4.4 LDU Factorization

Some factorization algorithms, referred to as LDU decompositions, derive three matrices L, D, and U from A such that

𝐋𝐃𝐔=𝐀\mathbf{LDU=A} (35)

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 uij = aij(i)a_{ij}^{(i)}, where i < j.

  • The elements of the diagonal matrix D are dii = 1aii(i-l)\frac{1}{a_{ii}^{(i-l)}}.

  • The elements of the unit lower triangular matrix L are lij = aij(j-1)a_{ij}^{(j-1)}, where i < j.

Figure 3 depicts the first two stages of Tinney’s factorization scheme.

Figure 3: Computational Sequence of Tinney’s LDU Decomposition
The computational sequence for used to create a LDU factorization (decomposition) of a matrix using Tinney's method