We present some accelerated variants of fixed point iterations for computing the minimal non-negative solution of the unilateral matrix equation associated with an M/G/1-type Markov chain. These schemes derive from certain staircase regular splittings of the block Hessenberg M-matrix associated with the Markov chain. By exploiting the staircase profile we introduce a two-step fixed point iteration. The iteration can be further accelerated by computing a weighted average between the approximations obtained in two consecutive steps. The convergence of the basic two-step fixed point iteration and of its relaxed modification is proved. Our theoretical analysis along with several numerical experiments show that the proposed variants generally outperform the classical iterations.
翻译:为计算与M/G/1-型马可夫链相连的单方矩阵方程式的最小非负式解决方案,我们提出了一些固定点迭代的加速变式,这些计划来自与马可夫链相连的赫森堡M-矩阵区块的某些楼梯常规分割。我们利用楼梯剖面引入了两步固定点迭代。通过计算连续两步取得的近似之间的加权平均值,可以进一步加速迭代。基本两步固定点迭代及其宽松修改的趋同得到了证明。我们的理论分析以及若干数字实验表明,拟议的变异通常超过传统的迭代。