 + \lambda_1 x + \lambda_2 x^2 + \cdots + \lambda_t x^t <~~/~~math>~~

	~~Conceptually, we may evaluate <math>\Lambda(\beta)<~~/~~math> for each non~~-~~zero <math>\beta<~~/~~math> in GF(<math>q</math>). Those resulting in 0 are roots of the polynomial~~.

	~~The Chien search~~ is ~~based on two observations:~~

	* Each non-zero <math>\beta</math> may be expressed as <math>\alpha^{i_\beta}</math> for some <math>i_\beta</math>, where ~~<math>\alpha</math>~~ is a [~~[primitive element (finite field)\|primitive element]] of <math>\mathrm{GF}(q)<~~/~~math>, <math>i_\beta<~~/~~math> is the power number of primitive element <math>\alpha<~~/~~math>. Thus the powers <math>\alpha^i<~~/~~math> for <math>0 \leq i < (q~~-~~1)<~~/~~math> cover the entire field (excluding the zero element)~~.

	* The following relationship exists:

	~~:<math>~~
	~~\begin{array}{lllllllllll}~~
	~~\Lambda(\alpha^i) &=& \lambda_0 &+& \lambda_1 (\alpha^i) &+& \lambda_2 (\alpha^i)^2 &+& \cdots &+& \lambda_t (\alpha^i)^~~t \\
	~~&\triangleq& \gamma_{0,i} &+& \gamma_{1,i} &+& \gamma_{2,i} &+& \cdots &+& \gamma_{t,i}~~
	~~\end{array}~~
	~~</math>~~

	~~:<math>~~
	~~\begin{array}{lllllllllll}~~
	~~\Lambda(\alpha^{i+1}) &=& \lambda_0 &+& \lambda_1 (\alpha^{i+1}) &+& \lambda_2 (\alpha^{i+1})^2 &+& \cdots &+& \lambda_t (\alpha^{i+1})^t \\~~
	~~&=& \lambda_0 &+& \lambda_1 (\alpha^i)\,\alpha &+& \lambda_2 (\alpha^i)^2\,\alpha^2 &+& \cdots &+& \lambda_t (\alpha^i)^t\,\alpha^t \\~~
	~~&=& \gamma_{0,i} &+& \gamma_{1,i}\,\alpha &+& \gamma_{2,i}\,\alpha^2 &+& \cdots &+& \gamma_{t,i}\,\alpha^t \\~~
	~~&\triangleq& \gamma_{0,i+1} &+& \gamma_{1,i+1} &+& \gamma_{2,i+1} &+& \cdots &+& \gamma_{t,i+1}~~
	~~\end{array}~~
	~~</math>~~

	~~In other words, we may define each <math>\Lambda(\alpha^i)</math> as the sum of~~ a ~~set of terms~~ <~~math~~>~~\{\gamma_{j,i} \| 0\leq j \leq t\}~~<~~/math~~>~~, from which the next set of coefficients may be derived thus~~:

	:~~<math>\ \gamma_{j,i+1} = \gamma_{j,i}\,\alpha^j<~~/~~math>~~

	~~In this way, we may start at <math>i = 0<~~/~~math> with <math>\gamma_{j,0} = \lambda_j</math>, and iterate through each value of <math>i</math> up to <math>(q-1)</math>~~. ~~If at any stage the resultant summation is zero, i~~.e.

	~~:<math>\ \sum_{j=0}^t \gamma_{j,i} = 0,</math>~~

	~~then <math>\Lambda(\alpha^i) = 0<~~/~~math> also, so <math>\alpha_i<~~/~~math> is a root. In this way, we check every element in the field.~~

	When implemented in hardware, this approach significantly reduces the complexity, as all multiplications consist of one variable and one constant, rather than two variables as in the brute-force approach.

