4.9 Complete Pivoting

If a complete pivoting strategy is observed (pivoting involves both row and column interchanges), factorization produces matrices L and U which satisfy the following equation.

$$\mathbf{LU=PAQ}$$ (57)

where P is a row permutation matrix and Q is a column permutation matrix Q is derived from column interchanges in the same way P is derived from row interchanges.

If A and its factors are related according to Equation 57, then Equation 35 can still be solved for A by the sequential solution of two triangular systems.

$$\displaystyle\mathbf{y}\displaystyle=\mathbf{Pb}$$ (58)
$$\displaystyle\mathbf{Lc}\displaystyle=\mathbf{y}$$ (59)
$$\displaystyle\mathbf{Uz}\displaystyle=\mathbf{c}$$ (60)
$$\displaystyle\mathbf{x}\displaystyle=\mathbf{Qz}$$ (61)

Since Equation 56 and Equation 58 are identical, P can still be implemented as a mapping that is applied to b before substitution begins. Since Equation 61 computes the product Qz after back substitution is finished, Q can be implemented as a mapping that is applied to A following the substitution process.

If A is symmetric, pivoting for numerical stability may destroy the symmetry of the LU decomposition of A. For a symmetric factorization of A, matching row and column interchanges are required. In other words, pivoting must be complete and the permutation matrices must be related as follows: