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.