Countably infinite systems of linear ODEs arise as forward equations for many continuous-time Markov processes. The standard recipe -- truncate to a finite cap N and exponentiate -- pays cubic cost in N and a time-growing boundary-feedback bias. We identify a structural condition on the rates, L_{n+r,n} = alpha_r n + beta_r ("linear-rate"), under which the generating function G(z,t) = sum_n x_n(t) z^n satisfies a first-order linear PDE in z, and the method of characteristics yields a composition-multiplier representation G(z,t) = K_t(z) G(Phi_t(z), 0). The Taylor coefficients of Phi_t and K_t on any output window {0,...,N} are determined exactly by a closed lower-triangular polynomial ODE on R^{2(N+1)}, independent of any coefficients above N. Truncation enters only through the support M_0 of the initial law, set independently of N. For binary birth-death the closure collapses to the geometric tail p_n(t) = p_1(t) rho(t)^{n-1} with rho(t) = lambda(1 - e^{-(mu-lambda)t})/(mu - lambda e^{-(mu-lambda)t}). The linear-rate class spans Markov branching with immigration, multi-type branching, matrix-valued telegraph and G/R elongation, and signed or non-stochastic hierarchies. When the generator decomposes as L = A + B with A linear-rate and B non-affine (Schlogl bistable, predator-prey, lattice reaction-diffusion), we pair the closure with Strang splitting on B; Richardson extrapolation lifts the time order to Delta-t^4 at ~3x wall clock. On the Schlogl problem at V=500, N=8,000, the split runs 6.3x faster than dense Pade and 20x faster than sparse Krylov expv. For the stationary regime, a closure-Strang power iteration extends the same machinery to multi-dimensional product-state-space generators where sparse LU hits OOM/OOT or boundary-projection bias at usable caps. Numerical experiments locate where each route wins and where it is dominated by standard tools.

翻译：可数无穷线性常微分方程组出现在许多连续时间马尔可夫过程的正向方程中。标准解法——截断至有限上限N并取矩阵指数——代价随N三次方增长，且存在随时间扩增的边界反馈偏差。我们识别了速率满足L_{n+r,n}=α_r n + β_r（"线性率"）的结构条件，在此条件下生成函数G(z,t)=∑_n x_n(t)z^n满足关于z的一阶线性偏微分方程，特征线法导出组合-乘子表示G(z,t)=K_t(z)G(Φ_t(z),0)。对任意输出窗口{0,...,N}，Φ_t与K_t的泰勒系数完全由R^{2(N+1)}上的封闭下三角多项式常微分方程确定，且不依赖高于N的系数。截断仅通过初始分布的支持集M_0引入，该集合独立于N选取。对于二元生灭过程，该闭合坍缩为几何尾部p_n(t)=p_1(t)ρ(t)^{n-1}，其中ρ(t)=λ(1-e^{-(μ-λ)t})/(μ-λe^{-(μ-λ)t})。线性率类涵盖带迁移的马尔可夫分支过程、多类型分支过程、矩阵值电报过程、G/R延伸过程，以及带符号或非随机层次结构。当生成元可分解为L=A+B（其中A为线性率，B为非仿射形式，如Schlögl双稳态、捕食-被捕食、晶格反应扩散系统），我们将该闭合与B上的Strang分裂相结合；Richardson外推将时间阶提升至Δt^4，壁钟时间约为3倍。在V=500、N=8000的Schlögl问题中，分裂方法运行速度比稠密Padé方法快6.3倍，比稀疏Krylov expv方法快20倍。对于稳态情形，闭合-Strang幂迭代法将相同机制扩展至多维乘积状态空间生成元，在这些场景中稀疏LU方法在可用截断条件下会遭遇内存溢出/超时或边界投影偏差。数值实验定位了各方法的优势区域及其被标准工具主导的情形。