The non-Hermitian Lanczos method is an oblique projection method
(see §3.2)
for solving the NHEP,
With two starting vectors and
, the Lanczos method
builds a pair of biorthogonal bases for the Krylov subspaces
and
, provided that
the matrix-vector multiplications
and
for an arbitrary vector
are available.
The inner loop uses two three-term recurrences.
These recurrences use less memory and fewer memory references than the
corresponding recurrences in the Arnoldi method discussed in
§7.5. The Lanczos
method provides approximations for both right and left eigenvectors.
When estimating errors and condition numbers of the computed eigenpairs,
it is crucial that both the left and right eigenvectors be available.
However, there are risks of breakdown and numerical instability
with the method, since it does not work with orthogonal transformations.
This section defines the basic Lanczos method and
its main properties, and presents algorithmic techniques that
enhance the method's numerical stability and accuracy.