nag_nearest_correlation_bounded (g02abc) (PDF version)
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NAG C Library Manual

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nag_nearest_correlation_bounded (g02abc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_nearest_correlation_bounded (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>
#include <nagg02.h>
void  nag_nearest_correlation_bounded (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)

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:     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[pdg×n]doubleInput/Output
Note: 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 set to I.
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: the order of the matrix G.
Constraint: n>0.
5:     optNag_NearCorr_ProbTypeInput
On entry: indicates the problem to be solved.
opt=Nag_LowerBound
The lower bound problem is solved.
opt=Nag_WeightedNorm
The weighted norm problem is solved.
opt=Nag_Both
Both problems are solved.
Constraint: opt=Nag_LowerBound, Nag_WeightedNorm or Nag_Both.
6:     alphadoubleInput
On entry: the value of α.
If opt=Nag_WeightedNorm, alpha need not be set.
Constraint: 0.0<alpha<1.0.
7:     w[n]doubleInput/Output
On entry: the square roots of the diagonal elements of W, that is the diagonal of W12.
If opt=Nag_LowerBound, w need not be set.
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:     errtoldoubleInput
On entry: the termination tolerance for the Newton iteration. If errtol0.0 then n×machine precision is used.
9:     maxitsIntegerInput
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:   maxitIntegerInput
On entry: specifies the maximum number of Newton iterations.
If maxit0, 200 is used.
11:   x[pdx×n]doubleOutput
Note: 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 nearest correlation matrix.
12:   pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraint: pdxn.
13:   iterInteger *Output
On exit: the number of Newton steps taken.
14:   fevalInteger *Output
On exit: the number of function evaluations of the dual problem.
15:   nrmgrddouble *Output
On exit: the norm of the gradient of the last Newton step.
16:   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
Machine precision is limiting convergence. In this instance the returned matrix X may be useful.
Newton iteration fails to converge in value iterations. Increase maxit or check the call to the function.
NE_EIGENPROBLEM
Failure to solve intermediate eigenproblem.
NE_INT
On entry, n=value.
Constraint: n>0.
NE_INT_2
On entry, pdg=value and n=value.
Constraint: pdgn.
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.
NE_REAL
On entry, alpha=value.
Constraint: 0.0<alpha<1.0.
NE_WEIGHTS_NOT_POSITIVE
On entry, all elements of w were not positive.

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_bounded (g02abc). The total size of these arrays is 12×n+3×n×n+max2×n×n+6×n+1,120+9×n double elements and 5×n+3 Integer elements. All allocated memory is freed before return of nag_nearest_correlation_bounded (g02abc).

9  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.

9.1  Program Text

Program Text (g02abce.c)

9.2  Program Data

Program Data (g02abce.d)

9.3  Program Results

Program Results (g02abce.r)


nag_nearest_correlation_bounded (g02abc) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG C Library Manual

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