An algorithm for constructing a $J$-orthogonal basis of the extended Krylov subspace $\mathcal{K}_{r,s}=\operatorname{range}\{u,Hu, H^2u,$ $ \ldots, $ $H^{2r-1}u, H^{-1}u, H^{-2}u, \ldots, H^{-2s}u\},$ where $H \in \mathbb{R}^{2n \times 2n}$ is a large (and sparse) Hamiltonian matrix is derived (for $r = s+1$ or $r=s$). Surprisingly, this allows for short recurrences involving at most five previously generated basis vectors. Projecting $H$ onto the subspace $\mathcal{K}_{r,s}$ yields a small Hamiltonian matrix. The resulting HEKS algorithm may be used in order to approximate $f(H)u$ where $f$ is a function which maps the Hamiltonian matrix $H$ to, e.g., a (skew-)Hamiltonian or symplectic matrix. Numerical experiments illustrate that approximating $f(H)u$ with the HEKS algorithm is competitive for some functions compared to the use of other (structure-preserving) Krylov subspace methods.
翻译：用于构建扩展 Krylov 子空间的 $J$- orthogoal 基数 $\ mathcal{K ⁇ r, s ⁇ operatorname{range{ ⁇ u, hu, H ⁇ 2u, H ⁇ -1}u, H ⁇ -2}u, H ⁇ -2}u, h ⁇ -2}u, h ⁇ 2{ldots, H ⁇ 2{R ⁇ 2n} 的算法, 用于构建一个大型( 稀释的) 汉密尔顿矩阵( $= s+1 $ 或 $$$ ) 。 令人惊讶的是, 这允许在大多数以前生成的基向矢量 $, $ $, $ H ⁇ 2r-1}, H ⁇ 1 美元, H ⁇ _ 美元, 或 美元 美元 美元 汉密尔密尔顿矩阵( 美元) 函数绘制汉密尔密尔顿矩阵矩阵 $( e. g) 或 美元 亚空基系统 演示 。
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