The function may be called by the names: g02aac, nag_correg_corrmat_nearest or nag_nearest_correlation.
3Description
A correlation matrix may be characterised as a real square matrix that is symmetric, has a unit diagonal and is positive semidefinite.
g02aac applies an inexact Newton method to a dual formulation of the problem, as described by Qi and Sun (2006). It applies the improvements suggested by Borsdorf and Higham (2010).
4References
Borsdorf R and Higham N J (2010) A preconditioned (Newton) algorithm for the nearest correlation matrix IMA Journal of Numerical Analysis30(1) 94–107
Qi H and Sun D (2006) A quadratically convergent Newton method for computing the nearest correlation matrix SIAM J. Matrix AnalAppl29(2) 360–385
5Arguments
1: – Nag_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 . 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:
or .
2: – doubleInput/Output
Note: the dimension, dim, of the array
g
must be at least
.
On entry: , the initial matrix.
On exit: a symmetric matrix with the diagonal set to .
3: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
g.
Constraint:
.
4: – IntegerInput
On entry: the size of the matrix .
Constraint:
.
5: – doubleInput
On entry: the termination tolerance for the Newton iteration. If , is used.
6: – IntegerInput
On entry: maxits specifies the maximum number of iterations used for the iterative scheme used to solve the linear algebraic equations at each Newton step.
If , is used.
7: – IntegerInput
On entry: specifies the maximum number of Newton iterations.
If , is used.
8: – doubleOutput
Note: the dimension, dim, of the array
x
must be at least
.
On exit: contains the nearest correlation matrix.
9: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
x.
Constraint:
.
10: – Integer *Output
On exit: the number of Newton steps taken.
11: – Integer *Output
On exit: the number of function evaluations of the dual problem.
12: – double *Output
On exit: the norm of the gradient of the last Newton step.
13: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_CONVERGENCE
Machine precision is limiting convergence. The array returned in x may still be of interest.
Newton iteration fails to converge in iterations.
NE_EIGENPROBLEM
An intermediate eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.
NE_INT
On entry, .
Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
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.
7Accuracy
The returned accuracy is controlled by errtol and limited by machine precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g02aac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02aac 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.
9Further Comments
Arrays are internally allocated by g02aac. The total size of these arrays is real elements and integer elements.
10Example
This example finds the nearest correlation matrix to the matrix , where
The example also demonstrates how a modified Cholesky factorization (computed using f01mdc and f01mec) can be used to obtain a bound on the distance to the nearest correlation matrix prior to computing the NCM itself.