The Block Rational Arnoldi Method

Abstract

The block version of the rational Arnoldi method is a widely used procedure for generating an orthonormal basis of a block rational Krylov space. We study block rational Arnoldi decompositions associated with this method and prove an implicit Q theorem. We relate these decompositions to nonlinear eigenvalue problems. We show how to choose parameters to prevent a premature breakdown of the method and improve its numerical stability. We explain how rational matrix-valued functions are encoded in rational Arnoldi decompositions and how they can be evaluated numerically. Two different types of deflation strategies are discussed. Numerical illustrations using the MATLAB Rational Krylov Toolbox are included.

Computing low-rank approximations of the Fréchet derivative of a matrix function using Krylov subspace methods

“…The usage of rational Krylov subspaces can of course be combined with a block Krylov approach similar to that of section 2.4, leading to a rational block Krylov method, see, e.g. 40 . A combination of rational Krylov subspaces with a twosided approach as in section 2.3 is in principle also possible, but as there are no short recurrences even in the Hermitian case, there also do not exist short-recurrence two-sided rational methods for the non-Hermitian case.…”

“…Figure 4 shows the actual convergence of the approximation ( 16) and the estimate (39) and the estimate (40) for = 1. For later iterations, both estimates are very accurate, while for early iterations, the estimate (39) overestimates the actual error norm, while (40) underestimates it. In particular in situations where it is crucial to reach a certain accuracy, it is advisable to be careful when using estimate (40) as stopping criterion as it might severly underestimate the actual error when convergence is slow.…”

“…Example 9. Consider the following simple numerical example to illustrate the estimates ( 39) and (40). Set = 5, = diag(1, −2, 1) + 1.5 ⋅ diag(−1, 0, 1) ∈ ℝ × , and take randomly y ∈ ℝ and z ∈ ℝ .…”

Divide and conquer methods for functions of matrices with banded or hierarchical low-rank structure

“…The orthonormal bases U m , V m of RK m (A, U R , ξ), RK m (A T , V R , ξ) are computed with the block rational Arnoldi method described in [11,21]. This computation is performed incrementally with respect to m and yields the compressed matrices U T m AU m and V T m (A + R)V m nearly for free.…”

Low-rank updates of matrix functions II: Rational Krylov methods

“…Adapting the usual rational Arnoldi method [19] to (6), Algorithm 2 is used to compute an orthonormal basis U m = U 1 , . .…”