On the Convergence of Rational Ritz Values

Abstract

Abstract. Ruhe's rational Krylov method is a popular tool for approximating eigenvalues of a given matrix, though its convergence behavior is far from being fully understood. Under fairly general assumptions we characterize in an asymptotic sense which eigenvalues of a Hermitian matrix are approximated by rational Ritz values and how fast this approximation takes place. Our main tool is a constrained extremal problem from logarithmic potential theory, where an additional external field is required for taking into account the poles of the underlying rational Krylov space. Several examples illustrate our analytic results.

Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

“…When the matrix is not unitary, then the underlying theory is for polynomials and ORFs that are orthogonal with respect to a measure supported on the real line or a real interval and convergence has been studied based on the asymptotic behaviour of the ORFs and the approximation error (see for example [5,13]). For the unitary case such a theory is much less developed.…”

A black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functions

“…In view of the behavior of our adaptive rational Arnoldi method for the above symmetric matrices, we believe that the convergence can be asymptotically (that is, for a sequence of symmetric matrices growing larger in size and having a joint limit eigenvalue distribution) compared to min-max rational functions with poles on Γ and zeros on Σ being constrained Leja points in the sense of [11]. The constraint for the zeros is given by the interlacing property of Ritz values associated with symmetric matrices (see, e.g., [7]). …”

“…In fact, our new method typically even outperforms such methods due to the spectral adaption of the poles during the iteration. A rigorous convergence analysis, perhaps involving tools from potential theory as in [7,6], for explaining spectral adaption of this rational Arnoldi variant applied with a symmetric matrix, may be an interesting research problem.…”

A unification of unitary similarity transforms to compressed representations

“…An arbitrary unitary matrix U can be factored in sequences of rotations as the left matrix in Scheme (7), in fact one just computes the Q R-factorization and R becomes the identity matrix. 1 Applying successive shift-through operations can change the ∧-form on the left in a ∨-form on the right.…”

“…, v). In [1,7,15,16] the convergence of (rational) Ritz-values to eigenvalues is studied and it is proven that under mild conditions on, e.g., the starting vector, the Ritz-values approximate those eigenvalues well-separated from the rest of the spectrum first. In some circumstances these are not the eigenvalues one wants to find first, as a result shift and invert techniques are used resulting in rational Ritz-values.…”