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
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 g02aaf is called.
3: – IntegerInput
On entry: the size of the matrix .
4: – Real (Kind=nag_wp)Input
On entry: the termination tolerance for the Newton iteration. If , is used.
5: – 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.
6: – IntegerInput
On entry: specifies the maximum number of Newton iterations.
If , is used.
7: – Real (Kind=nag_wp) arrayOutput
On exit: contains the nearest correlation matrix.
8: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02aaf is called.
9: – IntegerOutput
On exit: the number of Newton steps taken.
10: – IntegerOutput
On exit: the number of function evaluations of the dual problem.
11: – Real (Kind=nag_wp)Output
On exit: the norm of the gradient of the last Newton step.
12: – 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, and .
On entry, and .
On entry, .
Newton iteration fails to converge in iterations.
Machine precision is limiting convergence. The array returned in x may still be of interest.
An intermediate eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.
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 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.
g02aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02aaf 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 g02aaf. The total size of these arrays is real elements and integer elements.
This example finds the nearest correlation matrix to the matrix , where
The example also demonstrates how a modified Cholesky factorization (computed using f01mdf and f01mef) can be used to obtain a bound on the distance to the nearest correlation matrix prior to computing the NCM itself.