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en>R'n'B: Disambiguating links to Backlog (link removed) using DisamAssist.

2014-08-13T19:28:35Z

Disambiguating links to Backlog (link removed) using DisamAssist.

← Older revision		Revision as of 21:28, 13 August 2014
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	Nice to satisfy you, my name is ~~Refugia~~. He is ~~truly fond of performing ceramics but~~ he is ~~having difficulties to find time~~ for ~~it. Her family members life in Minnesota~~. Managing people ~~has been~~ his ~~working day job for a while~~.<br><br>~~Feel free to surf to~~ my web-site :~~: at home std testing (~~[http://~~www~~.~~animecontent~~.~~com~~/~~blog~~/~~436250 visit the following web site~~])		Nice to satisfy you, my name is Figures Held although I don't truly like being called like that. To play baseball is the hobby he will never stop doing. Minnesota is exactly where he's been living for many years. Managing people is his profession.<br><br>Review my web-site: [http://netwk.hannam.ac.kr/xe/data_2/38191 at home std testing]

en>P199: unreferenced

2014-02-24T18:10:46Z

unreferenced

← Older revision		Revision as of 20:10, 24 February 2014
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	~~In [[mathematics]], in particular [[linear algebra]], the '''Sherman–Morrison formula''',<ref name="SM1949">{{cite journal~~		Nice to satisfy you, my name is Refugia. He is truly fond of performing ceramics but he is having difficulties to find time for it. Her family members life in Minnesota. Managing people has been his working day job for a while.<br><br>Feel free to surf to my web-site :: at home std testing ([http://www.animecontent.com/blog/436250 visit the following web site])
	~~\|first=Jack \|last=Sherman~~
	~~\|first2=Winifred J. \|last2=Morrison~~
	~~\|title=Adjustment of an Inverse Matrix Corresponding~~ to ~~Changes in the Elements of a Given Column or a Given Row of the Original Matrix (abstract)~~
	~~\|journal=Annals of Mathematical Statistics~~
	~~\|volume=20 \|pages=621 \|year=1949~~
	~~\|doi=10.1214/aoms/1177729959 }}</ref><ref name="SM1950">{{cite journal~~
	~~\|first=Jack \|last=Sherman~~
	~~\|first2=Winifred J. \|last2=Morrison~~
	~~\|title=Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix~~
	~~\|journal=[[Annals of Mathematical Statistics]]~~
	~~\|volume=21 \|issue=1 \|pages=124–127 \|year=1950~~
	~~\|doi=10.1214/aoms/1177729893 \|mr=35118 \| zbl=0037.00901 }}</ref><ref name="PFTV1992">{{Citation~~
	~~\|last1=Press \|first1=William H.~~
	~~\|last2=Teukolsky \|first2=Saul A.~~
	~~\|last3=Vetterling \|first3=William T.~~
	~~\|last4=Flannery \|first4=Brian P.~~
	~~\|title=Numerical Recipes: The Art of Scientific Computing \|edition=3rd \|year=2007 \|publisher=Cambridge University Press \|publication-place=New York~~
	~~\|isbn=978-0-521-88068-8~~
	~~\|chapter=Section 2.7.1 Sherman–Morrison Formula~~
	~~\|chapter-url=http://apps.nrbook.com/empanel/index.html?pg=76 }}</ref> named after Jack Sherman and Winifred J. Morrison~~, ~~computes the inverse of the sum of an [[inverse matrix\|invertible]] [[matrix (mathematics)\|matrix]] <math>A</math>~~
	and the [[outer product]], <math>u v^T</math>, of [[vector (mathematics)\|vectors]] <math>u</math> and <math>v</math>. The Sherman–Morrison formula is a special case of the [[Woodbury matrix identity\|Woodbury formula]].
	~~Though named after Sherman and Morrison, it appeared already in earlier publications.<ref~~ name~~="hager">{{cite journal~~
	~~\|first=William W. \|last=Hager~~
	~~\|title=Updating the inverse of a matrix~~
	~~\|journal=SIAM Review~~
	~~\|volume=31 \|year=1989 \|pages=221–239 \|issue=2~~
	~~\|doi=10.1137/1031049 \|mr=997457 \|jstor=2030425 }}</ref>~~

