g02 Chapter Contents
g02 Chapter Introduction
NAG C 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 characterized 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 ${\mathrm{min}\phantom{\rule{0.25em}{0ex}}‖G-X{X}^{\mathrm{T}}+\mathrm{diag}\left(X{X}^{\mathrm{T}}-I\right)‖}_{F}$ 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.

## 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:     orderNag_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 Nag_ColMajor.
2:     g[$\mathit{dim}$]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:     pdgIntegerInput
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:     nIntegerInput
On entry: $n$, the order of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
5:     kIntegerInput
On entry: $k$, the number of factors and columns of $X$.
Constraint: $0<{\mathbf{k}}\le {\mathbf{n}}$.
6:     errtoldoubleInput
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:     maxitIntegerInput
On entry: specifies the maximum number of iterations in the spectral projected gradient method.
If ${\mathbf{maxit}}\le 0$, $40000$ is used.
8:     x[$\mathit{dim}$]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:     pdxIntegerInput
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:   iterInteger *Output
On exit: the number of steps taken in the spectral projected gradient method.
11:   fevalInteger *Output
On exit: the number of evaluations ${\mathrm{min}\phantom{\rule{0.25em}{0ex}}‖G-X{X}^{\mathrm{T}}+\mathrm{diag}\left(X{X}^{\mathrm{T}}-I\right)‖}_{F}$.
12:   nrmpgddouble *Output
On exit: the norm of the projected gradient at the final iteration.
13:   failNagError *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.
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.

## 7  Accuracy

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

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.

## 9  Example

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

### 9.1  Program Text

Program Text (g02aece.c)

### 9.2  Program Data

Program Data (g02aece.d)

### 9.3  Program Results

Program Results (g02aece.r)