V-Invariant Methods for Generalised Least Squares Problems

An important consideration in generalised least squares problems is that the dimension of the covariance matrix V is the dimension of the data set and is large when the data set is large. Also, the problem solution can be well determined in cases where V is illconditioned or singular. Here aspects of a class of methods which factorize the design matrix while leaving V invariant, and which can be expected to be well behaved exactly when the original problem solution is well behaved, are considered. Implementation is most satisfactory when V is diagonal. This can be achieved by a preprocessing step in which V is replaced by the diagonal matrix D which results from the modified Cholesky factorization P V P T → LDL T where L is unit lower triangular and P is the permutation matrix associated with diagonal pivoting. Conditions under which this replacement is satisfactory are investigated.