IMA Journal of Numerical Analysis Advance Access published online on June 8, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn085
A preconditioned Newton algorithm for the nearest correlation matrix
Department of Mathematics, Chemnitz University of Technology, D-09107 Chemnitz, Germany
School of Mathematics, The University of Manchester, Manchester M13 9PL, UK
Email: ruediger.borsdorf{at}s2003.tu-chemnitz.de
Corresponding author. Email: higham{at}ma.man.ac.uk
Received on 21 April 2008. Accepted for publication 1 December 2008.
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Abstract |
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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 speed-up over the original algorithm and allow the solution of problems of dimension a few thousand in a few tens of minutes.
Key Words: correlation matrix; positive semidefinite matrix; Newton's method; preconditioning; rounding error; Armijo line search conditions; alternating projections method