nag_nearest_correlation_k_factor (g02aec) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_nearest_correlation_k_factor (g02aec)

+ Contents

    1  Purpose
    7  Accuracy

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 <nag.h>
#include <nagg02.h>
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=XXT+diagI-XXT, 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 minG-XXT+diagXXT-IF such that xiT21, for i=1,2,,n, where xi is the ith 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 order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     g[dim]doubleInput/Output
Note: the dimension, dim, of the array g must be at least pdg×n.
The i,jth element of the matrix G is stored in
  • g[j-1×pdg+i-1] when order=Nag_ColMajor;
  • g[i-1×pdg+j-1] when order=Nag_RowMajor.
On entry: G, the initial matrix.
On exit: a symmetric matrix 12G+GT 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: pdgn.
4:     nIntegerInput
On entry: n, the order of the matrix G.
Constraint: n>0.
5:     kIntegerInput
On entry: k, the number of factors and columns of X.
Constraint: 0<kn.
6:     errtoldoubleInput
On entry: the termination tolerance for the projected gradient norm. See references for further details. If errtol0.0 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 maxit0, 40000 is used.
8:     x[dim]doubleOutput
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×k when order=Nag_ColMajor;
  • max1,n×pdx when order=Nag_RowMajor.
The i,jth element of the matrix X is stored in
  • x[j-1×pdx+i-1] when order=Nag_ColMajor;
  • x[i-1×pdx+j-1] when order=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.
  • if order=Nag_ColMajor, pdxn;
  • if order=Nag_RowMajor, pdxk.
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 of G-XXT+diagXXT-IF.
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

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
Spectral gradient method fails to converge in value iterations.
On entry, n=value.
Constraint: n>0.
On entry, k=value and n=value.
Constraint: 0<kn.
On entry, pdg=value and n=value.
Constraint: pdgn.
On entry, pdx=value and k=value.
Constraint: pdxk.
On entry, pdx=value and n=value.
Constraint: pdxn.
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.

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 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 n×n+4×n×k+nb+3×n+n+50 double elements and 6×n Integer elements. There is an additional n×k double elements if order=Nag_RowMajor. Here 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)

nag_nearest_correlation_k_factor (g02aec) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014