1 - Implicitly restarted arnoldi/lanczos methods and large scale svd applications

Abstract

Implicit restarting is a technique for combining the implicitly shifted QR mechanism with a k-step Arnoldi or Lanczos factorization to obtain a truncated form of the implicitly shifted QR-iteration for eigenvalue problems. The software package ARPACK that is based upon this technique has been successfully used to solve large scale symmetric and nonsymmetric (generalized) eigenvalue problems arising from a variety of applications. This method only requires a pre-determined limited storage proportional to n times the desired number of eigenvalues. Numerical difficulties and storage problems normally associated with Arnoldi and Lanczos processes are avoided. This technique has also proven to be very useful for computing a few of the largest singular values and corresponding singular vectors of a very large matrix. Biological 3-D Image Reconstruction and Molecular Dynamics are two interesting applications. The singular value decomposition (SVD) plays a key roll in a classification procedure that is the computationally intensive portion of the 3-D image reconstruction of biological macromolecules. A relatively new application is to analyze the motions of the proteins using the SVD instead of normal mode analysis. The primary research goal is to pick out the non-harmonic phenomena that usually contains most of the interesting behavior of the protein's motion. This chapter reviews the Implicitly Restarted Arnoldi/Lanczos Method and briefly discusses the biological applications of the SVD.