Merico E. Argentati

Principal Angles between Subspaces in an A -Based Scalar Product: Algorithms and Perturbation Estimates

Computation of principal angles between subspaces is important in many applications, e.g., in statistics and information retrieval. In statistics, the angles are closely related to measures of dependency and covariance of random variables. When applied to column-spaces of matrices, the principal angles describe canonical correlations of a matrix pair. We highlight that all popular software codes for canonical correlations compute only cosine of principal angles, thus making impossible, because of round-off errors, finding small angles accurately. We review a combination of sine and cosine based algorithms that provide accurate results for all angles. We generalize the method to the computation of principal angles in an A-based scalar product for a symmetric and positive definite matrix A. We provide a comprehensive overview of interesting properties of principal angles. We prove basic perturbation theorems for absolute errors for sine and cosine of principal angles with improved constants. Numerical examples and a detailed description of our code are given.

Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in Hypre and PETSc

We describe our software package Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) recently publicly released. BLOPEX is available as a stand-alone serial library, as an external package to PETSc (Portable, Extensible Toolkit for Scientific Computation, a general purpose suite of tools developed by Argonne National Laboratory for the scalable solution of partial differential equations and related problems), and is also built into hypre (High Performance Preconditioners, a scalable linear solvers package developed by Lawrence Livermore National Laboratory). The present BLOPEX release includes only one solver—the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method for symmetric eigenvalue problems. hypre provides users with advanced high-quality parallel multigrid preconditioners for linear systems. With BLOPEX, the same preconditioners can now be efficiently used for symmetric eigenvalue problems. PETSc facilitates the integration of independently developed application modules, with strict attention to component interoperability, and makes BLOPEX extremely easy to compile and use with preconditioners that are available via PETSc. We present the LOBPCG algorithm in BLOPEX for hypre and PETSc. We demonstrate numerically the scalability of BLOPEX by testing it on a number of distributed and shared memory parallel systems, including a Beowulf system, SUN Fire 880, an AMD dual-core Opteron workstation, and IBM BlueGene/L supercomputer, using PETSc domain decomposition and hypre multigrid preconditioning. We test BLOPEX on a model problem, the standard 7-point finite-difference approximation of the 3-D Laplacian, with the problem size in the range of

Majorization for Changes in Angles Between Subspaces, Ritz Values, and Graph Laplacian Spectra

10^5

Rayleigh-Ritz Majorization Error Bounds with Applications to FEM

-

Angles between infinite dimensional subspaces with applications to the Rayleigh-Ritz and alternating projectors methods ✩

10^8

Preconditioned Eigensolver LOBPCG in hypre and PETSc

.

Bounds on Changes in Ritz Values for a Perturbed Invariant Subspace of a Hermitian Matrix

Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol

Bounds for the Rayleigh Quotient and the Spectrum of Self-Adjoint Operators

{\prec}_{w}

On proximity of Rayleigh quotients for different vectors and Ritz values generated by different trial subspaces

. We explain these ideas and apply them in several areas, angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related. Let

Rayleigh-Ritz majorization error bounds with applications to FEM and subspace iterations

\Theta({\mathcal X},{\mathcal Y})