The routine may be called by the names g02apf or nagf_correg_corrmat_target.
Starting from an approximate correlation matrix, , g02apf finds a correlation matrix, , which has the form
where and is a target matrix. denotes the matrix with elements . is a matrix of weights that defines the target matrix. The target matrix must be positive definite and thus have off-diagonal elements less than in magnitude. A value of in essentially fixes an element in so it is unchanged in .
g02apf utilizes a shrinking method to find the minimum value of such that is positive definite with unit diagonal and with a smallest eigenvalue of at least times the smallest eigenvalue of the target matrix.
Higham N J, Strabić N and Šego V (2014) Restoring definiteness via shrinking, with an application to correlation matrices with a fixed block MIMS EPrint 2014.54 Manchester Institute for Mathematical Sciences, The University of Manchester, UK
1: – Real (Kind=nag_wp) arrayInput/Output
On entry: , the initial matrix.
On exit: a symmetric matrix with the diagonal elements set to .
2: – IntegerInput
On entry: the first dimension of the array g as declared in the (sub)program from which g02apf is called.
3: – IntegerInput
On entry: the order of the matrix .
4: – Real (Kind=nag_wp)Input
On entry: the value of . If , is used.
5: – Real (Kind=nag_wp) arrayInput/Output
On entry: the matrix of weights .
On exit: a symmetric matrix with its diagonal elements set to .
6: – IntegerInput
On entry: the first dimension of the array h as declared in the (sub)program from which g02apf is called.
7: – Real (Kind=nag_wp)Input
On entry: the termination tolerance for the iteration.
On entry: the tolerance used in determining the definiteness of the target matrix .
If , where and denote the minimum and maximum eigenvalues of respectively, is positive definite.
If , machine precision is used.
9: – Real (Kind=nag_wp) arrayOutput
On exit: contains the matrix .
10: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02apf is called.
11: – Real (Kind=nag_wp)Output
On exit: the constant used in the formation of .
12: – IntegerOutput
On exit: the number of iterations taken.
13: – Real (Kind=nag_wp)Output
On exit: the smallest eigenvalue of the target matrix .
14: – Real (Kind=nag_wp)Output
On exit: the value of after the final iteration.
15: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, .
On entry, and .
On entry, .
On entry, and .
On entry, and .
The target matrix is not positive definite.
Failure to solve intermediate eigenproblem. This should not occur. Please contact NAG.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
The algorithm uses a bisection method. It is terminated when the computed is within errtol of the minimum value.
Note: when is zero is still positive definite, in that it can be successfully factorized with a call to f07fdf.
The number of iterations taken for the bisection will be:
8Parallelism and Performance
g02apf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02apf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
Arrays are internally allocated by g02apf. The total size of these arrays does not exceed real elements. All allocated memory is freed before return of g02apf.
This example finds the smallest such that is a correlation matrix. The leading principal submatrix of the input is preserved, and the last diagonal block is weighted to give some emphasis to the off diagonal elements.