The Block Rational Arnoldi Method
Steven Elsworth, Stefan Güttel · SIAM Journal on Matrix Analysis and Applications · 2020
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 show how to choose parameters to prevent a premature breakdown of the method and improve its numerical stability by relating the decompositions to nonlinear eigenvalue problems. We explain how rational matrix-valued functions are encoded in block 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.