Similar algorithms exist for updating LU decompositions of A. If U is upper triangular and L is unit lower triangular, an element uij from and U is related to the elements $u^{\prime}_{ij}$ and $d^{\prime}_{i}$ of the LDU decomposition as follows.
| $$u_{ij}=u^{\prime}_{ij}d^{\prime}_{i}$$ | (66) |
The recurrence relations of the inner loop of Algorithm 8 must change to reflect this relationship. The following statement updates the z vector for a unit upper triangular U.
| $$z_{j}=z_{j}-q{u}^{\prime}_{ij}$$ | (67) |
If U is upper triangular, the statement becomes
| $$z_{j}=z_{j}-p\frac{u_{ij}}{\delta}$$ | (68) |
where $\delta$ is the value of uii before it was changed during stage i of the procedure. Along the same lines, the factor update statement
| $$u^{\prime}_{ij}=u^{\prime}_{ij}+{\beta}_{2}{z}_{j}$$ | (69) |
becomes
| $$\frac{u_{ij}}{u_{ii}}=\frac{{u}_{ij}}{\delta}+{\beta}_{2}{z}_{j}$$ | (70) |
Solving for the updated value of uij yields
| $$u_{ij}=u_{ii}(\frac{{u}_{ij}}{\delta}+{\beta}_{2}{z}_{j})$$ | (71) |
Taking these observations into consideration and pulling operations on constants out of the inner loop, Algorithm 9 updates U based on a rank one change to A.
| for $i=1,\cdots,n$ |
| $\delta=u_{ii}$ |
| $p=y_{i}$ |
| $q=z_{i}$ |
| $u_{ii}=u_{ii}+\alpha pq$ |
| $\beta_{2}=\alpha q$ |
| $q=\displaystyle\frac{q}{\delta}$ |
| $\delta=\displaystyle\frac{u_{ii}}{\delta}$ |
| $\alpha=\displaystyle\frac{\alpha}{\delta}$ |
| for $j=i+1,\cdots,n$ |
| $y_{j}=y_{j}-pl_{ji}$ |
| $z_{j}=z_{j}-qu_{ij}$ |
| $l_{ji}=l_{ji}+\beta_{1}y_{j}$ |
| $u_{ij}=\delta u_{ij}+\beta_{2}z_{j}$ |
If U is unit upper triangular and L is lower triangular, a similar algorithm is derived from the observation that lij of L and $l^{\prime}_{ij}$, $d^{\prime}_{i}$ of the LDU decomposition are related as follows.
| $$l_{ij}=l^{\prime}_{ij}d^{\prime}_{j}$$ | (72) |
The resulting algorithm deviates from Algorithm 8 in a manner that parallels Algorithm 9.