Principal angles between subspaces as related to Rayleigh quotient and Raleigh Ritz inequalities with applications to eigenvalue accuracy and an eigenvalue solver

In this dissertation we focus on three related areas of research: (1) principal angles (sometimes denoted as canonical angles) between subspaces including theory and numerical results, (2) Rayleigh quotient and Raleigh Ritz perturbations and eigenvalue accuracy, and (3) parallel software implementation and numerical results concerning the eigenvalue solver LOBPCG (Locally Optimal Block Preconditioned Conjugate Gradient Method) [35], using parallel software libraries and interface specifications based on the Lawrence Livermore National Laboratory, Center for Applied Scientific Computing (LLNL-CASC) High Performance Preconditioners (Hypre) project.; Concerning principal angles or canonical angles between subspaces, we provide some theoretical results and discuss how current codes compute the cosine of principal angles, thus making impossible, because of round-off errors, finding small angles accurately. We propose several MATLAB based algorithms that compute both small and large angles accurately; and illustrate their practical robustness with numerical examples. We prove basic perturbation theorems for absolute errors for sine and cosine of principal angles with improved constants. MATLAB release 13 has implemented our SVD-based algorithm for the sine.; Secondly, we discuss Rayleigh quotient and Raleigh Ritz perturbations and eigenvalue accuracy. Several completely new results are presented. One of the interesting findings characterizes the perturbation of Ritz values for a symmetric positive definite matrix and two different subspaces, in terms of the spectral condition number and the largest principal angle between the subspaces.; The final area of research involves the parallel implementation of the LOBPCG algorithm using Hypre, which involves the computation of eigenvectors that are computed by optimizing Rayleigh quotients with the conjugate gradient method. We discuss a flexible "matrix-free" parallel algorithm and performance on several test problems. This LOBPCG Hypre software has been integrated into LLNL Hypre Beta Release Hypre-1.8.0b.