nag_dgbcon (f07bgc) estimates the condition number of a real band matrix
, where
has been factorized by
nag_dgbtrf (f07bdc).
nag_dgbcon (f07bgc) estimates the condition number of a real band matrix
, in either the
-norm or the
-norm:
The function should be preceded by a call to
nag_dgb_norm (f16rbc) to compute
or
, and a call to
nag_dgbtrf (f07bdc) to compute the
factorization of
. The function 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.
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.
A call to
nag_dgbcon (f07bgc) involves solving a number of systems of linear equations of the form
or
; the number is usually
or
and never more than
. Each solution involves approximately
floating-point operations (assuming
and
) but takes considerably longer than a call to
nag_dgbtrs (f07bec) with one right-hand side, because extra care is taken to avoid overflow when
is approximately singular.
The complex analogue of this function is
nag_zgbcon (f07buc).
This example estimates the condition number in the
-norm of the matrix
, where
Here
is nonsymmetric and is treated as a band matrix, which must first be factorized by
nag_dgbtrf (f07bdc). The true condition number in the
-norm is
.