On the computation and updating of the modified Cholesky decomposition of a covariance matrix

Methods for obtaining and updating the modified Cholesky decomposition (MCD) for the particular case of a covariance matrix when one is given only the original data are described. These methods are the standard method of forming the covariance matrix K then solving for the MCD, L and D (where K=LDLT); a method based on Householder reflections; and lastly, a method employing the composite-t algorithm. For many cases in the analysis of remotely sensed data, the composite-t method is the superior method despite the fact that it is the slowest one, since (1) the relative amount of time computing MCD's is often quite small, (2) the stability properties of it are the best of the three, and (3) it affords an efficient and numerically stable procedure for updating the MCD. The properties of these methods are discussed and FORTRAN programs implementing these algorithms are listed.