NAG Library Routine Document
G02AJF
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
SUBROUTINE G02AJF ( |
G, LDG, N, ALPHA, H, LDH, ERRTOL, MAXIT, X, LDX, ITER, NORM, IFAIL) |
INTEGER |
LDG, N, LDH, MAXIT, LDX, ITER, IFAIL |
REAL (KIND=nag_wp) |
G(LDG,N), ALPHA, H(LDH,N), ERRTOL, X(LDX,N), NORM |
|
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=A∘B 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 (To appear) An inexact accelerated proximal gradient method for large scale linearly constrained convex SDP
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 Parameters
- 1: G(LDG,N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: G, the initial matrix.
On exit: a symmetric matrix 12G+GT with the diagonal set to I.
- 2: LDG – INTEGERInput
On entry: the first dimension of the array
G as declared in the (sub)program from which G02AJF is called.
Constraint:
LDG≥N.
- 3: N – INTEGERInput
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: H(LDH,N) – REAL (KIND=nag_wp) arrayInput/Output
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:
Hij≥0.0, for all i and j=1,2,…,n, i≠j.
- 6: LDH – INTEGERInput
On entry: the first dimension of the array
H as declared in the (sub)program from which G02AJF is called.
Constraint:
LDH≥N.
- 7: ERRTOL – REAL (KIND=nag_wp)Input
-
On entry: the termination tolerance for the iteration. If
ERRTOL≤0.0 then
N×machine precision is used. See
Section 7 for further details.
- 8: MAXIT – INTEGERInput
-
On entry: specifies the maximum number of iterations to be used.
If MAXIT≤0, 200 is used.
- 9: X(LDX,N) – REAL (KIND=nag_wp) arrayOutput
On exit: contains the nearest correlation matrix.
- 10: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which G02AJF is called.
Constraint:
LDX≥N.
- 11: ITER – INTEGEROutput
On exit: the number of iterations taken.
- 12: NORM – REAL (KIND=nag_wp)Output
On exit: the value of H∘G-XF after the final iteration.
- 13: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
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: LDG≥N.
- IFAIL=3
-
On entry, LDH=value and N=value.
Constraint: LDH≥N.
- IFAIL=4
-
On entry, LDX=value and N=value.
Constraint: LDX≥N.
- 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 fails 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=-999
-
Dynamic memory allocation failed.
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
where
H has been scaled as described above.
Then the algorithm terminates when:
8 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.
9 Example
This example finds the nearest correlation matrix to:
weighted by:
with minimum eigenvalue
0.04.
9.1 Program Text
Program Text (g02ajfe.f90)
9.2 Program Data
Program Data (g02ajfe.d)
9.3 Program Results
Program Results (g02ajfe.r)