A way to obtain Monte Carlo matrix inversion with minimal error

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Abstract

In this paper, we present a novel Monte Carlo algorithm for obtaining the inverse of a given nonsingular matrix A with high accuracy. The method is performed by splitting the given matrix A=D-B where D is a strictly diagonally dominant matrix and it can be easily constructed based on matrix A and B=kA such that k can be selected as an arbitrary number greater than or equal to 1. The accuracy of presented algorithm is compatible with any numerical method. The theory of the method and its numerical results are also presented.

Introduction

The problem of inverting a real square n×n matrix is an unquestionable importance in many scientific and engineering applications, e.g. communications, stochastic modeling, and many physical problems involving partial differential equations. The basic Monte Carlo method and its experimental results for obtaining approximate inverse presented in [1] and for diagonally dominant matrices using a fast maker parameter to estimate accurate inverse matrix by Monte Carlo method also presented in [4]. Recently, in [5], [6] a new Monte Carlo algorithm introduced so that it obtains not only more accurate inverse matrix than [1], [4] but also it covers the all types of matrices even non-diagonally dominant matrices to compute their inversions.

In this novel work, we introduce another Monte Carlo algorithm that it can be employed to obtain a high accurate inverse of the given nonsingular matrix. We start with reviewing the previous work to establish our main algorithm present in this paper. The algorithm to obtain the inverse of a diagonally dominant matrices presented in [5] and we briefly review it in Section 2.3. In [5] the authors presented a fast Monte Carlo algorithm based on simulated matrix data in parallel matrix computations, especially for solving large systems with a coefficient sparse matrix.

Section snippets

Stochastic method for solving linear systems

SupposeBx=b,where B is a given nonsingular system, bT=(b1,,bn) is a known vector and xT=(x1,x2,,xn) is the solution vector that we are looking for evaluating it. If we consider a nonsingular matrix Mn×n such that MB=I-C, Mb=f, then the linear system (1) is converted tox(k+1)=Cx(k)+f,where Cn×n is a given nonsingular matrix. In Monte Carlo calculations for solving linear systems, we usually use the maximum norm of matrix C given by C=maxij=1n|cij| then withρ(C)C<1,where ρ(C) is the

New algorithm for obtaining inverse matrix

In this section, we introduce a new efficient algorithm to obtain the inverse of a general nonsingular matrix with maximal rate of convergence.

Conclusion

The introduced Monte Carlo algorithm for obtaining the inverse of a general non singular matrix is a flexible algorithm to reduce the norm in condition (3) to approach to zero. This algorithm computes the accurate inverse of a general matrix when the introduced algorithms [1], [4] are not applicable. In fact, this performed algorithm not only is a new Monte Carlo algorithm to obtain the accurate inverse of diagonally dominant but also covers the obtaining of the inverse of a general nonsingular

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