Rapidly convergent series expansions for a class of resolvents
| Autor: | Milton, Graeme W. |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 19 Pages, No figures |
| Databáze: | arXiv |
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