nag_nearest_correlation_k_factor (g02aec) (PDF version)
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NAG C 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 characterized 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 3.2.1.3 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.
Constraints:
  • 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 minG-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

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
Spectral gradient method fails to converge in value iterations.
NE_INT
On entry, n=value.
Constraint: n>0.
NE_INT_2
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.
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.

8  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.

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)


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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012