V-Invariant Methods for Generalised Least Squares Problems

摘要

An important consideration in solving generalised least squares problems is the dimension of the covariance matrix V. This has the dimension of the data set and is large when the data set is large. In addition the problem can be formulated to have a well determined solution in cases where V is illconditioned or singular, a class of problems that includes the case of equality constrained least squares. This paper considers a class of methods which factorize the design matrix A while leaving V invariant, and which can be expected to be well behaved exactly when the original problem solution is well behaved. 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 PVP~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.