NAG CL Interface
g02ajc (corrmat_​h_​weight)

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1 Purpose

g02ajc computes the nearest correlation matrix, using element-wise weighting in the Frobenius norm and optionally with bounds on the eigenvalues, to a given square, input matrix.

2 Specification

#include <nag.h>
void  g02ajc (double g[], Integer pdg, Integer n, double alpha, double h[], Integer pdh, double errtol, Integer maxit, double x[], Integer pdx, Integer *iter, double *norm, NagError *fail)
The function may be called by the names: g02ajc, nag_correg_corrmat_h_weight or nag_nearest_correlation_h_weight.

3 Description

g02ajc finds the nearest correlation matrix, X, to an approximate correlation matrix, G, using element-wise weighting, this minimizes H(G-X) F , where C=AB denotes the matrix C with elements Cij=Aij×Bij.
You can optionally specify a lower bound on the eigenvalues, α, of the computed correlation matrix, forcing the matrix to be strictly positive definite, if 0<α<1.
Zero elements in H should be used when you wish to put no emphasis on the corresponding element of G. The algorithm scales H so that the maximum element is 1. It is this scaled matrix that is used in computing the norm above and for the stopping criteria described in Section 7.
Note that if the elements in H vary by several orders of magnitude from one another the algorithm may fail to converge.

4 References

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
Jiang K, Sun D and Toh K-C (2012) An inexact accelerated proximal gradient method for large scale linearly constrained convex SDP SIAM J. Optim. 22(3) 1042–1064
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

5 Arguments

1: g[dim] double Input/Output
Note: the dimension, dim, of the array g must be at least pdg×n.
On entry: G, the initial matrix.
On exit: G is overwritten.
2: pdg Integer Input
On entry: the stride separating row elements of the matrix G in the array g.
Constraint: pdgn.
3: n Integer Input
On entry: the order of the matrix G.
Constraint: n>0.
4: alpha double Input
On entry: the value of α.
If alpha<0.0, 0.0 is used.
Constraint: alpha<1.0.
5: h[dim] double Input/Output
Note: the dimension, dim, of the array h must be at least pdh×n.
The (i,j)th element of the matrix H is stored in h[(j-1)×pdh+i-1].
On entry: the matrix of weights H.
On exit: a symmetric matrix 12 (H+HT) with its diagonal elements set to zero and the remaining elements scaled so that the maximum element is 1.0.
Constraint: H[(j-1)×pdh+i-1]0.0, for all i and j=1,2,,n, ij.
6: pdh Integer Input
On entry: the stride separating matrix row elements in the array h.
Constraint: pdhn.
7: errtol double Input
On entry: the termination tolerance for the iteration. If errtol0.0, n×machine precision is used. See Section 7 for further details.
8: maxit Integer Input
On entry: specifies the maximum number of iterations to be used.
If maxit0, 200 is used.
9: x[dim] double Output
Note: the dimension, dim, of the array x must be at least pdx×n.
On exit: contains the nearest correlation matrix.
10: pdx Integer Input
On entry: the stride separating row elements of the matrix X in the array x.
Constraint: pdxn.
11: iter Integer * Output
On exit: the number of iterations taken.
12: norm double * Output
On exit: the value of H(G-X)F after the final iteration.
13: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
Function failed to converge in value iterations.
Increase maxit or check the call to the function.
Failure to solve intermediate eigenproblem. This should not occur. Please contact NAG with details of your call.
On entry, n=value.
Constraint: n>0.
On entry, pdg=value and n=value.
Constraint: pdgn.
On entry, pdh=value and n=value.
Constraint: pdhn.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, alpha=value.
Constraint: alpha<1.0.
On entry, one or more of the off-diagonal elements of H were negative.

7 Accuracy

The returned accuracy is controlled by errtol and limited by machine precision. If ei is the value of norm at the ith iteration, that is
ei = H(G-X)F ,  
where H has been scaled as described above, then the algorithm terminates when:
|ei-ei-1| 1+ max(ei,ei-1) errtol .  

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g02ajc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02ajc 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.

9 Further Comments

Arrays are internally allocated by g02ajc. The total size of these arrays is 15×n+5×n×n+max(2×n×n+6×n+1,120+9×n) double elements and 5×n+3 Integer elements. All allocated memory is freed before return of g02ajc.

10 Example

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 )  
weighted by:
H = ( 0.0 10.0 0.0 0.0 10.0 0.0 1.5 1.5 0.0 1.5 0.0 0.0 0.0 1.5 0.0 0.0 )  
with minimum eigenvalue 0.04.

10.1 Program Text

Program Text (g02ajce.c)

10.2 Program Data

Program Data (g02ajce.d)

10.3 Program Results

Program Results (g02ajce.r)