NAG CL Interface
g02abc (corrmat_​nearest_​bounded)

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1 Purpose

g02abc computes the nearest correlation matrix, in the Frobenius norm or weighted Frobenius norm, and optionally with bounds on the eigenvalues, to a given square, input matrix.

2 Specification

#include <nag.h>
void  g02abc (Nag_OrderType order, double g[], Integer pdg, Integer n, Nag_NearCorr_ProbType opt, double alpha, double w[], double errtol, Integer maxits, Integer maxit, double x[], Integer pdx, Integer *iter, Integer *feval, double *nrmgrd, NagError *fail)
The function may be called by the names: g02abc, nag_correg_corrmat_nearest_bounded or nag_nearest_correlation_bounded.

3 Description

Finds the nearest correlation matrix X by minimizing 12G-X2 where G is an approximate correlation matrix.
The norm can either be the Frobenius norm or the weighted Frobenius norm 12 W12(G-X)W12 F 2 .
You can optionally specify a lower bound on the eigenvalues, α, of the computed correlation matrix, forcing the matrix to be positive definite, 0<α<1.
Note that if the weights vary by several orders of magnitude from one another the algorithm may fail to converge.

4 References

Borsdorf R and Higham N J (2010) A preconditioned (Newton) algorithm for the nearest correlation matrix IMA Journal of Numerical Analysis 30(1) 94–107
Qi H and Sun D (2006) A quadratically convergent Newton method for computing the nearest correlation matrix SIAM J. Matrix AnalAppl 29(2) 360–385

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: G is overwritten. The initial change is to obtain a starting matrix G by symmetrizing and setting its diagonal elements 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: the order of the matrix G.
Constraint: n>0.
5: opt Nag_NearCorr_ProbType Input
On entry: indicates the problem to be solved.
The lower bound problem is solved.
The weighted norm problem is solved.
Both problems are solved.
Constraint: opt=Nag_LowerBound, Nag_WeightedNorm or Nag_Both.
6: alpha double Input
On entry: the value of α.
If opt=Nag_WeightedNorm, alpha need not be set.
Constraint: 0.0<alpha<1.0.
7: w[dim] double Input/Output
Note: the dimension, dim, of the array w must be at least
  • n when optNag_LowerBound;
  • otherwise w may be NULL.
On entry: the square roots of the diagonal elements of W, that is the diagonal of W12.
If opt=Nag_LowerBound, w is not referenced and may be NULL.
On exit: if opt=Nag_WeightedNorm or Nag_Both, the array is scaled so 0<w[i-1]1, for i=1,2,,n.
Constraint: w[i-1]>0.0, for i=1,2,,n.
8: errtol double Input
On entry: the termination tolerance for the Newton iteration. If errtol0.0, n×machine precision is used.
9: maxits Integer Input
On entry: specifies the maximum number of iterations to be used by the iterative scheme to solve the linear algebraic equations at each Newton step.
If maxits0, 2×n is used.
10: maxit Integer Input
On entry: specifies the maximum number of Newton iterations.
If maxit0, 200 is used.
11: x[dim] double Output
Note: the dimension, dim, of the array x must be at least pdx×n.
On exit: contains the nearest correlation matrix.
12: pdx Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix X in the array x.
Constraint: pdxn.
13: iter Integer * Output
On exit: the number of Newton steps taken.
14: feval Integer * Output
On exit: the number of function evaluations of the dual problem.
15: nrmgrd double * Output
On exit: the norm of the gradient of the last Newton step.
16: 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.
Newton iteration fails to converge in value iterations. Increase maxit or check the call to the function.
The machine precision is limiting convergence. In this instance the returned value of x may be useful.
An intermediate eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.
On entry, n=value.
Constraint: n>0.
On entry, pdg=value and n=value.
Constraint: pdgn.
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.
On entry, alpha=value.
Constraint: 0.0<alpha<1.0.
On entry, all elements of w were not positive.
Constraint: w[i-1]>0.0, for all i.

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.
g02abc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02abc 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 g02abc. The total size of these arrays is 12×n+3×n×n+max(2×n×n+6×n+1,120+9×n) double elements and 5×n+3 Integer elements. All allocated memory is freed before return of g02abc.

10 Example

This example finds the nearest correlation matrix to:
G = ( 2 −1 0 0 −1 2 −1 0 0 −1 2 −1 0 0 −1 2 )  
weighted by W12 = diag(100,20,20,20) with minimum eigenvalue 0.02.

10.1 Program Text

Program Text (g02abce.c)

10.2 Program Data

Program Data (g02abce.d)

10.3 Program Results

Program Results (g02abce.r)