	~~==References==~~
	*{{Citation
	~~\|first= R. T.~~
	~~\|last= Chien~~
	~~\|title= Cyclic Decoding Procedures for the Bose-Chaudhuri-Hocquenghem Codes~~
	~~\|journal= IEEE Transactions on Information Theory~~
	~~\|volume= IT-10~~
	~~\|issue= 4~~
	~~\|pages=357–363~~
	~~\|date= October 1964~~
	~~\|url=~~
	~~\|issn= 0018-9448~~
	~~\|doi=}}~~
	*{{Citation
	~~\|first= S.~~
	~~\|last= Lin~~
	~~\|first2= D~~.
	~~\|last2= Costello~~
	~~\|title= Error Control Coding: Fundamentals and Applications~~
	~~\|publisher= Prentice-Hall~~
	~~\|location= Englewood Cliffs, NJ~~
	~~\|year= 2004~~
	~~\|isbn=~~
	~~\|doi= }}~~
	*{{Citation
	~~\|last=Gill~~
	~~\|first= John~~
	~~\|title= EE387 Notes #7, Handout #28~~
	~~\|year= unknown~~
	~~\|accessdate= April 21, 2010~~
	~~\|pages=42–45~~
	~~\|publisher= Stanford University~~
	~~\|url~~= http://www.~~stanford~~.~~edu/class/ee387~~/~~handouts~~/~~notes7~~.~~pdf~~
	~~\|doi~~=}}

	~~[[Category:Error detection and correction]]~~
	~~[[Category:Finite fields]~~]

en>Tobias Bergemann: Reverted 1 edit by 182.72.118.226 (talk). (TW)

2013-12-18T11:05:52Z

Reverted 1 edit by 182.72.118.226 (talk). (TW)

← Older revision		Revision as of 13:05, 18 December 2013
Line 1:		Line 1:
	~~Hello~~. ~~Allow me introduce the author~~. ~~Her name~~ is ~~Refugia Shryock~~. ~~What I love performing~~ is ~~to gather badges but I've been using~~ over the ~~counter std test on new issues~~ ~~over~~ the ~~counter std test lately~~. ~~In her professional lifestyle she~~ is a ~~payroll clerk but she's usually~~ [~~http:~~//~~www~~.~~youporntime.com~~/~~user~~/~~KBiaggini home std test kit] needed her own business~~. ~~Her spouse~~ ~~[http~~://~~www.ddaybeauty.com~~/~~node~~/~~14766 home std test]~~ and ~~her reside in Puerto Rico but she will have~~ to ~~[http:~~/~~/Answers~~.~~yahoo~~.~~com~~/~~question~~/~~index?qid=20061008193252AA7Bbdv transfer] 1 day or another.~~<br><br>~~Have~~ a ~~look at my blog post~~: ~~home std test kit ([~~http://www.~~kindrasdesigns-testing~~.~~com~~/~~index.php?do=~~/~~profile-12578~~/~~info~~/ ~~go to this website~~])		In [[abstract algebra]], the '''Chien search''', named after R. T. Chien, is a fast algorithm for determining [[Root of a function\|root]]s of [[polynomial]]s defined over a [[finite field]]. The most typical use of the Chien search is in finding the roots of error-locator polynomials encountered in decoding [[Reed-Solomon code]]s and [[BCH code]]s.

			==Algorithm==
			We denote the polynomial (over the finite field GF(<math>q</math>)) whose roots we wish to determine as:

			:<math>\ \Lambda(x) = \lambda_0 + \lambda_1 x + \lambda_2 x^2 + \cdots + \lambda_t x^t </math>

			Conceptually, we may evaluate <math>\Lambda(\beta)</math> for each non-zero <math>\beta</math> in GF(<math>q</math>). Those resulting in 0 are roots of the polynomial.

			The Chien search is based on two observations:

			* Each non-zero <math>\beta</math> may be expressed as <math>\alpha^{i_\beta}</math> for some <math>i_\beta</math>, where <math>\alpha</math> is a [[primitive element (finite field)\|primitive element]] of <math>\mathrm{GF}(q)</math>, <math>i_\beta</math> is the power number of primitive element <math>\alpha</math>. Thus the powers <math>\alpha^i</math> for <math>0 \leq i < (q-1)</math> cover the entire field (excluding the zero element).