	~~== Statement ==~~
	~~Suppose <math>A</math>~~ is an [[invertible]] [[square matrix]] and <math>u</math>, <math>v</math> are [[Vector (geometric)\|vectors]]. Suppose furthermore that <math>1 + v^T A^{-1}u \neq 0</math>. Then the Sherman–Morrison formula states that
	~~:<math>(A+uv^T)^{-1} = A^{-1} - {A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u}~~.~~</math>~~

	~~Here, <math>uv^T</math>~~ is ~~the [[outer product]]~~ of ~~two vectors <math>u</math> and <math>v</math>. The general form shown here~~ is ~~the one published by Bartlett.<ref name="B1951">{{cite journal~~
	~~\|first=Maurice S~~. ~~\|last=Bartlett~~
	~~\|title=An Inverse Matrix Adjustment Arising~~ in ~~Discriminant Analysis~~
	~~\|journal=[[Annals of Mathematical Statistics]]~~
	~~\|volume=22 \|issue=1 \|pages=107–111 \|year=1951~~
	~~\|doi=10~~.~~1214/aoms/1177729698 \|mr=40068 \| zbl = 0042~~.~~38203~~
	}}<~~/ref~~>

	~~==Application==~~

	~~If the inverse of~~ <~~math~~>~~A</math> is already known, the formula provides a~~
	~~[[Computational complexity theory\|numerically]] [[Computationally expensive\|cheap]] way~~
	to ~~compute the inverse of <math>A</math> corrected by the matrix <math>uv^T</math>~~
	~~(depending on the point of view, the correction may be seen as a~~
	~~[[perturbation theory\|perturbation]] or as a [[rank (linear algebra)\|rank]]-1 update).~~
	~~The computation is relatively cheap because the inverse of <math>A+uv^T</math>~~
	~~does not have~~ to ~~be computed from scratch (which in general is expensive),~~
	~~but can be computed by correcting (or perturbing) <math>A^{~~-~~1}</math>.~~

	~~Using [[unit column]]s~~ (~~columns from the [[identity matrix]]) for <math>u</math> or <math>v</math>, individual columns or rows of <math>A</math> may be manipulated~~
	and a correspondingly updated inverse computed relatively cheaply in this way.<ref>Langville, Amy N.; and Meyer, Carl D.; "Google's PageRank and Beyond: The Science of Search Engine Rankings", Princeton University Press, 2006, p. 156</ref> In the general case, where <math>A^{-1}</math> is a <math>n</math> times <math>n</math> matrix
	and <math>u</math> and <math>v</math> are arbitrary vectors of dimension <math>n</math>, the whole matrix is updated<ref name="B1951" /> and the computation takes <math>3n^2</math> scalar multiplications.<ref>[http://www.~~alglib~~.~~net/matrixops~~/~~general~~/~~invupdate.php Update of the inverse matrix by~~ the ~~Sherman–Morrison formula~~]</ref> If <math>u</math> is a unit column, the computation takes only <math>2n^2</math> scalar multiplications. The same goes if <math>v</math> is a unit column. If both <math>u</math> and <math>v</math> are unit columns, the computation takes only <math>n^2</math> scalar multiplications.

