f07ggc estimates the condition number of a real symmetric positive definite matrix
$A$, where
$A$ has been factorized by
f07gdc, using packed storage.
f07ggc estimates the condition number (in the
$1$-norm) of a real symmetric positive definite 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 function should be preceded by a call to
f16rdc to compute
${\Vert A\Vert}_{1}$ and a call to
f07gdc to compute the Cholesky factorization of
$A$. The function 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.
Background information to multithreading can be found in the
Multithreading documentation.
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
f07ggc 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
$2{n}^{2}$ floating-point operations but takes considerably longer than a call to
f07gec with one right-hand side, because extra care is taken to avoid overflow when
$A$ is approximately singular.
The complex analogue of this function is
f07guc.
This example estimates the condition number in the
$1$-norm (or
$\infty $-norm) of the matrix
$A$, where
Here
$A$ is symmetric positive definite, stored in packed form, and must first be factorized by
f07gdc. The true condition number in the
$1$-norm is
$97.32$.