Improvements on the hybrid Monte Carlo algorithms for matrix computations
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In this paper, we present some improvements on the Markov chain Monte Carlo and hybrid Markov chain Monte Carlo algorithms for matrix computations. We discuss the convergence of the Monte Carlo method using the Ulam–von Neumann approach related to selecting the transition probability matrix. Specifically, we show that if the norm of the iteration matrix T is less than 1 then the Monte Carlo Almost Optimal method is convergent. Moreover, we suggest a new technique to approximate the inverse of the strictly diagonally dominant matrix and we exert some modifications and corrections on the hybrid Monte Carlo algorithm to obtain the inverse matrix in general. Finally, numerical experiments are discussed to illustrate the efficiency of the theoretical results.
KeywordsSystem of linear algebraic equations Markov chain Monte Carlo convergence analysis transition probability matrix inversion
The authors would like to thank the reviewer(s) and the corresponding editor, Prof. Manoj Kumar Tiwari, for their valuable comments and suggestions in the significant improvement of the manuscript.
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