The routine may be called by the names g02anf or nagf_correg_corrmat_shrinking.
g02anf finds a correlation matrix, , starting from an approximate correlation matrix, , with positive definite leading principal submatrix of order . The returned correlation matrix, , has the following structure:
where is the leading principal submatrix of the input matrix and positive definite, and .
g02anf utilizes a shrinking method to find the minimum value of such that is positive definite with unit diagonal.
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 set to .
2: – IntegerInput
On entry: the first dimension of the array g as declared in the (sub)program from which g02anf is called.
3: – IntegerInput
On entry: the order of the matrix .
4: – IntegerInput
On entry: , the order of the leading principal submatrix .
5: – Real (Kind=nag_wp)Input
On entry: the termination tolerance for the iteration.
On entry: the tolerance used in determining the definiteness of .
If , where and denote the minimum and maximum eigenvalues of respectively, is positive definite.
If , machine precision is used.
7: – Real (Kind=nag_wp) arrayOutput
On exit: contains the matrix .
8: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02anf is called.
9: – Real (Kind=nag_wp)Output
On exit: .
10: – IntegerOutput
On exit: the number of iterations taken.
11: – Real (Kind=nag_wp)Output
On exit: the smallest eigenvalue of the leading principal submatrix .
12: – Real (Kind=nag_wp)Output
On exit: the value of after the final iteration.
13: – 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, and .
On entry, and .
The principal leading submatrix of the initial 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. The positive definiteness of is such that it can be successfully factorized with a call to f07fdf.
The number of iterations taken for the bisection will be:
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g02anf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02anf 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 g02anf. The total size of these arrays does not exceed real elements. All allocated memory is freed before return of g02anf.
This example finds the smallest uniform perturbation to , such that the output is a correlation matrix and the leading principal submatrix of the input is preserved,