# pivoting

## 1 Pivoting

Pivoting is a process performed on a matrix in order to improve numerical stability.

*Partial pivoting* of an $n\times n$ matrix is the sorting of the rows of the matrix so that row $i$ contains the maximum absolute column value for column $i$, among all rows $i,\mathrm{\dots},n$. That is, we begin by swapping row 1 with the row that has the largest absolute value^{} for the first column, then swap row 2 with the row that has the largest magnitude for the second column (among rows 2 and below), and so on.

*Complete pivoting* is a reordering of both rows and columns, using the same method as above. It is usually not necessary to ensure numerical stability.

Pivoting can be represented as multiplication by permutation matrices^{}.

## References

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

Title | pivoting |
---|---|

Canonical name | Pivoting |

Date of creation | 2013-03-22 12:06:54 |

Last modified on | 2013-03-22 12:06:54 |

Owner | akrowne (2) |

Last modified by | akrowne (2) |

Numerical id | 8 |

Author | akrowne (2) |

Entry type | Algorithm^{} |

Classification | msc 65F35 |

Classification | msc 65F30 |

Synonym | total pivoting |

Synonym | complete pivoting |

Synonym | partial pivoting |