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.

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 10 5 -10 8 .

The Rayleigh-Ritz (RR) method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is A-invariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is A-invariant, the absolute changes in the Ritz values of A with respect to X compared to the Ritz values with respect to Y represent the RR absolute eigenvalue approximation error. Our first main result is a sharp majorization-type RR error bound in terms of the principal angles between X and Y for an arbitrary A-invariant X , which was a conjecture in [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 548-559]. Second, we prove a novel type of RR error bound that deals with the products of the errors, rather than the sums. Third, we establish majorization bounds for the relative errors. We extend our bounds to the case dimX ≤ dimY < ∞ in Hilbert spaces and apply them in the context of the finite element method.

Abstract. Many inequality relations between real vector quantities can be succinctly expressed as "weak (sub)majorization" relations using the symbol ≺ w . 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 Θ(X , Y) be the vector of principal angles in nondecreasing order between subspaces X and Y of a finite dimensional space H with a scalar product. We consider the change in principal angles between subspaces X and Z, where we let X be perturbed to give Y. We measure the change using weak majorization. We prove that, and give similar results for differences of cosines, i.e.,, and of sines and sines squared, assuming dimX = dimY. We observe that cos 2 Θ(X , Z) can be interpreted as a vector of Ritz values, where the RayleighRitz method is applied to the orthogonal projector on Z using X as a trial subspace. Thus, our result for the squares of cosines can be viewed as a bound on the change in the Ritz values of an orthogonal projector. We then extend it to prove a general result for Ritz values for an arbitrary Hermitian operator A, not necessarily a projector: let Λ (P X A| X ) be the vector of Ritz values in nonincreasing order for A on a trial subspace X , which is perturbed to give another trial subspace, where the constant is the difference between the largest and the smallest eigenvalues of A. This establishes our conjecture that the root two factor in our earlier estimate may be eliminated. Our present proof is based on a classical but rarely used technique of extending a Hermitian operator in H to an orthogonal projector in the "double" space H 2 . An application of our Ritz values weak majorization result for Laplacian graph spectra comparison is suggested, based on the possibility of interpreting eigenvalues of the edge Laplacian of a given graph as Ritz values of the edge Laplacian of the complete graph. We prove that k |λ 1 k − λ 2 k | ≤ nl, where λ 1 k and λ 2 k are all ordered elements of the Laplacian spectra of two graphs with the same n vertices and with l equal to the number of differing edges.

We define angles for infinite dimensional subspaces of Hilbert spaces, inspired by the work of E.J. Hannan, 1961Hannan, /1962. The angles of Dixmier and Friedrichs, and the gaps are characterized. We establish connections between the angles corresponding to orthogonal complements. The sensitivity of angles with respect to subspaces is estimated. We show that the squared cosines of the angles from one subspace to another can be interpreted as Ritz values in the Rayleigh-Ritz method. The Hausdorff distance between the Ritz values, corresponding to different trial subspaces, is shown to be bounded by a constant times the gap between the subspaces. We prove a similar eigenvalue perturbation bound that involves the gap squared. An ultimate acceleration of the classical alternating projectors method is proposed. Its convergence rate is estimated in terms of the angles. We illustrate the acceleration for a domain decomposition method with a small overlap for the 1D diffusion equation. (A. Knyazev), Abram.Jujunashvili@na-net.ornl.gov (A. Jujunashvili), Merico.Argentati@na-net.ornl.gov (M. Argentati).