Positive definite completions and determinant maximization

A method is described for determining whether a positive definite completio n of a given partial Hermitian matrix exists and, if so, for finding the de terminant maximizing positive definite completion. No assumption is made ab out the arrangement of the specified entries. The method employs iterative application to a modified problem of an explicit formula for the maximum de terminant in case there is only one symmetrically placed pair of unspecifie d entries. A robust and fast algorithm based upon this approach is shown to have global convergence to the necessarily unique solution. (C) 1999 Elsev ier Science Inc. All rights reserved.