and x2 = 1.000. Partial, rook, and complete pivoting In partial pivoting, the algorithm selects the entry with largest absolute value from the column of the matrix that is currently being considered as the pivot element. More specifically, when reducing a matrix to row echelon form, partial pivoting swaps rows before the column's row reduction to make the pivot element have the largest absolute value compared to the elements below in the same column. Partial pivoting is generally sufficient to adequately reduce round-off error. However, for certain systems and algorithms, complete pivoting (or maximal pivoting) may be required for acceptable accuracy. Complete pivoting interchanges both rows and columns in order to use the largest (by absolute value) element in the matrix as the pivot. Complete pivoting is usually not necessary to ensure numerical stability and, due to the additional cost of searching for the maximal element, the improvement in numerical stability that it provides is typically outweighed by its reduced efficiency for all but the smallest matrices. Hence, it is rarely used. Another strategy, known as rook pivoting also interchanges both rows and columns but only guarantees that the chosen pivot is simultaneously the largest possible entry in its row and the largest possible entry in its column, as opposed to the largest possible in the entire remaining submatrix. When implemented on serial computers this strategy has expected cost of only about three times that of partial pivoting and is therefore cheaper than complete pivoting. Rook pivoting has been proved to be more stable than partial pivoting both theoretically and in practice. Scaled pivoting A variation of the partial pivoting strategy is scaled pivoting. In this approach, the algorithm selects as the pivot element the entry that is largest relative to the entries in its row. This strategy is desirable when entries' large differences in magnitude lead to the propagation of round-off error. Scaled pivoting should be used in a system like the one below where a row's entries vary greatly in magnitude. In the example below, it would be desirable to interchange the two rows because the current pivot element 30 is larger than 5.291 but it is relatively small compared with the other entries in its row. Without row interchange in this case, rounding errors will be propagated as in the previous example. Pivot position A pivot position in a matrix, A, is a position in the matrix