F07VGF (DTBCON) estimates the condition number of a real triangular band matrix.
F07VGF (DTBCON) estimates the condition number of a real triangular band matrix
A, in either the
1-norm or the
∞-norm:
The routine computes
A1 or
A∞ exactly, and uses Higham's implementation of Hager's method (see
Higham (1988)) to estimate
A-11 or
A-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
ρ, and in practice is nearly always less than
10ρ, although examples can be constructed where
RCOND is much larger.
A call to F07VGF (DTBCON) involves solving a number of systems of linear equations of the form
Ax=b or
ATx=b; the number is usually
4 or
5 and never more than
11. Each solution involves approximately
2nk floating point operations (assuming
n≫k) but takes considerably longer than a call to
F07VEF (DTBTRS) 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
F07VUF (ZTBCON).
This example estimates the condition number in the
1-norm of the matrix
A, where
Here
A is treated as a lower triangular band matrix with one subdiagonal. The true condition number in the
1-norm is
69.62.