Pivot element

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The pivot or pivot element is the element of a matrix, which is selected first by an algorithm (e.g. Gaussian elimination, Quicksort, Simplex algorithm), to do certain calculations with the matrix. These matrix algorithms require an entry distinct from zero in pivot position to work properly or at all respectively. Finding this element is called pivoting.

In the case of Gaussian elimination, it is best to choose a pivot element with large absolute value. This improves the numerical stability. In partial pivoting, the algorithm considers all entries in the column of the matrix that is currently being considered, picks the entry with largest absolute value, and finally swaps rows such that this entry is the pivot in question. Complete pivoting considers all entries in the whole matrix. Complete pivoting is usually not necessary to ensure numerical stability.

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

Pivot element in Quicksort means the element that is selected as boundary for partitioning. Quicksort sorts all elements "left" and "right" of the pivot element recursively.

[edit] References

• G. H. Golub, C. F. Loan, Matrix Computations, 3rd edition, Johns Hopkins, 1996. ISBN 0801854148.

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This article incorporates material from Pivoting on PlanetMath, which is licensed under the GFDL.