en>Whoop whoop pull up at 21:20, 13 November 2013

2013-11-13T21:20:29Z

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	~~Nice~~ to ~~meet you~~, ~~my name~~ is ~~Ling~~ and ~~I totally dig~~ that ~~name~~. The ~~occupation he~~'~~s been occupying~~ for ~~years~~ is ~~a messenger~~. I'~~ve usually loved residing~~ in ~~Idaho~~. ~~What he really enjoys doing~~ is to ~~perform handball~~ but he is ~~struggling~~ to ~~discover time for it~~.<br><br>~~Feel free to visit my website~~ ... [http://~~Kjhkkb~~.~~net~~/xe/~~notice~~/~~409153 Kjhkkb~~.~~net~~]		{{hatnote\|This article is about pivots in matrices. For the pivot in the quicksort algorithm, see [[quicksort]].}}
			The '''pivot''' or '''pivot element''' is the element of a [[Matrix (mathematics)\|matrix]], or an [[array data structure\|array]], which is selected first by an [[algorithm]] (e.g. [[Gaussian elimination]], [[simplex algorithm]], etc.), to do certain calculations. In the case of matrix algorithms, a pivot entry is usually required to be at least distinct from zero, and often distant from it; in this case finding this element is called '''pivoting'''. Pivoting may be followed by an interchange of rows or columns to bring the pivot to a fixed position and allow the algorithm to proceed successfully, and possibly to reduce round-off error.

			Pivoting might be thought of as swapping or sorting rows or columns in a matrix, and thus it can be represented as [[matrix multiplication\|multiplication]] by [[permutation matrix\|permutation matrices]]. However, algorithms rarely move the matrix elements because this would cost too much time; instead, they just keep track of the permutations.

			Overall, pivoting adds more operations to the computational cost of an algorithm. These additional operations are sometimes necessary for the algorithm to work at all. Other times these additional operations are worthwhile because they add [[numerical stability]] to the final result.

			==Examples of systems that require pivoting==
			In the case of Gaussian elimination, the algorithm requires that pivot elements not be zero.
			Interchanging rows or columns in the case of a zero pivot element is necessary. The system below requires the interchange of rows 2 and 3 to perform elimination.

			:<math>
			\left[ \begin{array}{ccc\|c}
			1 & -1 & 2 & 8 \\
			0 & 0 & -1 & -11 \\
			0 & 2 & -1 & -3
			\end{array} \right]
			</math>

			The system that results from pivoting is as follows and will allow the elimination algorithm and backwards substitution to output the solution to the system.

			:<math>
			\left[ \begin{array}{ccc\|c}
			1 & -1 & 2 & 8 \\
			0 & 2 & -1 & -3 \\
			0 & 0 & -1 & -11
			\end{array} \right]
			</math>
			Furthermore, in Gaussian elimination it is generally desirable to choose a pivot element with large [[absolute value]]. This improves the [[numerical stability]]. The following system is dramatically affected by round-off error when Gaussian elimination and backwards substitution are performed.

			:<math>
			\left[ \begin{array}{cc\|c}
			0.00300 & 59.14 & 59.17 \\
			5.291 & -6.130 & 46.78 \\
			\end{array} \right]
			</math>
			This system has the exact solution of x<sub>1</sub> = 10.00 and x<sub>2</sub> = 1.000, but when the elimination algorithm and backwards substitution are performed using four-digit arithmetic, the small value of a<sub>11</sub> causes small round-off errors to be propagated. The algorithm without pivoting yields the approximation of x<sub>1</sub> ≈ 9873.3 and x<sub>2</sub> ≈ 4. In this case it is desirable that we interchange the two rows so that a<sub>21</sub> is in the pivot position

			:<math>
			\left[ \begin{array}{cc\|c}
			5.291 & -6.130 & 46.78 \\
			0.00300 & 59.14 & 59.17 \\
			\end{array} \right].
			</math>

			Considering this system, the elimination algorithm and backwards substitution using four-digit arithmetic yield the correct values x<sub>1</sub> = 10.00 and x<sub>2</sub> = 1.000.

