F07HGF (DPBCON) estimates the condition number of a real symmetric positive definite band matrix
$A$, where
$A$ has been factorized by
F07HDF (DPBTRF).
F07HGF (DPBCON) estimates the condition number (in the
$1$-norm) of a real symmetric positive definite band matrix
$A$:
Since
$A$ is symmetric,
${\kappa}_{1}\left(A\right)={\kappa}_{\infty}\left(A\right)={\Vert A\Vert}_{\infty}{\Vert {A}^{-1}\Vert}_{\infty}$.
The routine should be preceded by a call to
F06REF to compute
${\Vert A\Vert}_{1}$ and a call to
F07HDF (DPBTRF) to compute the Cholesky factorization of
$A$. The routine then uses Higham's implementation of Hager's method (see
Higham (1988)) to estimate
${\Vert {A}^{-1}\Vert}_{1}$.
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 F07HGF (DPBCON) involves solving a number of systems of linear equations of the form
$Ax=b$; the number is usually
$4$ or
$5$ and never more than
$11$. Each solution involves approximately
$4nk$ floating point operations (assuming
$n\gg k$) but takes considerably longer than a call to
F07HEF (DPBTRS) 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
F07HUF (ZPBCON).
This example estimates the condition number in the
$1$-norm (or
$\infty $-norm) of the matrix
$A$, where
Here
$A$ is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by
F07HDF (DPBTRF). The true condition number in the
$1$-norm is
$74.15$.