NAG Library Routine Document

G02AEF

+− 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

G02AEF computes the factor loading matrix associated with the nearest correlation matrix with

k

-factor structure, in the Frobenius norm, to a given square, input matrix.

2 Specification

SUBROUTINE G02AEF (

G, LDG, N, K, ERRTOL, MAXIT, X, LDX, ITER, FEVAL, NRMPGD, IFAIL)

INTEGER	LDG, N, K, MAXIT, LDX, ITER, FEVAL, IFAIL
REAL (KIND=nag_wp)	G(LDG,N), ERRTOL, X(LDX,K), NRMPGD

3 Description

A correlation matrix

C

with

k

-factor structure may be characterised as a real square matrix that is symmetric, has a unit diagonal, is positive semidefinite and can be written as

C = X X^{T} + diag (I - X X^{T})

, where

I

is the identity matrix and

X

has

n

rows and

k

columns.

X

is often referred to as the factor loading matrix.

G02AEF applies a spectral projected gradient method to the modified problem

{\min ‖G - X X^{T} + diag (X X^{T} - I)‖}_{F}

such that

{‖x_{i}^{T}‖}_{2} \leq 1

, for

i = 1, 2, \dots, n

, where

x_{i}

is the

i

th row of the factor loading matrix,

X

, which gives us the solution.

4 References

Birgin E G, Martínez J M and Raydan M (2001) Algorithm 813: SPG–software for convex-constrained optimization ACM Trans. Math. Software 27 340–349

Borsdorf R, Higham N J and Raydan M (2010) Computing a nearest correlation matrix with factor structure. SIAM J. Matrix Anal. Appl. 31(5) 2603–2622

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 elements set to unity.
2: LDG – INTEGERInput: On entry: the first dimension of the array G as declared in the (sub)program from which G02AEF is called.
Constraint: $LDG \geq N$ .
3: N – INTEGERInput: On entry: $n$ , the order of the matrix $G$ .
Constraint: $N > 0$ .
4: K – INTEGERInput: On entry: $k$ , the number of factors and columns of $X$ .
Constraint: $0 < K \leq N$ .
5: ERRTOL – REAL (KIND=nag_wp)Input: On entry: the termination tolerance for the projected gradient norm. See references for further details. If $ERRTOL \leq 0.0$ then $0.01$ is used. This is often a suitable default value.
6: MAXIT – INTEGERInput: On entry: specifies the maximum number of iterations in the spectral projected gradient method.
If $MAXIT \leq 0$ , $40000$ is used.
7: X(LDX,K) – REAL (KIND=nag_wp) arrayOutput: On exit: contains the matrix $X$ .
8: LDX – INTEGERInput: On entry: the first dimension of the array X as declared in the (sub)program from which G02AEF is called.
Constraint: $LDX \geq N$ .
9: ITER – INTEGEROutput: On exit: the number of steps taken in the spectral projected gradient method.
10: FEVAL – INTEGEROutput: On exit: the number of evaluations ${\min ‖G - X X^{T} + diag (X X^{T} - I)‖}_{F}$ .
11: NRMPGD – REAL (KIND=nag_wp)Output: On exit: the norm of the projected gradient at the final iteration.
12: 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	$K \leq 0$ ,
or	$K > N$ ,

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

$IFAIL = - 999$: 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 G02AEF. The total size of these arrays is

N \times N + 4 \times N \times K + (nb + 3) \times N + N + 50

real elements and

6 \times N

integer elements. Here

nb

is the block size required for optimal performance by F08FEF (DSYTRD) and F08FGF (DORMTR) which are called internally. All allocated memory is freed before return of G02AEF.

See G03CAF for constructing the factor loading matrix from a known correlation matrix.

9 Example

This example finds the nearest correlation matrix with

k = 2

factor structure to:

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

NAG Library Routine DocumentG02AEF