The routine may be called by the names f07jgf, nagf_lapacklin_dptcon or its LAPACK name dptcon.
f07jgf should be preceded by a call to f07jdf, 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. f07jgf 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: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array d
must be at least
On entry: must contain the diagonal elements of the diagonal matrix from the factorization of .
3: – Real (Kind=nag_wp) arrayInput
Note: the dimension 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: – Real (Kind=nag_wp)Input
On entry: the -norm of the original matrix , which may be computed by calling f06rpf with its argument . anorm must be computed either before calling f07jdf or else from a copy of the original matrix .
5: – Real (Kind=nag_wp)Output
On exit: the reciprocal condition number, .
6: – Real (Kind=nag_wp) arrayWorkspace
7: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
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.
f07jgf 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.
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.