F07BGF (DGBCON) estimates the condition number of a real band matrix
$A$, where
$A$ has been factorized by
F07BDF (DGBTRF).
F07BGF (DGBCON) estimates the condition number of a real band matrix
$A$, in either the
$1$-norm or the
$\infty $-norm:
The routine should be preceded by a call to
F06RBF to compute
${\Vert A\Vert}_{1}$ or
${\Vert A\Vert}_{\infty}$, and a call to
F07BDF (DGBTRF) to compute the
$LU$ factorization of
$A$. The routine then uses Higham's implementation of Hager's method (see
Higham (1988)) to estimate
${\Vert {A}^{-1}\Vert}_{1}$ or
${\Vert {A}^{-1}\Vert}_{\infty}$.
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
$\rho $, and in practice is nearly always less than
$10\rho $, although examples can be constructed where
RCOND is much larger.
A call to F07BGF (DGBCON) involves solving a number of systems of linear equations of the form
$Ax=b$ or
${A}^{\mathrm{T}}x=b$; the number is usually
$4$ or
$5$ and never more than
$11$. Each solution involves approximately
$2n\left(2{k}_{l}+{k}_{u}\right)$ floating point operations (assuming
$n\gg {k}_{l}$ and
$n\gg {k}_{u}$) but takes considerably longer than a call to
F07BEF (DGBTRS) with one right-hand side, because extra care is taken to avoid overflow when
$A$ is approximately singular.
The complex analogue of this routine is
F07BUF (ZGBCON).
This example estimates the condition number in the
$1$-norm of the matrix
$A$, where
Here
$A$ is nonsymmetric and is treated as a band matrix, which must first be factorized by
F07BDF (DGBTRF). The true condition number in the
$1$-norm is
$56.40$.