NAG Library Function Document
nag_nearest_correlation_bounded (g02abc)
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 by minimizing where is an approximate correlation matrix.
The norm can either be the Frobenius norm or the weighted Frobenius norm .
You can optionally specify a lower bound on the eigenvalues, , of the computed correlation matrix, forcing the matrix to be positive definite, .
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_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
g[] – doubleInput/Output
-
Note: the
th element of the matrix
is stored in
- when ;
- when .
On entry: , the initial matrix.
On exit: is overwritten.
- 3:
pdg – IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
g.
Constraint:
.
- 4:
n – IntegerInput
On entry: the order of the matrix .
Constraint:
.
- 5:
opt – Nag_NearCorr_ProbTypeInput
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:
, or .
- 6:
alpha – doubleInput
On entry: the value of
.
If
,
alpha need not be set.
Constraint:
.
- 7:
w[n] – doubleInput/Output
On entry: the square roots of the diagonal elements of
, that is the diagonal of
.
If
,
w need not be set.
On exit: if or , the array is scaled so
, for .
Constraint:
, for .
- 8:
errtol – doubleInput
On entry: the termination tolerance for the Newton iteration. If then is used.
- 9:
maxits – IntegerInput
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 , is used.
- 10:
maxit – IntegerInput
On entry: specifies the maximum number of Newton iterations.
If , is used.
- 11:
x[] – doubleOutput
-
Note: the
th element of the matrix
is stored in
- when ;
- when .
On exit: contains the nearest correlation matrix.
- 12:
pdx – IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraint:
.
- 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 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE
-
Newton iteration fails to converge in
iterations. Increase
maxit or check the call to the function.
The machine precision is limiting convergence. In this instance the returned matrix may be useful.
- NE_EIGENPROBLEM
-
Failure to solve intermediate eigenproblem.
- NE_INT
-
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.
- NE_REAL
-
On entry, .
Constraint: .
- 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 Parallelism and Performance
nag_nearest_correlation_bounded (g02abc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nearest_correlation_bounded (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
Users' Note for your implementation for any additional implementation-specific information.
Arrays are internally allocated by nag_nearest_correlation_bounded (g02abc). The total size of these arrays is double elements and Integer elements. All allocated memory is freed before return of nag_nearest_correlation_bounded (g02abc).
10 Example
This example finds the nearest correlation matrix to:
weighted by
with minimum eigenvalue
.
10.1 Program Text
Program Text (g02abce.c)
10.2 Program Data
Program Data (g02abce.d)
10.3 Program Results
Program Results (g02abce.r)