A Preconditioned Newton Algorithm for the Nearest Correlation Matrix

Borsdorf, Rudiger and Higham, Nicholas J. (2008) A Preconditioned Newton Algorithm for the Nearest Correlation Matrix. IMA Journal of Numerical Analysis, 30 (1). pp. 94-107. ISSN 1464-3642

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Abstract

Various methods have been developed for computing the correlation matrix nearest in the Frobenius norm to a given matrix. We focus on a quadratically convergent Newton algorithm recently derived by Qi and Sun. Various improvements to the efficiency and reliability of the algorithm are introduced. Several of these relate to the linear algebra: the Newton equations are solved by minres instead of the conjugate gradient method, as it more quickly satisfies the inexact Newton condition; we apply a Jacobi preconditioner, which can be computed efficiently even though the coefficient matrix is not explicitly available; an efficient choice of eigensolver is identified; and a final scaling step is introduced to ensure that the returned matrix has unit diagonal. Potential difficulties caused by rounding errors in the Armijo line search are avoided by altering the step selection strategy. These and other improvements lead to a significant speedup over the original algorithm and allow the solution of problems of dimension a few thousand in a few tens of minutes.

Item Type: Article
Additional Information: Advance Access published on June 8, 2009.
Uncontrolled Keywords: correlation matrix, positive semidefinite matrix, Newton's method, preconditioning, rounding error, Armijo line search conditions, alternating projections method
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
MSC 2010, the AMS's Mathematics Subject Classification > 90 Operations research, mathematical programming
Depositing User: Nick Higham
Date Deposited: 28 Jan 2010
Last Modified: 20 Oct 2017 14:12
URI: https://eprints.maths.manchester.ac.uk/id/eprint/1201

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