NAG CL Interface
g02aec (corrmat_​nearest_​kfactor)

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1 Purpose

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>
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.

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+diag(I-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.
g02aec applies a spectral projected gradient method to the modified problem min(G-XXT+diag(XXT-I))F 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: 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 order=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: order=Nag_RowMajor or Nag_ColMajor.
2: g[dim] double Input/Output
Note: the dimension, dim, of the array g must be at least pdg×n.
On entry: G, the initial matrix.
On exit: a symmetric matrix 12(G+GT) with the diagonal elements set to unity.
3: 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: pdgn.
4: n Integer Input
On entry: n, the order of the matrix G.
Constraint: n>0.
5: k Integer Input
On entry: k, the number of factors and columns of X.
Constraint: 0<kn.
6: errtol double Input
On entry: the termination tolerance for the projected gradient norm. See references for further details. If errtol0.0, 0.01 is used. This is often a suitable default value.
7: maxit Integer Input
On entry: specifies the maximum number of iterations in the spectral projected gradient method.
If maxit0, 40000 is used.
8: x[dim] double Output
Note: the dimension, dim, of the array x must be at least
  • max(1,pdx×k) when order=Nag_ColMajor;
  • max(1,n×pdx) when order=Nag_RowMajor.
The (i,j)th 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: pdx Integer Input
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: iter Integer * Output
On exit: the number of steps taken in the spectral projected gradient method.
11: feval Integer * Output
On exit: the number of evaluations of G-XXT+diag(XXT-I)F.
12: nrmpgd double * Output
On exit: the norm of the projected gradient at the final iteration.
13: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
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.

7 Accuracy

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
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.

9 Further Comments

Arrays are internally allocated by 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 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.

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)