Let be a positive-definite Hermitian symmetric Toeplitz matrix defined with elements for a set of scalars , where . Without loss of generality, assume .
We have the Cholesky factor of the matrix such that . The inverse Cholesky factor is then given by . A closely related decomposition known as the LDL decomposition is given as follows: , , and therefore, . Matrix has ones along the principal diagonal. Let .
We have , where is the Hermitian transpose of the inverse with ones along the diagonal. As is lower triangular, we can solve for the strictly upper triangular parts of as the R.H.S. is known to be zero.
For the -th column of we have the first elements , upon simplification, as:
For Hermitian symmetric Toeplitz matrix we have , where is complex conjugate of the matrix and is the exchange matrix. Using this property, and setting we have:
,
which may be solved efficiently using the Durbin iterations [7]. The algorithm for finding is given as follows:
The algorithm requires flops [7, Algorithm 4.7.1] for the Durbin iterations, flops for scaling with and inverse square root computations; and may be easily adapted to find and of the LDL decomposition. A MATLAB reference code is available in [14] as well as reproduced below.
First published: 4th Sep. 2015 on aravindhk-math.blogspot.com
Modified: 16th Dec. 2023 – Style updates for LaTeX