NAG Library Routine Document

G02ABF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G02ABF computes the nearest correlation matrix, in the Frobenius norm or weighted Frobenius norm, and optionally with bounds on the eigenvalues, to a given square, input matrix.

2 Specification

SUBROUTINE G02ABF (

G, LDG, N, OPT, ALPHA, W, ERRTOL, MAXITS, MAXIT, X, LDX, ITER, FEVAL, NRMGRD, IFAIL)

INTEGER	LDG, N, MAXITS, MAXIT, LDX, ITER, FEVAL, IFAIL
REAL (KIND=nag_wp)	G(LDG,N), ALPHA, W(N), ERRTOL, X(LDX,N), NRMGRD
CHARACTER(1)	OPT

3 Description

Finds the nearest correlation matrix

X

by minimizing

\frac{1}{2} {‖G - X‖}^{2}

where

G

is an approximate correlation matrix.

The norm can either be the Frobenius norm or the weighted Frobenius norm

\frac{1}{2} {‖W^{\frac{1}{2}} (G - X) W^{\frac{1}{2}}‖}_{F}^{2}

You can optionally specify a lower bound on the eigenvalues,

α

, of the computed correlation matrix, forcing the matrix to be positive definite,

0 < α < 1

Note that if the weights 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

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

\frac{1}{2} (G + G^{T})

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 G02ABF is called.

Constraint:

LDG \geq N

3: N – INTEGERInput

On entry: the order of the matrix

G

Constraint:

N > 0

4: OPT – CHARACTER(1)Input

On entry: indicates the problem to be solved.

$OPT ='A'$: The lower bound problem is solved.
$OPT ='W'$: The weighted norm problem is solved.
$OPT ='B'$: Both problems are solved.

Constraint:

OPT ='A'

'W'

'B'

5: ALPHA – REAL (KIND=nag_wp)Input

On entry: the value of

α

OPT ='W'

, ALPHA need not be set.

Constraint:

0.0 < ALPHA < 1.0

6: W(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the square roots of the diagonal elements of

W

, that is the diagonal of

W^{\frac{1}{2}}

OPT ='A'

, W need not be set.

On exit: if

OPT ='W'

'B'

, the array is scaled so

0 < W (i) \leq 1

, for

i = 1, 2, \dots, n

Constraint:

W (i) > 0.0

, for

i = 1, 2, \dots, n

7: ERRTOL – REAL (KIND=nag_wp)Input

On entry: the termination tolerance for the Newton iteration. If

ERRTOL \leq 0.0

then

N \times \sqrt{machine precision}

is used.

8: MAXITS – INTEGERInput

On entry: specifies the maximum number of iterations to be used by the iterative scheme to solve the linear algebraic equations at each Newton step.

MAXITS \leq 0

2 \times N

is used.

9: MAXIT – INTEGERInput

On entry: specifies the maximum number of Newton iterations.

MAXIT \leq 0

200

is used.

10: X(LDX,N) – REAL (KIND=nag_wp) arrayOutput

On exit: contains the nearest correlation matrix.

11: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G02ABF is called.

Constraint:

LDX \geq N

12: ITER – INTEGEROutput

On exit: the number of Newton steps taken.

13: FEVAL – INTEGEROutput

On exit: the number of function evaluations of the dual problem.

14: NRMGRD – REAL (KIND=nag_wp)Output

On exit: the norm of the gradient of the last Newton step.

15: 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

- 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,	$N \leq 0$ ,
or	$LDG < N$ ,
or	$LDX < N$ .
or	$OPT \neq'A'$ , $'W'$ or $'B'$ .
or	$ALPHA \leq 0.0$ or $ALPHA \geq 1.0$ .
or	$W (i) \leq 0$ for at least one $i = 1, 2, \dots, n$ .

$IFAIL = 2$: The routine fails to converge in MAXIT iterations. Increase MAXIT or check the call to the routine.

$IFAIL = 3$: machine precision is limiting convergence. In this instance the returned value of X may be useful.

$IFAIL = 4$: An internal eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.

$IFAIL = - 999$: Internal memory allocation failed.

7 Accuracy

The returned accuracy is controlled by ERRTOL and limited by machine precision.

8 Further Comments

Arrays are internally allocated by G02ABF. The total size of these arrays is

12 \times N + 3 \times N \times N + \max (2 \times N \times N + 6 \times N + 1, 120 + 9 \times N)

real elements and

5 \times N + 3

integer elements. All allocated memory is freed before return of G02ABF.

9 Example

This example finds the nearest correlation matrix to:

G = (\begin{matrix} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 2 \end{matrix})

weighted by

W^{\frac{1}{2}} = diag (100, 20, 20, 20)

with minimum eigenvalue

0.02

NAG Library Routine DocumentG02ABF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G02ABF