一类预解式的快速收敛级数展开
Rapidly convergent series expansions for a class of resolvents
AI总结：
本文针对一类由正交投影构成的预解式，利用子空间替换技术构造了关于无穷远点的快速收敛级数展开，其收敛速度在实数情形下与共轭梯度法相当。
摘要：
继复合材料抽象理论的进展之后，我们针对预解式 ${\bf R}(z)=[z{\bf I}-{\bf P}^\dagger{\bf Q}{\bf P}]^{-1}$ 发展了关于 $z=\infty$ 的快速收敛级数展开，其中 ${\bf Q}$ 是一个正交投影，而 ${\bf P}$ 满足 ${\bf P}{\bf P}^\dagger$ 是一个正交投影。假设 ${\bf P}^\dagger{\bf Q}{\bf P}$ 的谱位于区间 $[z^-,z^+]$ 内，其中 $z^+\leq 1$ 和 $z^-\geq 0$ 为已知，并且投影 ${\bf Q}$ 和 ${\bf P}{\bf P}^\dagger$ 的作用易于计算。该级数在整个 $z$ 平面内收敛，但不包括割线 $[z^-,z^+]$。该级数通过子空间替换得到，其中所需的预解式与更大空间中的某个预解式相关联，并且 ${\bf Q}$ 被一个不再是正交的投影 $\underline{\bf Q}$ 所替代。当 $z$ 为实数时，新方法的收敛速度与共轭梯度法相匹配。
英文摘要：
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.
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