On a fast Arnoldi method for BML matrices
This work provides a theoretical extension of the Faber-Manteuffel theorem to a broader class of matrices, enabling efficient Krylov subspace methods for practitioners dealing with such structured matrices.
The paper develops a fast Arnoldi method for BML matrices, where the adjoint is a low-rank perturbation of a rational function of the matrix, enabling a short recurrence for generating an orthonormal Krylov basis using GMRES residual vectors.
Matrices whose adjoint is a low rank perturbation of a rational function of the matrix naturally arise when trying to extend the well known Faber-Manteuffel theorem, which provides necessary and sufficient conditions for the existence of a short Arnoldi recurrence. We show that an orthonormal Krylov basis for this class of matrices can be generated by a short recurrence relation based on GMRES residual vectors. These residual vectors are computed by means of an updating formula. Furthermore, the underlying Hessenberg matrix has an accompanying low rank structure, which we will investigate closely.