# NAG Library Routine Document

## 1Purpose

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.

## 2Specification

Fortran Interface
 Subroutine g02aef ( g, ldg, n, k, x, ldx, iter,
 Integer, Intent (In) :: ldg, n, k, maxit, ldx Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iter, feval Real (Kind=nag_wp), Intent (In) :: errtol Real (Kind=nag_wp), Intent (Inout) :: g(ldg,n), x(ldx,k) Real (Kind=nag_wp), Intent (Out) :: nrmpgd
#include nagmk26.h
 void g02aef_ (double g[], const Integer *ldg, const Integer *n, const Integer *k, const double *errtol, const Integer *maxit, double x[], const Integer *ldx, Integer *iter, Integer *feval, double *nrmpgd, Integer *ifail)

## 3Description

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 $C=X{X}^{\mathrm{T}}+\mathrm{diag}\left(I-X{X}^{\mathrm{T}}\right)$, 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 ${\mathrm{min}\phantom{\rule{0.25em}{0ex}}‖G-X{X}^{\mathrm{T}}+\mathrm{diag}\left(X{X}^{\mathrm{T}}-I\right)‖}_{F}$ such that ${‖{x}_{\mathit{i}}^{\mathrm{T}}‖}_{2}\le 1$, 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.

## 4References

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

## 5Arguments

1:     $\mathbf{g}\left({\mathbf{ldg}},{\mathbf{n}}\right)$ – 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:     $\mathbf{ldg}$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
4:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of factors and columns of $X$.
Constraint: $0<{\mathbf{k}}\le {\mathbf{n}}$.
5:     $\mathbf{errtol}$ – 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:     $\mathbf{maxit}$ – IntegerInput
On entry: specifies the maximum number of iterations in the spectral projected gradient method.
If ${\mathbf{maxit}}\le 0$, $40000$ is used.
7:     $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{k}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: contains the matrix $X$.
8:     $\mathbf{ldx}$ – 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:     $\mathbf{iter}$ – IntegerOutput
On exit: the number of steps taken in the spectral projected gradient method.
10:   $\mathbf{feval}$ – IntegerOutput
On exit: the number of evaluations of ${‖G-X{X}^{\mathrm{T}}+\mathrm{diag}\left(X{X}^{\mathrm{T}}-I\right)‖}_{F}$.
11:   $\mathbf{nrmpgd}$ – Real (Kind=nag_wp)Output
On exit: the norm of the projected gradient at the final iteration.
12:   $\mathbf{ifail}$ – 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).

## 6Error 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, ${\mathbf{k}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $0<{\mathbf{k}}\le {\mathbf{n}}$.
On entry, ${\mathbf{ldg}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldg}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{ldx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=2$
Spectral gradient method fails to converge in $〈\mathit{\text{value}}〉$ iterations.
${\mathbf{ifail}}=-99$
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.

## 7Accuracy

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

## 8Parallelism 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 implementation-specific information.

Arrays are internally allocated by g02aef. The total size of these arrays is ${\mathbf{n}}×{\mathbf{n}}+4×{\mathbf{n}}×{\mathbf{k}}+\left(\mathit{nb}+3\right)×{\mathbf{n}}+{\mathbf{n}}+50$ real elements and $6×{\mathbf{n}}$ 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.

## 10Example

This example finds the nearest correlation matrix with $k=2$ factor structure to:
 $G = 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 2$

### 10.1Program Text

Program Text (g02aefe.f90)

### 10.2Program Data

Program Data (g02aefe.d)

### 10.3Program Results

Program Results (g02aefe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017