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Positive definite completions and determinant maximization. (English) Zbl 0973.15009

Summary: A method is described for determining whether a positive definite completion of a given partial Hermitian matrix exists and, if so, for finding the determinant maximizing positive definite completion. No assumption is made about the arrangement of the specified entries. The method employs iterative application to a modified problem of an explicit formula for the maximum determinant in case there is only one symmetrically placed pair of unspecified entries. A robust and fast algorithm based upon this approach is shown to have global convergence to the necessarily unique solution.

MSC:

15A29 Inverse problems in linear algebra
65F40 Numerical computation of determinants
15A15 Determinants, permanents, traces, other special matrix functions
15B48 Positive matrices and their generalizations; cones of matrices
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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References:

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