g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_nearest_correlation_k_factor (g02aec)

## 1  Purpose

nag_nearest_correlation_k_factor (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.

## 2  Specification

 #include #include
 void nag_nearest_correlation_k_factor (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)

## 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 $C=X{X}^{\mathrm{T}}+\mathrm{diag}\left(I-X{X}^{\mathrm{T}}\right)$, where $I$ is the identity matrix and $X$ has $n$ rows and $k$ columns. $X$ is often referred to as the factor loading matrix.
nag_nearest_correlation_k_factor (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.

## 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  Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{g}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array g must be at least ${\mathbf{pdg}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $G$ is stored in
• ${\mathbf{g}}\left[\left(j-1\right)×{\mathbf{pdg}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{g}}\left[\left(i-1\right)×{\mathbf{pdg}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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:    $\mathbf{pdg}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array g.
Constraint: ${\mathbf{pdg}}\ge {\mathbf{n}}$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
5:    $\mathbf{k}$IntegerInput
On entry: $k$, the number of factors and columns of $X$.
Constraint: $0<{\mathbf{k}}\le {\mathbf{n}}$.
6:    $\mathbf{errtol}$doubleInput
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.
7:    $\mathbf{maxit}$IntegerInput
On entry: specifies the maximum number of iterations in the spectral projected gradient method.
If ${\mathbf{maxit}}\le 0$, $40000$ is used.
8:    $\mathbf{x}\left[\mathit{dim}\right]$doubleOutput
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:    $\mathbf{pdx}$IntegerInput
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 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction 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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

## 7  Accuracy

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

## 8  Parallelism and Performance

nag_nearest_correlation_k_factor (g02aec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nearest_correlation_k_factor (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.

## 9  Further Comments

Arrays are internally allocated by nag_nearest_correlation_k_factor (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 nag_dsytrd (f08fec) and nag_dormtr (f08fgc) which are called internally. All allocated memory is freed before return of nag_nearest_correlation_k_factor (g02aec).
See nag_mv_factor (g03cac) 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:
 $G = 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 2$

### 10.1  Program Text

Program Text (g02aece.c)

### 10.2  Program Data

Program Data (g02aece.d)

### 10.3  Program Results

Program Results (g02aece.r)