If the LU decomposition of the matrix A exists and the factorization of a related matrix
is needed, it is sometimes advantageous to compute the factorization of A by modifying the factors of A rather than explicitly decomposing A. Implementations of this factor update operation should have the following properties:
Arithmetic is minimized,
Numerical stability is maintained, and
Sparsity is preserved.
The current discussion outlines procedures for updating the factors of A following a rank one modification. A rank one modification of A is defined as
where is a scalar and the vectors y and zT are dimensionally correct. The terminology comes from the observation that the product zT is a matrix whose rank is one.
Computationally, a rank one factor update to a dense matrix is an O( n2 ) operation. Recall that decomposing a matrix from scratch is O( n3 ).