Some factorization algorithms, referred to as LDU decompositions, derive three matrices L, D, and U from A such that
$\mathbf{LDU=A}$ | (51) |
where L is unit upper triangular, D is diagonal, and U is unit lower triangular. It should be obvious that the storage requirements of LDU decompositions and LU decompositions are the same.
A procedure proposed by Tinnney and Walker [18] provides a concrete example of an LDU decomposition that is based on Gaussian elimination. One row of the subdiagonal portion of A is eliminated at each stage of the computation. Tinney refers to the LDU decomposition as a “table of factors”. He constructs the factorization as follows:
The elements of the unit upper triangular matrix U are u_{ij} = $a_{ij}^{(i)}$, where i < j.
The elements of the diagonal matrix D are d_{ii} = $\frac{1}{a_{ii}^{(i-l)}}$.
The elements of the unit lower triangular matrix L are l_{ij} = $a_{ij}^{(j-1)}$, where i < j.
Figure 3 depicts the first two stages of Tinney’s factorization scheme.