Rapidly convergent series expansions for a class of resolvents
<|reference_start|>Rapidly convergent series expansions for a class of resolvents: Following advances in the abstract theory of composites, we develop rapidly converging series expansions about $z=\infty$ for the resolvent ${\bf R}(z)=[z{\bf I}-{\bf P}^\dagger{\bf Q}{\bf P}]^{-1}$ where ${\bf Q}$ is an orthogonal projection and ${\bf P}$ is such that ${\bf P}{\bf P}^\dagger$ is an orthogonal projection. It is assumed that the spectrum of ${\bf P}^\dagger{\bf Q}{\bf P}$ lies within the interval $[z^-,z^+]$ for some known $z^+\leq 1$ and $z^-\geq 0$ and that the actions of the projections ${\bf Q}$ and ${\bf P}{\bf P}^\dagger$ are easy to compute. The series converges in the entire $z$-plane excluding the cut $[z^-,z^+]$. It is obtained using subspace substitution, where the desired resolvent is tied to a resolvent in a larger space and ${\bf Q}$ gets replaced by a projection $\underline{{\bf Q}}$ that is no longer orthogonal. When $z$ is real the rate of convergence of the new method matches that of the conjugate gradient method.<|reference_end|>
@article{milton2024rapidly,
    title={Rapidly convergent series expansions for a class of resolvents},
    author={Graeme W. Milton},
    journal={arXiv preprint arXiv:2407.15751},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.15751},
    primaryClass={math.NA cs.NA math-ph math.AP math.CA math.MP}
}