Determine stable strategies for solving the linear systems in extended Krylov methods

Determine how linear systems involving the matrix A should be solved, including the required accuracy, when constructing extended Krylov subspaces in floating-point arithmetic, beyond the assumption of access to a highly accurate stored LU, LDL, or Cholesky factorization.

Background

The paper analyzes the first three implementation choices in extended Krylov subspace methods: the order in which powers and inverse powers are added, whether vectors are generated individually or in batches, and how the projected matrix is formed. The fourth implementation choice concerns the solution of the linear systems required to apply A{-1}.

The analysis assumes that a highly accurate factorization of A is already available and can be stored and reused. Consequently, the stability and accuracy implications of the linear-system solver itself—including how accurately the systems should be solved—remain unresolved.

References

The fourth point remains for future work but necessarily depends on the outcome of the previous questions.

— On the stability of basis construction in extended Krylov subspace methods  (2610.21603 - Lund et al., 15 Oct 2026) in Section 1, Introduction

Block versions are implemented in our code ektoolbox and discussed in \Cref{sec:experiments}, but their careful analysis remains for future work.

— On the stability of basis construction in extended Krylov subspace methods  (2610.21603 - Lund et al., 15 Oct 2026) in Remark 2.4, Section 2