			* The following relationship exists:

			:<math>
			\begin{array}{lllllllllll}
			\Lambda(\alpha^i) &=& \lambda_0 &+& \lambda_1 (\alpha^i) &+& \lambda_2 (\alpha^i)^2 &+& \cdots &+& \lambda_t (\alpha^i)^t \\
			&\triangleq& \gamma_{0,i} &+& \gamma_{1,i} &+& \gamma_{2,i} &+& \cdots &+& \gamma_{t,i}
			\end{array}
			</math>

			:<math>
			\begin{array}{lllllllllll}
			\Lambda(\alpha^{i+1}) &=& \lambda_0 &+& \lambda_1 (\alpha^{i+1}) &+& \lambda_2 (\alpha^{i+1})^2 &+& \cdots &+& \lambda_t (\alpha^{i+1})^t \\
			&=& \lambda_0 &+& \lambda_1 (\alpha^i)\,\alpha &+& \lambda_2 (\alpha^i)^2\,\alpha^2 &+& \cdots &+& \lambda_t (\alpha^i)^t\,\alpha^t \\
			&=& \gamma_{0,i} &+& \gamma_{1,i}\,\alpha &+& \gamma_{2,i}\,\alpha^2 &+& \cdots &+& \gamma_{t,i}\,\alpha^t \\
			&\triangleq& \gamma_{0,i+1} &+& \gamma_{1,i+1} &+& \gamma_{2,i+1} &+& \cdots &+& \gamma_{t,i+1}
			\end{array}
			</math>

			In other words, we may define each <math>\Lambda(\alpha^i)</math> as the sum of a set of terms <math>\{\gamma_{j,i} \| 0\leq j \leq t\}</math>, from which the next set of coefficients may be derived thus:

			:<math>\ \gamma_{j,i+1} = \gamma_{j,i}\,\alpha^j</math>

			In this way, we may start at <math>i = 0</math> with <math>\gamma_{j,0} = \lambda_j</math>, and iterate through each value of <math>i</math> up to <math>(q-1)</math>. If at any stage the resultant summation is zero, i.e.

			:<math>\ \sum_{j=0}^t \gamma_{j,i} = 0,</math>

			then <math>\Lambda(\alpha^i) = 0</math> also, so <math>\alpha_i</math> is a root. In this way, we check every element in the field.

			When implemented in hardware, this approach significantly reduces the complexity, as all multiplications consist of one variable and one constant, rather than two variables as in the brute-force approach.

			==References==
			*{{Citation
			\|first= R. T.
			\|last= Chien
			\|title= Cyclic Decoding Procedures for the Bose-Chaudhuri-Hocquenghem Codes
			\|journal= IEEE Transactions on Information Theory
			\|volume= IT-10
			\|issue= 4
			\|pages=357–363
			\|date= October 1964
			\|url=
			\|issn= 0018-9448
			\|doi=}}
			*{{Citation
			\|first= S.
			\|last= Lin
			\|first2= D.
			\|last2= Costello
			\|title= Error Control Coding: Fundamentals and Applications
			\|publisher= Prentice-Hall
			\|location= Englewood Cliffs, NJ
			\|year= 2004
			\|isbn=
			\|doi= }}
			*{{Citation
			\|last=Gill
			\|first= John
			\|title= EE387 Notes #7, Handout #28
			\|year= unknown
			\|accessdate= April 21, 2010
			\|pages=42–45
			\|publisher= Stanford University
			\|url= http://www.stanford.edu/class/ee387/handouts/notes7.pdf
			\|doi=}}

			[[Category:Error detection and correction]]
			[[Category:Finite fields]]

en>Headbomb: Various citation cleanup + AWB fixes using AWB (8062)

2012-07-30T20:40:58Z

Various citation cleanup + AWB fixes using AWB (8062)

New page

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