Triangular form with pivoting

We bring again to a triangular form the system

but this time we use pivoting; the dimensional vector stores the row permutations, while stores the multipliers as before.

For the first step, since the maximal element in the first column of is we permute rows one and three, and store this permutation as .

As before, we multiply the first row by () and subtract it from the second (third) row in order to zero the elements and respectively; the multipliers , are stored in in the appropriate positions.

Now we process the second column. The maximum between and is ; no row permutation is necessary here, hence . The second row is multiplied by and subtracted from the third to get

Now the product gives

that is, the initial matrix with rows 1 and 3 interchanged. In matrix language,

Note that, when pivoting is used, all the multipliers are less than or equal to 1. Compactly we can represent the LU decomposition of as