Thick-restart block Lanczos method for large-scale shell-model calculations

Thick-restart Lanczos method for electronic structure calculations

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...

Large scale electronic structure calculations using the Lanczos method

The orthogonality requirement in either iterative diagonalization or conjugate gradient approaches to the single particle Schr6dinger equation/~/~u = E~u leads to an overall N 3 scaling of the effort with the number N of atoms. We show that the Lanczos method circumvents this problem even when applied to all occupied states. Our implementation shows that the method is stable, exact, scales as N...

Implicitly Restarted Arnoldi/lanczos Methods for Large Scale Eigenvalue Calculations

This report provides an introductory overview of the numerical solution of large scale algebraic eigenvalue problems. The main focus is on a class of methods called Krylov subspace projection methods. The Lanczos method is the premier member of this class and the Arnoldi method is a generalization to the nonsymmetric case. A recently developed and very promising variant of the Arnoldi/Lanczos s...