NAG FL Interface
g02ajf (corrmat_​h_​weight)

1 Purpose

g02ajf 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

Fortran Interface
Subroutine g02ajf ( g, ldg, n, alpha, h, ldh, errtol, maxit, x, ldx, iter, norm, ifail)
Integer, Intent (In) :: ldg, n, ldh, maxit, ldx
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: iter
Real (Kind=nag_wp), Intent (In) :: alpha, errtol
Real (Kind=nag_wp), Intent (Inout) :: g(ldg,*), h(ldh,*), x(ldx,*)
Real (Kind=nag_wp), Intent (Out) :: norm
C Header Interface
#include <nag.h>
void  g02ajf_ (double g[], const Integer *ldg, const Integer *n, const double *alpha, double h[], const Integer *ldh, const double *errtol, const Integer *maxit, double x[], const Integer *ldx, Integer *iter, double *norm, Integer *ifail)
The routine may be called by the names g02ajf or nagf_correg_corrmat_h_weight.

3 Description

g02ajf 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: gldg* Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array g must be at least n.
On entry: G, the initial matrix.
On exit: G is overwritten.
2: ldg Integer Input
On entry: the first dimension of the array g as declared in the (sub)program from which g02ajf is called.
Constraint: ldgn.
3: n Integer Input
On entry: the order of the matrix G.
Constraint: n>0.
4: alpha Real (Kind=nag_wp) Input
On entry: the value of α.
If alpha<0.0, 0.0 is used.
Constraint: alpha<1.0.
5: hldh* Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array h must be at least n.
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: Hij0.0, for all i and j=1,2,,n, ij.
6: ldh Integer Input
On entry: the first dimension of the array h as declared in the (sub)program from which g02ajf is called.
Constraint: ldhn.
7: errtol Real (Kind=nag_wp) 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: xldx* Real (Kind=nag_wp) array Output
Note: the second dimension of the array x must be at least n.
On exit: contains the nearest correlation matrix.
10: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02ajf is called.
Constraint: ldxn.
11: iter Integer Output
On exit: the number of iterations taken.
12: norm Real (Kind=nag_wp) Output
On exit: the value of HG-XF after the final iteration.
13: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of -1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value -1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, n=value.
Constraint: n>0.
ifail=2
On entry, ldg=value and n=value.
Constraint: ldgn.
ifail=3
On entry, ldh=value and n=value.
Constraint: ldhn.
ifail=4
On entry, ldx=value and n=value.
Constraint: ldxn.
ifail=5
On entry, alpha=value.
Constraint: alpha<1.0.
ifail=6
On entry, one or more of the off-diagonal elements of H were negative.
ifail=7
Routine failed to converge in value iterations.
Increase maxit or check the call to the routine.
ifail=8
Failure to solve intermediate eigenproblem. This should not occur. Please contact NAG with details of your call.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 = HG-XF ,  
where H has been scaled as described above, then the algorithm terminates when:
ei-ei-1 1+ maxei,ei-1 errtol .  

8 Parallelism and Performance

g02ajf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02ajf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

Arrays are internally allocated by g02ajf. The total size of these arrays is 15×n+5×n×n+max2×n×n+6×n+1,120+9×n real elements and 5×n+3 integer elements. All allocated memory is freed before return of g02ajf.

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 (g02ajfe.f90)

10.2 Program Data

Program Data (g02ajfe.d)

10.3 Program Results

Program Results (g02ajfe.r)