6.1 LDU Factor Update

Algorithm 8 follows the lead of our sources (see Section 6.3 for details) and implements a technique for updating the factors L, D, and U of A following a rank one change to A.

Algorithm 8: LDU Factor Update
 for $i=1,\cdots,n$ $\delta=d_{i}$ $p=y_{i}$ $q=z_{i}$ $d_{i}=d_{i}+\alpha pq$ $\beta_{1}=\displaystyle\frac{\alpha p}{d_{i}}$ $\beta_{2}=\displaystyle\frac{\alpha q}{d_{i}}$ $\alpha=\displaystyle\frac{\alpha\delta}{d_{i}}$ for $j=i+1,\cdots,n$ $y_{j}=y_{j}-pl_{ji}$ $z_{j}=z_{j}-qu_{ij}$ $l_{ji}=l_{ji}+\beta_{1}y_{j}$ $u_{ij}=u_{ij}+\beta_{2}y_{j}$

The scalar $\alpha$ and the vectors y and zT are destroyed by this procedure. The factors of A are overwritten by their new values.

The outer loop of Algorithm 8 does not have to begin at one unless y is full. If y has any leading zeros, the initial value of i should be the index of yi the first nonzero element of y. If there is no a priori information about the structure of y but there is a high probability of leading zeros, testing yi for zero at the beginning of the loop might save a lot of work. However, you must remember to cancel the test as soon as a nonzero yi is encountered.