NAG CL Interface
g02asc (corrmat_fixed)
1
Purpose
g02asc computes the nearest correlation matrix, in the Frobenius norm, while fixing elements and optionally with bounds on the eigenvalues, to a given square input matrix.
2
Specification
void |
g02asc (double g[],
Integer pdg,
Integer n,
double alpha,
const Integer h[],
Integer pdh,
double errtol,
Integer maxit,
Integer m,
double x[],
Integer pdx,
Integer *its,
double *fnorm,
NagError *fail) |
|
The function may be called by the names: g02asc or nag_correg_corrmat_fixed.
3
Description
g02asc finds the nearest correlation matrix, , to a matrix, , in the Frobenius norm. It uses an alternating projections algorithm with Anderson acceleration. Elements in the input matrix can be fixed by supplying the value in the corresponding element of the matrix . However, note that the algorithm may fail to converge if the fixed elements do not form part of a valid correlation matrix. You can optionally specify a lower bound, , on the eigenvalues of the computed correlation matrix, forcing the matrix to be positive definite with .
4
References
Anderson D G (1965) Iterative Procedures for Nonlinear Integral Equations J. Assoc. Comput. Mach. 12 547–560
Higham N J and Strabić N (2016) Anderson acceleration of the alternating projections method for computing the nearest correlation matrix Numer. Algor. 72 1021–1042
5
Arguments
-
1:
– double
Input/Output
-
Note: the th element of the matrix is stored in .
On entry: , the initial matrix.
On exit: the symmetric matrix with the diagonal elements set to .
-
2:
– Integer
Input
-
On entry: the stride separating matrix row elements in the array
g.
Constraint:
.
-
3:
– Integer
Input
-
On entry: the order of the matrix .
Constraint:
.
-
4:
– double
Input
-
On entry: the value of
.
If , a value of is used.
Constraint:
.
-
5:
– const Integer
Input
-
Note: the th element of the matrix is stored in .
On entry: the symmetric matrix . If an element of is then the corresponding element in is fixed in the output . Only the strictly lower triangular part of need be set.
-
6:
– Integer
Input
-
On entry: the stride separating matrix row elements in the array
h.
Constraint:
.
-
7:
– double
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.
If , a value of is used.
-
9:
– Integer
Input
-
On entry: the number of previous iterates to use in the Anderson acceleration. If
, Anderson acceleration is not used. See
Section 7 for further details.
If , a value of is used.
Constraint:
.
-
10:
– double
Output
-
Note: the th element of the matrix is stored in .
On exit: contains the matrix .
-
11:
– Integer
Input
-
On entry: the stride separating matrix row elements in the array
x.
Constraint:
.
-
12:
– Integer *
Output
-
On exit: the number of iterations taken.
-
13:
– double *
Output
-
On exit: the value of after the final iteration.
-
14:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALG_FAIL
-
Failure during Anderson acceleration.
Consider setting and recomputing.
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE
-
Function failed to converge in
iterations.
A solution may not exist, however, try increasing
maxit.
- NE_INT
-
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
The fixed element , lies outside the interval ,
for and .
- NE_INTERNAL_ERROR
-
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.
- NE_NO_LICENCE
-
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.
- NE_REAL
-
On entry, .
Constraint: .
7
Accuracy
Alternating projections is an iterative process where at each iteration the new iterate,
, can be improved by using Anderson acceleration to reduce the overall number of iterations. The alternating projections algorithm terminates at the
th iteration when
where
is the result of the first of two projections computed at each step.
Without Anderson acceleration this algorithm is guaranteed to converge. There is no theoretical guarantee of convergence of Anderson acceleration and therefore, when it is used, no guarantee of convergence of
g02asc. However, in practice it can be seen to significantly reduce the number of alternating projection iterations. Anderson acceleration is not used when
m is set to zero. See
c05mdc and
Higham and Strabić (2016) and
Anderson (1965) for further information.
8
Parallelism and Performance
g02asc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02asc 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.
Arrays are internally allocated by g02asc. The total size of these arrays does not exceed real elements. All allocated memory is freed before return of g02asc.
10
Example
This example finds the nearest correlation matrix,
, to the input,
, whilst fixing two diagonal blocks as given by
. The minimum eigenvalue of
is stipulated to be
.
and
Only the strictly lower half of is supplied in the example.
10.1
Program Text
10.2
Program Data
10.3
Program Results