f07hgf estimates the condition number of a real symmetric positive definite band matrix
$A$, where
$A$ has been factorized by
f07hdf.
f07hgf 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 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.
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
f07hgf 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 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.
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. The true condition number in the
$1$-norm is
$74.15$.