The function may be called by the names: f07juc, nag_lapacklin_zptcon or nag_zptcon.
f07juc should be preceded by a call to f07jrc, which computes a modified Cholesky factorization of the matrix as
where is a unit lower bidiagonal matrix and is a diagonal matrix, with positive diagonal elements. f07juc then utilizes the factorization to compute by a direct method, from which the reciprocal of the condition number of , is computed as
is returned, rather than , since when is singular is infinite.
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
1: – IntegerInput
On entry: , the order of the matrix .
2: – const doubleInput
Note: the dimension, dim, of the array d
must be at least
On entry: must contain the diagonal elements of the diagonal matrix from the factorization of .
3: – const ComplexInput
Note: the dimension, dim, of the array e
must be at least
On entry: must contain the subdiagonal elements of the unit lower bidiagonal matrix . (e can also be regarded as the superdiagonal of the unit upper bidiagonal matrix from the factorization of .)
4: – doubleInput
On entry: the -norm of the original matrix , which may be computed as shown in Section 10. anorm must be computed either before calling f07jrc or else from a copy of the original matrix .
5: – double *Output
On exit: the reciprocal condition number, .
6: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
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.
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.
On entry, .
The computed condition number will be the exact condition number for a closely neighbouring matrix.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07juc 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.
The condition number estimation requires floating-point operations.
See Section 15.6 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.