	~~==Verification==~~

	~~We verify the properties of the inverse.~~
	~~A matrix <math>Y</math> (in this case the right-hand side of the Sherman–Morrison formula)~~
	~~is the inverse of a matrix <math>X</math> (in this case <math>A+uv^T</math>)~~
	~~if and only if <math>XY = YX = I</math>.~~

	~~We first verify that the right hand side (<math>Y</math>) satisfies <math>XY = I</math>.~~
	~~:<math>XY = (A + uv^T)\left( A^{-1} - {A^{-1} uv^T A^{-1} \over 1 + v^T A^{-1}u}\right)</math>~~

	~~::<math>= AA^{-1} + uv^T A^{-1} - {AA^{-1}uv^T A^{-1} + uv^T A^{-1}uv^T A^{-1} \over 1 + v^TA^{-1}u}</math>~~

	~~::<math>= I + uv^T A^{-1} - {uv^T A^{-1} + uv^T A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u}</math>~~

	~~::<math>= I + uv^T A^{-1} - {u(1 + v^T A^{-1}u) v^T A^{-1} \over 1 + v^T A^{-1}u}</math>~~
	~~Note that <math>v^T A^{-1}u</math> is a scalar, so <math>(1+v^T A^{-1}u)</math> can be factored out, leading to:~~
	~~:<math>XY= I + uv^T A^{-1} - uv^T A^{-1} = I.\,</math>~~

	~~In the same way, it is verified that~~

	~~:<math>YX = \left(A^{-1} - {A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u}\right)(A + uv^T~~) ~~= I.</math>~~

	~~Following is an alternate verification of the Sherman–Morrison formula using the easily verifiable identity~~

	~~: <math>( I+wv^T )^{-1}=I-\frac{wv^T}{1+v^Tw}</math>~~

	~~Let <math>u=Aw</math> and <math>A+uv^T=A\left( I+wv^T \right)</math>, then~~

	~~: <math>( A+uv^T )^{-1}=( I+wv^T )^{-1}{A^{-1}}=\left( I-\frac{wv^T}{1+v^Tw} \right)A^{-1}</math>~~

	~~Substituting <math>w={{A}^{-1}}u</math> gives~~

	~~: <math>( A+uv^T )^{-1}=\left( I-\frac{A^{-1}uv^T}{1+v^TA^{-1}u} \right)A^{-1}= {A^{-1}}-\frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u}</math>~~

	~~== See also ==~~
	* The [[matrix determinant lemma]] performs a rank-1 update to a [[determinant]].
	* [[Woodbury matrix identity]]
	* [[Quasi-Newton method]]
	* [[Binomial inverse theorem]]
	* [[Bunch–Nielsen–Sorensen formula]]

	~~== References ==~~
	~~{{reflist}}~~

	~~== External links ==~~
	* {{MathWorld\|title=Sherman–Morrison formula\|urlname=Sherman-MorrisonFormula}}

	~~{{DEFAULTSORT:Sherman-Morrison formula}}~~
	~~[[Category:Linear algebra]]~~
	~~[[Category:Matrix theory]]~~

