NAG Library Routine Document

G02AJF

+− 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

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 \circ (G - X)‖}_{F}

, where

C = A \circ B

denotes the matrix

C

with elements

C_{i j} = A_{i j} \times B_{i j}

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 $\frac{1}{2} (G + G^{T})$ 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 G02AJF is called.
Constraint: $LDG \geq N$ .
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 $\frac{1}{2} (H + H^{T})$ with its diagonal elements set to zero and the remaining elements scaled so that the maximum element is $1.0$ .
Constraint: $H (i, j) \geq 0.0$ , for all $i$ and $j = 1, 2, \dots, n$ , $i \neq j$ .
6: LDH – INTEGERInput: On entry: the first dimension of the array H as declared in the (sub)program from which G02AJF is called.
Constraint: $LDH \geq N$ .
7: ERRTOL – REAL (KIND=nag_wp)Input: On entry: the termination tolerance for the iteration. If $ERRTOL \leq 0.0$ then $N \times \sqrt{machine precision}$ is used. See Section 7 for further details.
8: MAXIT – INTEGERInput: On entry: specifies the maximum number of iterations to be used.
If $MAXIT \leq 0$ , $200$ 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: $LDX \geq N$ .
11: ITER – INTEGEROutput: On exit: the number of iterations taken.
12: NORM – REAL (KIND=nag_wp)Output: On exit: the value of ${‖H \circ (G - X)‖}_{F}$ 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

- 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: $LDG \geq N$ .

$IFAIL = 3$: On entry, $LDH = ⟨value⟩$ and $N = ⟨value⟩$ .
Constraint: $LDH \geq N$ .

$IFAIL = 4$: On entry, $LDX = ⟨value⟩$ and $N = ⟨value⟩$ .
Constraint: $LDX \geq N$ .

$IFAIL = 5$: 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

e_{i}

is the value of NORM at the

i

th iteration, that is

e_{i} = {‖H \circ (G - X)‖}_{F},

where

H

has been scaled as described above.

Then the algorithm terminates when:

\frac{|e_{i} - e_{i - 1}|}{1 + \max (e_{i}, e_{i - 1})} \leq ERRTOL .

8 Further Comments

Arrays are internally allocated by G02AJF. The total size of these arrays is

15 \times N + 5 \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 G02AJF.

9 Example

This example finds the nearest correlation matrix to:

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

weighted by:

H = (\begin{array}{r} 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 \end{array})

with minimum eigenvalue

0.04

NAG Library Routine DocumentG02AJF