Abstract

-- Authors --
M. Ilic and I.W. Turner
m.ilic@fsc.qut.edu.au, i.turner@fsc.qut.edu.au
School of Mathematical Sciences
Queensland University of Technology
Brisbane, Australia

-- Abstract -- A subspace V is said to be an invariant subspace of an n x n matrix A when Ax V, for all x V. Some examples of invariant subspaces of A include Rⁿ, nullspace N(A), range R(A), span{x₁, ..., x_k} where x_i, i = 1, ..., k are eigenvectors of A and the Krylov subspace K_m(A, v), where m is the algebraic grade. Such subspaces are well documented in the literature and approximations for them exist. Approximate invariant subspaces have properties that are highly desirable for iterative solution strategies of large sparse matrix systems and for approximating Ritz values and Ritz vectors of such matrices. It is often a difficult task to identify an approximate invariant subspace numerically. In this work we propose a new definition that can assist us with the task of identifying when a subspace is approximately invariant by measuring the sine of the angle between the image of any vector in the subspace and its orthogonal projection onto the subspace. In particular we examine the effect that different bases have on this measure. Finally, we use this definition to estimate errors when solving systems of equations or the eigenvalue problem.

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