# User Contributed Dictionary

### Verb

pivoting- present participle of pivot

# Extensive Definition

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.

## References

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

pivoting in German: Pivotelement

pivoting in Swedish: Pivotelement

# Synonyms, Antonyms and Related Words

angular momentum, angular motion, angular
velocity, axial motion, bowling, centrifugation, circulation, circumgyration, circumrotation, full
circle, gyration,
reeling, revolution, roll, rolling, rotation, rotational motion,
spin, spinning, swinging, swirling, swiveling, trolling, trundling, turbination, turning, volutation, volution, wheeling, whir, whirling