F07PUF (ZHPCON) estimates the condition number of a complex Hermitian indefinite matrix
A, where
A has been factorized by
F07PRF (ZHPTRF), using packed storage.
F07PUF (ZHPCON) estimates the condition number (in the
1-norm) of a complex Hermitian indefinite matrix
A:
Since
A is Hermitian,
κ1A=κ∞A=A∞A-1∞.
The routine should be preceded by a call to
F06UDF to compute
A1 and a call to
F07PRF (ZHPTRF) to compute the Bunch–Kaufman factorization of
A. The routine then uses Higham's implementation of Hager's method (see
Higham (1988)) to estimate
A-11.
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
ρ, and in practice is nearly always less than
10ρ, although examples can be constructed where
RCOND is much larger.
A call to F07PUF (ZHPCON) involves solving a number of systems of linear equations of the form
Ax=b; the number is usually
5 and never more than
11. Each solution involves approximately
8n2 real floating point operations but takes considerably longer than a call to
F07PSF (ZHPTRS) with one right-hand side, because extra care is taken to avoid overflow when
A is approximately singular.
The real analogue of this routine is
F07PGF (DSPCON).
This example estimates the condition number in the
1-norm (or
∞-norm) of the matrix
A, where
Here
A is Hermitian indefinite, stored in packed form, and must first be factorized by
F07PRF (ZHPTRF). The true condition number in the
1-norm is
9.10.