12.18.100.126: /* Service level */

2014-01-27T19:28:27Z

Service level

← Older revision		Revision as of 21:28, 27 January 2014
Line 1:		Line 1:
	The ~~author~~ is ~~called Irwin Wunder but~~ it~~'s not~~ the ~~most masucline~~ title ~~out there~~. ~~Minnesota has always been his house but his spouse desires them~~ to ~~move~~. ~~He utilized~~ to be ~~unemployed~~ but ~~now he~~ is a ~~meter reader. To gather coins~~ is ~~what her family~~ and ~~her enjoy.~~<br><br>~~Feel free to visit my homepage ..~~. [http://www.~~streaming~~.~~iwarrior~~.~~net~~/~~user~~/~~WMcneil www~~.~~streaming~~.~~iwarrior~~.~~net~~]		In [[mathematics]], in particular [[linear algebra]], the '''Sherman–Morrison formula''',<ref name="SM1949">{{cite journal
			\|first=Jack \|last=Sherman
			\|first2=Winifred J. \|last2=Morrison
			\|title=Adjustment of an Inverse Matrix Corresponding to Changes in the Elements of a Given Column or a Given Row of the Original Matrix (abstract)
			\|journal=Annals of Mathematical Statistics
			\|volume=20 \|pages=621 \|year=1949
			\|doi=10.1214/aoms/1177729959 }}</ref><ref name="SM1950">{{cite journal
			\|first=Jack \|last=Sherman
			\|first2=Winifred J. \|last2=Morrison
			\|title=Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix
			\|journal=[[Annals of Mathematical Statistics]]
			\|volume=21 \|issue=1 \|pages=124–127 \|year=1950
			\|doi=10.1214/aoms/1177729893 \|mr=35118 \| zbl=0037.00901 }}</ref><ref name="PFTV1992">{{Citation
			\|last1=Press \|first1=William H.
			\|last2=Teukolsky \|first2=Saul A.
			\|last3=Vetterling \|first3=William T.
			\|last4=Flannery \|first4=Brian P.
			\|title=Numerical Recipes: The Art of Scientific Computing \|edition=3rd \|year=2007 \|publisher=Cambridge University Press \|publication-place=New York
			\|isbn=978-0-521-88068-8
			\|chapter=Section 2.7.1 Sherman–Morrison Formula
			\|chapter-url=http://apps.nrbook.com/empanel/index.html?pg=76 }}</ref> named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an [[inverse matrix\|invertible]] [[matrix (mathematics)\|matrix]] <math>A</math>
			and the [[outer product]], <math>u v^T</math>, of [[vector (mathematics)\|vectors]] <math>u</math> and <math>v</math>. The Sherman–Morrison formula is a special case of the [[Woodbury matrix identity\|Woodbury formula]].
			Though named after Sherman and Morrison, it appeared already in earlier publications.<ref name="hager">{{cite journal
			\|first=William W. \|last=Hager
			\|title=Updating the inverse of a matrix
			\|journal=SIAM Review
			\|volume=31 \|year=1989 \|pages=221–239 \|issue=2
			\|doi=10.1137/1031049 \|mr=997457 \|jstor=2030425 }}</ref>

			== Statement ==
			Suppose <math>A</math> is an [[invertible]] [[square matrix]] and <math>u</math>, <math>v</math> are [[Vector (geometric)\|vectors]]. Suppose furthermore that <math>1 + v^T A^{-1}u \neq 0</math>. Then the Sherman–Morrison formula states that
			:<math>(A+uv^T)^{-1} = A^{-1} - {A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u}.</math>

			Here, <math>uv^T</math> is the [[outer product]] of two vectors <math>u</math> and <math>v</math>. The general form shown here is the one published by Bartlett.<ref name="B1951">{{cite journal
			\|first=Maurice S. \|last=Bartlett
			\|title=An Inverse Matrix Adjustment Arising in Discriminant Analysis
			\|journal=[[Annals of Mathematical Statistics]]
			\|volume=22 \|issue=1 \|pages=107–111 \|year=1951
			\|doi=10.1214/aoms/1177729698 \|mr=40068 \| zbl = 0042.38203
			}}</ref>

			==Application==

			If the inverse of <math>A</math> is already known, the formula provides a
			[[Computational complexity theory\|numerically]] [[Computationally expensive\|cheap]] way
			to compute the inverse of <math>A</math> corrected by the matrix <math>uv^T</math>
			(depending on the point of view, the correction may be seen as a
			[[perturbation theory\|perturbation]] or as a [[rank (linear algebra)\|rank]]-1 update).
			The computation is relatively cheap because the inverse of <math>A+uv^T</math>
			does not have to be computed from scratch (which in general is expensive),
			but can be computed by correcting (or perturbing) <math>A^{-1}</math>.

