If the LU decomposition of the n × n symmetric matrix A exists and the factorization of a related matrix
is desired, factor update is often the procedure of choice.
Section 6 examines factor update techniques for dense, asymmetric matrices. The current section examines techniques that exploit computational efficiencies introduced by symmetry. Symmetry reduces the work required to update the factorization of A by half, just as it reduces the work required to decompose A in the first place.
More specifically, the current section examines procedures for updating the factors of A following a symmetric rank one modification
where is a scalar and y is an n vector.
Algorithm C1 of Gill et al.  updates the factors L and D of a LDU decomposition of A. The algorithm assumes that the upper triangular factors U are implicit. Algorithm 20 mimics Algorithm C1. The scalar and the y vector are destroyed by this procedure. The factors of A are overwritten by their new values.
See Section 6.1 for a discussion of updating asymmetric LDU factorizations following rank one changes to a matrix.
If the LU decomposition of the symmetric matrix A exists and U is stored explicitly, the recurrence relations of the inner loop of Algorithm 20 must change. Following a train of thought similar to the derivation of Algorithm 9 (see Section 6.2 for details) results in Algorithm 21 which updates U based on a rank one change to A.
If U is maintained in a zero-based linear array, Algorithm 21 changes in the normal manner, that is
The double subscript notation is replaced by the indexing rule defined in Equation 87.
The outer loop counter i ranges from zero to n-1.
The inner loop counter j ranges from i+1 to n-1.
The outer loops of the symmetric factor update algorithms do not have to begin at one unless y is full. If U has any leading zeros, the initial value of i (or j in Algorithm 20) 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 suspend the test as soon as the first nonzero value of yi is encountered.
For a fuller discussion of the derivation and implementation of LU factor update, see Section 6.2.