Adaptive Rational Krylov for Sylvester Equations

This repository contains the implementation of a block rational Krylov subspace method for the solution of Sylvester equations of the form $AX - XB = UV^T$, with, $U,V$ tall and skinny, and $A$ and $B$ large and square. The method is described in detail in [1], and has the following features:

• Builds on matrix-vector products and linear solves with shifted copies of $A$ and $B$. In particular, the methods can take advantage of sparse or data-sparse matrices.
• Adaptive pole selection: optimal poles are chosen during the construction of the projection subspace automatically. The pole selection is based on a novel criterion that enables a better understanding of the convergence of block Krylov methods, which is not just a straightforward extension of "scalar" Krylov methods.

The examples included in the repository can be used to test the method on two examples: the discretizations of a 2D Poisson problem, and a 2D convection-diffusion equation.

The algorithm computes rational Krylov subspaces using the rktoolbox described in [2]. The code has been tested using the version 2.9 of rktoolbox.

References

[1]. Casulli, A. & Robol, L., An effcient block rational Krylov solver for Sylvester equations with adaptive pole selection, arXiv.

[2]. Berljafa, M. & S. Elsworth, S. & Guttel, S., A rational Krylov toolbox for Matlab (2014).