A Thick-Restart Lanczos Algorithm with Polynomial Filtering for Hermitian Eigenvalue Problems

A Thick-Restart Lanczos Algorithm with Polynomial Filtering for Hermitian Eigenvalue Problems

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a ThickRestart version of the Lanczos algorithm with deflation (‘locking’) and a new type of polynomial filters obtained fro...

Thick-Restart Lanczos Method for Symmetric Eigenvalue Problems

For real 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 signiicantly fewer arithmetic operations per iteration. To limit the maximum memory usage, these algorithms are often restarted. In re...

TRPL+K: Thick-Restart Preconditioned Lanczos+K Method for Large Symmetric Eigenvalue Problems

The Lanczos method is one of the standard approaches for computing a few eigenpairs of a large, sparse, symmetric matrix. It is typically used with restarting to avoid unbounded growth of memory and computational requirements. Thick-restart Lanczos is a popular restarted variant because of its simplicity and numerically robustness. However, convergence can be slow for highly clustered eigenvalu...

Thick-restart Lanczos method for electronic structure calculations

ABLE: An Adaptive Block Lanczos Method for Non-Hermitian Eigenvalue Problems

This work presents an adaptive block Lanczos method for large-scale non-Hermitian Eigenvalue problems (henceforth the ABLE method). The ABLE method is a block version of the non-Hermitian Lanczos algorithm. There are three innovations. First, an adaptive blocksize scheme cures (near) breakdown and adapts the blocksize to the order of multiple or clustered eigenvalues. Second, stopping criteria ...