Novel Monte Carlo Algorithm for Linear Algebraic Systems

A novel Monte Carlo algorithm has been introduced and analyzed for solving linear algebraic equations. This hybrid algorithm is comparable to the ”Walk on Equations”; Monte Carlo approach developed by Ivan Dimov, Sylvain Maire, and Jean Michel Sellier. A comparison with the Gauss-Siedel method has been made for matrices up to a size of 10000. The algorithms’ performance is enhanced by choosing appropriate values for the relaxation parameters, resulting in a significant reduction in computation time and lower relative errors for a given number of iterations. A theorem proving the algorithms convergence has been presented. By balancing the iteration matrix, the original algorithm can be optimized. In addition, a sequential Monte Carlo method developed by John Halton based on an iterative application of the control variate method has been employed. The most notable numerical experiment involves a large system derived from a finite element approximation of a problem that describes a beam structure in constructive mechanics.