			Using [[unit column]]s (columns from the [[identity matrix]]) for <math>u</math> or <math>v</math>, individual columns or rows of <math>A</math> may be manipulated
			and a correspondingly updated inverse computed relatively cheaply in this way.<ref>Langville, Amy N.; and Meyer, Carl D.; "Google's PageRank and Beyond: The Science of Search Engine Rankings", Princeton University Press, 2006, p. 156</ref> In the general case, where <math>A^{-1}</math> is a <math>n</math> times <math>n</math> matrix
			and <math>u</math> and <math>v</math> are arbitrary vectors of dimension <math>n</math>, the whole matrix is updated<ref name="B1951" /> and the computation takes <math>3n^2</math> scalar multiplications.<ref>[http://www.alglib.net/matrixops/general/invupdate.php Update of the inverse matrix by the Sherman–Morrison formula]</ref> If <math>u</math> is a unit column, the computation takes only <math>2n^2</math> scalar multiplications. The same goes if <math>v</math> is a unit column. If both <math>u</math> and <math>v</math> are unit columns, the computation takes only <math>n^2</math> scalar multiplications.

			==Verification==

			We verify the properties of the inverse.
			A matrix <math>Y</math> (in this case the right-hand side of the Sherman–Morrison formula)
			is the inverse of a matrix <math>X</math> (in this case <math>A+uv^T</math>)
			if and only if <math>XY = YX = I</math>.

			We first verify that the right hand side (<math>Y</math>) satisfies <math>XY = I</math>.
			:<math>XY = (A + uv^T)\left( A^{-1} - {A^{-1} uv^T A^{-1} \over 1 + v^T A^{-1}u}\right)</math>

			::<math>= AA^{-1} + uv^T A^{-1} - {AA^{-1}uv^T A^{-1} + uv^T A^{-1}uv^T A^{-1} \over 1 + v^TA^{-1}u}</math>

			::<math>= I + uv^T A^{-1} - {uv^T A^{-1} + uv^T A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u}</math>

			::<math>= I + uv^T A^{-1} - {u(1 + v^T A^{-1}u) v^T A^{-1} \over 1 + v^T A^{-1}u}</math>
			Note that <math>v^T A^{-1}u</math> is a scalar, so <math>(1+v^T A^{-1}u)</math> can be factored out, leading to:
			:<math>XY= I + uv^T A^{-1} - uv^T A^{-1} = I.\,</math>

			In the same way, it is verified that

			:<math>YX = \left(A^{-1} - {A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u}\right)(A + uv^T) = I.</math>

			Following is an alternate verification of the Sherman–Morrison formula using the easily verifiable identity

			: <math>( I+wv^T )^{-1}=I-\frac{wv^T}{1+v^Tw}</math>

			Let <math>u=Aw</math> and <math>A+uv^T=A\left( I+wv^T \right)</math>, then

			: <math>( A+uv^T )^{-1}=( I+wv^T )^{-1}{A^{-1}}=\left( I-\frac{wv^T}{1+v^Tw} \right)A^{-1}</math>

			Substituting <math>w={{A}^{-1}}u</math> gives

			: <math>( A+uv^T )^{-1}=\left( I-\frac{A^{-1}uv^T}{1+v^TA^{-1}u} \right)A^{-1}= {A^{-1}}-\frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u}</math>

			== See also ==
			* The [[matrix determinant lemma]] performs a rank-1 update to a [[determinant]].
			* [[Woodbury matrix identity]]
			* [[Quasi-Newton method]]
			* [[Binomial inverse theorem]]
			* [[Bunch–Nielsen–Sorensen formula]]

			== References ==
			{{reflist}}

			== External links ==
			* {{MathWorld\|title=Sherman–Morrison formula\|urlname=Sherman-MorrisonFormula}}

			{{DEFAULTSORT:Sherman-Morrison formula}}
			[[Category:Linear algebra]]
			[[Category:Matrix theory]]

en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Controlling

2012-08-08T04:27:35Z

Repairing links to disambiguation pages - You can help! - Controlling

New page

The author is called Irwin Wunder but it's not the most masucline title out there. Minnesota has always been his house but his spouse desires them to move. He utilized to be unemployed but now he is a meter reader. To gather coins is what her family and her enjoy.<br><br>Feel free to visit my homepage ... [http://www.streaming.iwarrior.net/user/WMcneil www.streaming.iwarrior.net]