NAG Library Function Document

nag_nearest_correlation (g02aac)


    1  Purpose
    7  Accuracy


nag_nearest_correlation (g02aac) computes the nearest correlation matrix, in the Frobenius norm, to a given square, input matrix.


#include <nag.h>
#include <nagg02.h>
void  nag_nearest_correlation (Nag_OrderType order, double g[], Integer pdg, Integer n, double errtol, Integer maxits, Integer maxit, double x[], Integer pdx, Integer *iter, Integer *feval, double *nrmgrd, NagError *fail)


A correlation matrix may be characterised as a real square matrix that is symmetric, has a unit diagonal and is positive semidefinite.
nag_nearest_correlation (g02aac) applies an inexact Newton method to a dual formulation of the problem, as described by Qi and Sun (2006). It applies the improvements suggested by Borsdorf and Higham (2010).


Borsdorf R and Higham N J (2010) A preconditioned (Newton) algorithm for the nearest correlation matrix IMA Journal of Numerical Analysis 30(1) 94–107
Qi H and Sun D (2006) A quadratically convergent Newton method for computing the nearest correlation matrix SIAM J. Matrix AnalAppl 29(2) 360–385


1:     order Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     g[dim] doubleInput/Output
Note: the dimension, dim, of the array g must be at least pdg×n.
On entry: G, the initial matrix.
On exit: a symmetric matrix 12G+GT with the diagonal set to I.
3:     pdg IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix G in the array g.
Constraint: pdgn.
4:     n IntegerInput
On entry: the size of the matrix G.
Constraint: n>0.
5:     errtol doubleInput
On entry: the termination tolerance for the Newton iteration. If errtol0.0 then n×machine precision is used.
6:     maxits IntegerInput
On entry: maxits specifies the maximum number of iterations used for the iterative scheme used to solve the linear algebraic equations at each Newton step.
If maxits0, 100 is used.
7:     maxit IntegerInput
On entry: specifies the maximum number of Newton iterations.
If maxit0, 200 is used.
8:     x[dim] doubleOutput
Note: the dimension, dim, of the array x must be at least pdx×n.
On exit: contains the nearest correlation matrix.
9:     pdx IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix X in the array x.
Constraint: pdxn.
10:   iter Integer *Output
On exit: the number of Newton steps taken.
11:   feval Integer *Output
On exit: the number of function evaluations of the dual problem.
12:   nrmgrd double *Output
On exit: the norm of the gradient of the last Newton step.
13:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
Machine precision is limiting convergence.
The array returned in x may still be of interest.
Newton iteration fails to converge in value iterations.
An intermediate eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.
On entry, n=value.
Constraint: n>0.
On entry, pdg=value.
Constraint: pdg>0.
On entry, pdx=value.
Constraint: pdx>0.
On entry, pdg=value and n=value.
Constraint: pdgn.
On entry, pdx=value and n=value.
Constraint: pdxn.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.


The returned accuracy is controlled by errtol and limited by machine precision.

Parallelism and Performance

nag_nearest_correlation (g02aac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nearest_correlation (g02aac) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Further Comments

Arrays are internally allocated by nag_nearest_correlation (g02aac). The total size of these arrays is 11×n+3×n×n+max2×n×n+6×n+1,120+9×n real elements and 5×n+3 integer elements.


This example finds the nearest correlation matrix to:
G = 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 2  

Program Text

Program Text (g02aace.c)

Program Data

Program Data (g02aace.d)

Program Results

Program Results (g02aace.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017