Characterize convergence of mixed-channel Lanczos methods for nonsymmetric problems

Characterize whether mixed-channel extended Krylov methods with a Lanczos orthogonalization backbone converge reliably for nonsymmetric matrices and identify the conditions governing such convergence across broader classes of matrices and matrix functions.

Background

The numerical experiments show that mixed-channel methods using a Lanczos backbone appear to converge for some nonsymmetric problems, even though Lanczos-type methods can rapidly lose orthogonality in nonsymmetric settings. The observed behavior is inconsistent across the tested configurations and is based on only two matrices and two functions.

The authors explicitly identify this as an open question and suggest that the modular ektoolbox could support broader investigation. A general explanation or reliable convergence characterization is not established in the paper.