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##

Variants

The description of the Arnoldi procedure given earlier was based on the
modified Gram-Schmidt process. Other orthogonalization
algorithms could be used as well. One improvement is
to reorthogonalize when necessary.
Whenever the final vector obtained at the end of the second loop in
the above algorithm has been computed, a test is performed to compare
its norm with the norm of the initial (which is ).
If the reduction falls below a certain threshold (an indication that
severe cancelation might have occurred), a second orthogonalization is
made. It is known from a result by Kahan that
more than two
orthogonalizations are superfluous (see, for example, Parlett
[353]).

One of the most reliable orthogonalization techniques, from
the numerical point of view, is the Householder algorithm
[198]. This has been implemented for the
Arnoldi procedure by Walker [455].
The Householder algorithm is numerically more reliable
than the Gram-Schmidt or modified Gram-Schmidt versions, but it is also
more expensive,
requiring roughly the same storage as modified Gram-Schmidt
but about twice as many operations.
The Householder orthogonalization is a reasonable
choice when developing general purpose, reliable software packages where
robustness is a critical criterion.

** Next:** Explicit Restarts
** Up:** Arnoldi Method Y. Saad
** Previous:** Basic Algorithm
** Contents**
** Index**
Susan Blackford
2000-11-20