# 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 $i,\ldots,n$. 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.
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