F07BUF (ZGBCON) estimates the condition number of a complex band matrix
, where
has been factorized by
F07BRF (ZGBTRF).
F07BUF (ZGBCON) estimates the condition number of a complex band matrix
, in either the
-norm or the
-norm:
The routine should be preceded by a call to
F06UBF
to compute
or
, and a call to
F07BRF (ZGBTRF) to compute the
factorization of
. The routine then uses Higham's implementation of Hager's method (see
Higham (1988)) to estimate
or
.
Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396
The computed estimate
RCOND is never less than the true value
, and in practice is nearly always less than
, although examples can be constructed where
RCOND is much larger.
F07BUF (ZGBCON) is not threaded by NAG in any implementation.
F07BUF (ZGBCON) 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.
A call to F07BUF (ZGBCON) involves solving a number of systems of linear equations of the form
or
; the number is usually
and never more than
. Each solution involves approximately
real floating-point operations (assuming
and
) but takes considerably longer than a call to
F07BSF (ZGBTRS) with one right-hand side, because extra care is taken to avoid overflow when
is approximately singular.
The real analogue of this routine is
F07BGF (DGBCON).
This example estimates the condition number in the
-norm of the matrix
, where