Majorization for Changes in Ritz Values and Canonical Angles Between Subspaces (Part I and Part II)
by
Merico E. Argentati
Department of Mathematics and Center for Computational Mathematics, University of Colorado Denver
Coauthors: Merico E. Argentati, Abram Jujunashvili, Ilya Lashuk, Andrew Knyazev

Part I

We review a combination of sine and cosine based algorithms to compute principal (also called canonical) angles between subspaces and discuss their accuracy. We develop a new accurate method to compute principal angles and the corresponding vectors in an A-based scalar product for a Hermitian and positive definite operator A. The algorithm does not require that A is explicitly available as a matrix, but rather only through a function call that multiplies A by a vector. The code is publicly available at http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do? objectId=55&objectType=file. We prove basic theorems on absolute errors for sine and cosine of principal angles with improved constants. We present some numerical results that demonstrate the accuracy of the code.

Part II

We extend the material of Part I using majorization and unitarily invariant norms. The absolute value of the difference of the squares of the cosines/sines is majorized by the sines of the angles between the perturbed subspaces, with a constant of one. We show that this result can be interpreted as a bound on the proximity of the Ritz values in the Rayleigh-Ritz method with the change of the trial subspace, in a particular case where the Rayleigh-Ritz method is applied to an orthogonal projector. We then prove the general result for an arbitrary Hermitian operator, not necessarily a projector, where the constant becomes the difference between the largest and the smallest eigenvalues of the operator. This confirms our previous conjecture that the square root of two factor, which has been present in our earlier estimate, can be eliminated. Our proof is based on a novel idea of an extension of an arbitrary Hermitian operator to an orthogonal projector.

Some of our results can be generalized to cover infinite dimensional subspaces.

Date received: April 30, 2005