#
Thick-Restart Lanczos Method for Symmetric Eigenvalue
Problems

##
*Kesheng Wu and Horst Simon*

For symmetric eigenvalue problems, there are a number of algorithms that
are mathematically equivalent, for example, the Lanczos algorithm, the
Arnoldi method and the unpreconditioned Davidson method. The Lanczos
algorithm is often preferred because it uses significantly fewer
arithmetic operations per iteration. To limit the maximum memory usage,
these algorithms are often restarted. In recent years, a number of
effective restarting schemes have been developed for the Arnoldi method
and the Davidson method. This paper describes an simple restarting
scheme for the Lanczos algorithm. This restarted Lanczos algorithm uses
as many arithmetic operations as the original algorithm. Theoretically,
this restarted Lanczos method is equivalent to the implicitly restarted
Arnoldi method and the thick-restart Davidson method. Because it uses
less arithmetic operations than the others, it is an attractive
alternative for solving symmetric eigenvalue problems.