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.
Original language  English 

Pages (fromto)  94107 
Number of pages  13 
Journal  IMA Journal of Numerical Analysis 
Volume  30 
Issue number  1 
DOIs  
Publication status  Published  Jan 2010 
Keywords
 Alternating projections method
 Armijo line search conditions
 Correlation matrix
 Newton's method
 Positive semidefinite matrix
 Preconditioning
 Rounding error
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Mathematical Software for the Nearest Correlation Matrix
Nicholas Higham (Participant) & Rudiger Borsdorf (Participant)
Impact: Technological, Economic