使用循环归约进行逐分量精确的布朗运动计算
Componentwise accurate Brownian motion computations using Cyclic Reduction
摘要：
马尔可夫调制的布朗运动是一个流行的工具，在随机环境中模拟连续时间现象。感兴趣的主要量是不变密度，其满足与二次矩阵多项式$P(z) = Vz^2-Dz +Q$相关联的微分方程，其中矩阵$V$和$D$是对角的，并且$Q$是离散时间马尔可夫链的转移矩阵。其解通常通过计算与左半平面中的特征值相关联的$P(z)$的不变对来构造，或者通过求解矩阵方程$X^2V-XD+Q=0$来构造。我们表明，这些任务可以解决使用基于循环减少的组件精确算法，推广最近出现的算法的线性情况下（$V=0$）。我们给出了证明我们的算法的数值稳定性在组件的意义上，同样的证明适用于循环减少在一个更一般的M-矩阵设置出现在其他应用程序，如QBD过程的建模。
英文摘要：
Markov-modulated Brownian motion is a popular tool to model continuous-time phenomena in a stochastic context. The main quantity of interest is the invariant density, which satisfies a differential equation associated with the quadratic matrix polynomial $P(z) = Vz^2-Dz +Q$, where the matrices $V$ and $D$ are diagonal and $Q$ is the transition matrix of a discrete-time Markov chain. Its solution is typically constructed by computing an invariant pair of $P(z)$ associated with its eigenvalues in the left half-plane, or by solving the matrix equation $X^2V-XD+Q=0$. We show that these tasks can be solved using a componentwise accurate algorithm based on Cyclic Reduction, generalizing the recently appeared algorithms for the linear case ($V=0$). We give a proof of the numerical stability of our algorithm in the componentwise sense; the same proof applies to Cyclic Reduction in a more general M-matrix setting which appears in other applications such as the modelling of QBD processes.
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