			==Partial and complete pivoting==
			In '''partial pivoting''', the algorithm selects the entry with largest absolute value from the column of the matrix that is currently being considered as the pivot element. Partial pivoting is generally sufficient to adequately reduce round-off error. However for certain systems and algorithms, '''complete pivoting''' (or maximal pivoting) may be required for acceptable accuracy. Complete pivoting considers all entries in the whole matrix, interchanging rows and columns to achieve the highest accuracy. Complete pivoting is usually not necessary to ensure numerical stability and, due to the additional computations it introduces, it may not always be the most appropriate pivoting strategy.

			==Scaled pivoting==
			A variation of the partial pivoting strategy is '''scaled partial pivoting'''. In this approach, the algorithm selects as the pivot element the entry that is largest relative to the entries in its row. This strategy is desirable when entries' large differences in magnitude lead to the propagation of round-off error. Scaled pivoting should be used in a system like the one below where a row's entries vary greatly in magnitude. In the example below, it would be desirable to interchange the two rows because the current pivot element 30 is larger than 5.291 but it is relatively small compared with the other entries in its row. Without row interchange in this case, rounding errors will be propagated as in the previous example.

			:<math>
			\left[ \begin{array}{cc\|c}
			30 & 591400 & 591700 \\
			5.291 & -6.130 & 46.78 \\
			\end{array} \right]
			</math>

			== Pivot position ==
			A pivot position in a matrix, A, is a position in the matrix that corresponds to a row–leading 1 in the [[reduced row echelon form]] of A. Since the reduced row echelon form of A is unique, the pivot positions are uniquely determined and do not depend on whether or not row interchanges are performed in the reduction process.

			==References==
			{{PlanetMath attribution\|id=1243\|title=Pivoting}}
			* R. L. Burden, J. D. Faires, ''Numerical Analysis'', 8th edition, Thomson Brooks/Cole, 2005. ISBN 0-534-39200-8
			* G. H. Golub, C. F. Loan, ''Matrix Computations'', 3rd edition, Johns Hopkins, 1996. ISBN 0-8018-5414-8.
			* {{cite journal\|first1=Komei\|last1=Fukuda\|<!-- authorlink1=Komei Fukuda -->\|first2=Tamás\|last2=Terlaky\|<!-- authorlink2=Tamás Terlaky -->\|title=Criss-cross methods: A fresh view on pivot algorithms \|doi=10.1007/BF02614325\|journal=Mathematical Programming: Series B\|volume=79
			\|issue=1—3\|pages=369–395\|issue=Papers from the 16th International Symposium on Mathematical Programming held in Lausanne, 1997\|editors=Thomas M. Liebling and Dominique de Werra\|publisher=North-Holland Publishing Co. \|location=Amsterdam\|year=1997\|doi=10.1016/S0025-5610(97)00062-2\|mr=1464775\|ref=harv\|id=[http://www.cas.mcmaster.ca/~terlaky/files/crisscross.ps Postscript preprint]}}
			* {{cite journal\|last1=Terlaky\|first1=Tamás\|<!-- authorlink1=Tamás Terlaky -->\|last2=Zhang\|first2=Shu Zhong\|title=Pivot rules for linear programming: A Survey on recent theoretical developments\|issue=Degeneracy in optimization problems\|journal=Annals of Operations Research\|volume=46–47\|year=1993\|issue=1\|pages=203–233\|doi=10.1007/BF02096264\|mr=1260019\| id = {{citeseerx\|10.1.1.36.7658}} \|publisher=Springer Netherlands\|issn=0254-5330\|ref=harv}}
			{{Numerical linear algebra}}

			[[Category:Numerical linear algebra]]
			[[Category:Exchange algorithms]]

			[[sv:Pivotelement]]

en>EmausBot: r2.7.3) (Robot: Adding fi:Reaktiomoottori

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Nice to meet you, my name is Ling and I totally dig that name. The occupation he's been occupying for years is a messenger. I've usually loved residing in Idaho. What he really enjoys doing is to perform handball but he is struggling to discover time for it.<br><br>Feel free to visit my website ... [http://Kjhkkb.net/xe/notice/409153 Kjhkkb.net]

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en>Whoop whoop pull up at 21:20, 13 November 2013

en>EmausBot: r2.7.3) (Robot: Adding fi:Reaktiomoottori