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) |
|
C++ Header Interface
#include <nag.h> extern "C" {
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, , to an approximate correlation matrix, , using element-wise weighting, this minimizes , where denotes the matrix with elements .
You can optionally specify a lower bound on the eigenvalues, , of the computed correlation matrix, forcing the matrix to be strictly positive definite, if .
Zero elements in
should be used when you wish to put no emphasis on the corresponding element of
. The algorithm scales
so that the maximum element is
. 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 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:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
g
must be at least
.
On entry: , the initial matrix.
On exit: is overwritten.
-
2:
– Integer
Input
-
On entry: the first dimension of the array
g as declared in the (sub)program from which
g02ajf is called.
Constraint:
.
-
3:
– Integer
Input
-
On entry: the order of the matrix .
Constraint:
.
-
4:
– Real (Kind=nag_wp)
Input
-
On entry: the value of
.
If , is used.
Constraint:
.
-
5:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
h
must be at least
.
On entry: the matrix of weights .
On exit: a symmetric matrix with its diagonal elements set to zero and the remaining elements scaled so that the maximum element is .
Constraint:
, for all and , .
-
6:
– Integer
Input
-
On entry: the first dimension of the array
h as declared in the (sub)program from which
g02ajf is called.
Constraint:
.
-
7:
– Real (Kind=nag_wp)
Input
-
On entry: the termination tolerance for the iteration. If
,
is used. See
Section 7 for further details.
-
8:
– Integer
Input
-
On entry: specifies the maximum number of iterations to be used.
If , is used.
-
9:
– Real (Kind=nag_wp) array
Output
-
Note: the second dimension of the array
x
must be at least
.
On exit: contains the nearest correlation matrix.
-
10:
– Integer
Input
-
On entry: the first dimension of the array
x as declared in the (sub)program from which
g02ajf is called.
Constraint:
.
-
11:
– Integer
Output
-
On exit: the number of iterations taken.
-
12:
– Real (Kind=nag_wp)
Output
-
On exit: the value of after the final iteration.
-
13:
– Integer
Input/Output
-
On entry:
ifail must be set to
,
. If you are unfamiliar with this argument you should refer to
Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
.
When the value is used it is essential to test the value of ifail on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
-
On entry, .
Constraint: .
-
On entry, and .
Constraint: .
-
On entry, and .
Constraint: .
-
On entry, and .
Constraint: .
-
On entry, .
Constraint: .
-
On entry, one or more of the off-diagonal elements of were negative.
-
Routine failed to converge in
iterations.
Increase
maxit or check the call to the routine.
-
Failure to solve intermediate eigenproblem. This should not occur. Please contact
NAG with details of your call.
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.
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.
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
is the value of
norm at the
th iteration, that is
where
has been scaled as described above, then the algorithm terminates when:
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.
Arrays are internally allocated by g02ajf. The total size of these arrays is real elements and integer elements. All allocated memory is freed before return of g02ajf.
10
Example
This example finds the nearest correlation matrix to:
weighted by:
with minimum eigenvalue
.
10.1
Program Text
10.2
Program Data
10.3
Program Results