- - - Department of Statistics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box: ۴۱۳۳۵-۱۹۱۴۱, Rasht, Iran
- - - Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box: ۴۱۳۳۵-۱۹۱۴۱, Rasht, Iran

In this paper, we firstly study the employing of the Monte Carlo method for solving system of linear algebraic equations and then analyze on convergence of this method. We propound new results related to the convergence of the Monte Carlo method. Additionally, we introduce a new Monte Carlo algorithm with effective techniques. Finally, we compare the efficiency of new Monte Carlo algorithm with its old version in the numerical experiments.

System of linear algebraic equations, Monte Carlo method, Transition probability matrix, Spectral radius, Ergodic Markov chain