# NAG CL Interfaceg02aec (corrmat_​nearest_​kfactor)

## 1Purpose

g02aec 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

 #include
 void g02aec (Nag_OrderType order, double g[], Integer pdg, Integer n, Integer k, double errtol, Integer maxit, double x[], Integer pdx, Integer *iter, Integer *feval, double *nrmpgd, NagError *fail)
The function may be called by the names: g02aec, nag_correg_corrmat_nearest_kfactor or nag_nearest_correlation_k_factor.

## 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.
g02aec applies a spectral projected gradient method to the modified problem ${\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(‖G-X{X}^{\mathrm{T}}+\mathrm{diag}\left(X{X}^{\mathrm{T}}-I\right)‖\right)}_{F}$ 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.
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: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{g}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array g must be at least ${\mathbf{pdg}}×{\mathbf{n}}$.
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.
3: $\mathbf{pdg}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $G$ in the array g.
Constraint: ${\mathbf{pdg}}\ge {\mathbf{n}}$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
5: $\mathbf{k}$Integer Input
On entry: $k$, the number of factors and columns of $X$.
Constraint: $0<{\mathbf{k}}\le {\mathbf{n}}$.
6: $\mathbf{errtol}$double Input
On entry: the termination tolerance for the projected gradient norm. See references for further details. If ${\mathbf{errtol}}\le 0.0$, $0.01$ is used. This is often a suitable default value.
7: $\mathbf{maxit}$Integer Input
On entry: specifies the maximum number of iterations in the spectral projected gradient method.
If ${\mathbf{maxit}}\le 0$, $40000$ is used.
8: $\mathbf{x}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdx}}×{\mathbf{k}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdx}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $X$ is stored in
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: contains the matrix $X$.
9: $\mathbf{pdx}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge {\mathbf{n}}$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge {\mathbf{k}}$.
10: $\mathbf{iter}$Integer * Output
On exit: the number of steps taken in the spectral projected gradient method.
11: $\mathbf{feval}$Integer * Output
On exit: the number of evaluations of ${‖G-X{X}^{\mathrm{T}}+\mathrm{diag}\left(X{X}^{\mathrm{T}}-I\right)‖}_{F}$.
12: $\mathbf{nrmpgd}$double * Output
On exit: the norm of the projected gradient at the final iteration.
13: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
Spectral gradient method fails to converge in $〈\mathit{\text{value}}〉$ iterations.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
NE_INT_2
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $0<{\mathbf{k}}\le {\mathbf{n}}$.
On entry, ${\mathbf{pdg}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdg}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{k}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

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

## 8Parallelism and Performance

g02aec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02aec 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Arrays are internally allocated by g02aec. The total size of these arrays is ${\mathbf{n}}×{\mathbf{n}}+4×{\mathbf{n}}×{\mathbf{k}}+\left(\mathit{nb}+3\right)×{\mathbf{n}}+{\mathbf{n}}+50$ double elements and $6×{\mathbf{n}}$ Integer elements. There is an additional ${\mathbf{n}}×{\mathbf{k}}$ double elements if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. Here $\mathit{nb}$ is the block size required for optimal performance by f08fec and f08fgc which are called internally. All allocated memory is freed before return of g02aec.
See g03cac 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 (g02aece.c)

### 10.2Program Data

Program Data (g02aece.d)

### 10.3Program Results

Program Results (g02aece.r)