单生过程第一特征对子的逼近定理

Approximation Theorem for the First Eigenpair of Single Birth Processes

中文关键词:  单生过程  最小特征对子  加速的反幂法

英文关键词:single birth processes  minimal eigenpair  accelerated inverse power method

中文摘要:

      本文首先获得单生过程Q矩阵对应的Poisson方程的显示解,由此直接给出此类Q矩阵的逆矩阵的显示表达.受反幂法的启发,作为上述结果的应用,结合Collatz-Wielandt公式,本文进一步给出了单生过程Q矩阵的最大特征对子的高效逼近定理.不同于经典的固定迭代算法,这里的逼近定理中每一步迭代的推移是变化的,每一步的推移构成的数列单调递增收敛到需要的特征值,此想法有效地减少了迭代步数.本文还计算了几个例子验证了上述结果的有效性.

英文摘要:

      The explicit solution to the Poisson equation corresponding to the $Q$-matrix of a single birth process is obtained, thus the explicit inverse (if exists) is presented directly. As an application, inspired by the inverse power method, combining the explicit inverse with Collatz-Wielandt formula, a powerful approximation theorem for the maximal eigenpair corresponding to the $Q$-matrix of a single birth process is presented. Different from the classical acceleration method using some fixed shift in the iteration, the shift in each iteration step is varying and the sequence formed by these shifts is strictly monotone and increases to the eigenvalue needed, which effectively reduces the number of iterations. Some examples are studied to illustrate the power of these results.