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.

翻译：基于复合材料抽象理论的进展，我们针对预解式 ${\bf R}(z)=[z{\bf I}-{\bf P}^\dagger{\bf Q}{\bf P}]^{-1}$ 发展了在 $z=\infty$ 处快速收敛的级数展开，其中 ${\bf Q}$ 为正交投影，${\bf P}$ 满足 ${\bf P}{\bf P}^\dagger$ 为正交投影。假设 ${\bf P}^\dagger{\bf Q}{\bf P}$ 的谱位于区间 $[z^-,z^+]$ 内，其中已知 $z^+\leq 1$ 且 $z^-\geq 0$，且投影 ${\bf Q}$ 与 ${\bf P}{\bf P}^\dagger$ 的作用易于计算。该级数在整个 $z$ 平面上收敛，仅排除割线 $[z^-,z^+]$。其通过子空间代换方法获得，即将目标预解式与更大空间中的预解式关联，并将 ${\bf Q}$ 替换为非正交投影 $\underline{{\bf Q}}$。当 $z$ 为实数时，新方法的收敛速率与共轭梯度法相匹配。