NAG Library Routine Document
G02AEF
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 $C=X{X}^{\mathrm{T}}+\mathrm{diag}\left(IX{X}^{\mathrm{T}}\right)$, 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 ${\mathrm{min}\phantom{\rule{0.25em}{0ex}}\Vert GX{X}^{\mathrm{T}}+\mathrm{diag}\left(X{X}^{\mathrm{T}}I\right)\Vert}_{F}$ such that ${\Vert {x}_{\mathit{i}}^{\mathrm{T}}\Vert}_{2}\le 1$, for $\mathit{i}=1,2,\dots ,n$, 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 convexconstrained 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 Arguments
 1: $\mathrm{G}\left({\mathbf{LDG}},{\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: $G$, the initial matrix.
On exit: a symmetric matrix $\frac{1}{2}\left(G+{G}^{\mathrm{T}}\right)$ with the diagonal elements set to unity.
 2: $\mathrm{LDG}$ – INTEGERInput

On entry: the first dimension of the array
G as declared in the (sub)program from which G02AEF is called.
Constraint:
${\mathbf{LDG}}\ge {\mathbf{N}}$.
 3: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrix $G$.
Constraint:
${\mathbf{N}}>0$.
 4: $\mathrm{K}$ – INTEGERInput

On entry: $k$, the number of factors and columns of $X$.
Constraint:
$0<{\mathbf{K}}\le {\mathbf{N}}$.
 5: $\mathrm{ERRTOL}$ – REAL (KIND=nag_wp)Input

On entry: the termination tolerance for the projected gradient norm. See references for further details. If ${\mathbf{ERRTOL}}\le 0.0$ then $0.01$ is used. This is often a suitable default value.
 6: $\mathrm{MAXIT}$ – INTEGERInput

On entry: specifies the maximum number of iterations in the spectral projected gradient method.
If ${\mathbf{MAXIT}}\le 0$, $40000$ is used.
 7: $\mathrm{X}\left({\mathbf{LDX}},{\mathbf{K}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: contains the matrix $X$.
 8: $\mathrm{LDX}$ – INTEGERInput

On entry: the first dimension of the array
X as declared in the (sub)program from which G02AEF is called.
Constraint:
${\mathbf{LDX}}\ge {\mathbf{N}}$.
 9: $\mathrm{ITER}$ – INTEGEROutput

On exit: the number of steps taken in the spectral projected gradient method.
 10: $\mathrm{FEVAL}$ – INTEGEROutput

On exit: the number of evaluations of ${\Vert GX{X}^{\mathrm{T}}+\mathrm{diag}\left(X{X}^{\mathrm{T}}I\right)\Vert}_{F}$.
 11: $\mathrm{NRMPGD}$ – REAL (KIND=nag_wp)Output

On exit: the norm of the projected gradient at the final iteration.
 12: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

On entry, ${\mathbf{K}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $0<{\mathbf{K}}\le {\mathbf{N}}$.
On entry, ${\mathbf{LDG}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{LDG}}\ge {\mathbf{N}}$.
On entry, ${\mathbf{LDX}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{LDX}}\ge {\mathbf{N}}$.
On entry, ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{N}}>0$.
 ${\mathbf{IFAIL}}=2$

Spectral gradient method fails to converge in $\u2329\mathit{\text{value}}\u232a$ iterations.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7 Accuracy
The returned accuracy is controlled by
ERRTOL and limited by
machine precision.
8 Parallelism and Performance
G02AEF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
G02AEF 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 routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
Arrays are internally allocated by G02AEF. The total size of these arrays is
${\mathbf{N}}\times {\mathbf{N}}+4\times {\mathbf{N}}\times {\mathbf{K}}+\left(\mathit{nb}+3\right)\times {\mathbf{N}}+{\mathbf{N}}+50$ real elements and
$6\times {\mathbf{N}}$ integer elements.
Here
$\mathit{nb}$ 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.
10 Example
This example finds the nearest correlation matrix with
$k=2$ factor structure to:
10.1 Program Text
Program Text (g02aefe.f90)
10.2 Program Data
Program Data (g02aefe.d)
10.3 Program Results
Program Results (g02aefe.r)