Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Matrix computations (3rd ed.)
Smoothness and dimension reduction in Quasi-Monte Carlo methods
Mathematical and Computer Modelling: An International Journal
A Parallel Quasi-Monte Carlo Method for Solving Systems of Linear Equations
ICCS '02 Proceedings of the International Conference on Computational Science-Part II
Solving Systems of Linear Algebraic Equations Using Quasirandom Numbers
LSSC '01 Proceedings of the Third International Conference on Large-Scale Scientific Computing-Revised Papers
A Monte Carlo Approach for Finding More than One Eigenpair
NMA '02 Revised Papers from the 5th International Conference on Numerical Methods and Applications
Sourcebook of parallel computing
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The convergence of Monte Carlo method for numerical integration can often be improved by replacing pseudorandom numbers (PRNs) with more uniformly distributed numbers known as quasirandom numbers (QRNs). Standard Monte Carlo methods use pseudorandom sequences and provide a convergence rate of O(N-1/2) using N samples. Quasi-Monte Carlo methods use quasirandom sequences with the resulting convergence rate for numerical integration as good as O((logN)k) N-1).In this paper we study the possibility of using QRNs for computing matrix-vector products, solving systems of linear algebraic equations and calculating the extreme eigenvalues of matrices. Several algorithms using the same Markov chains with different random variables are described. We have shown, theoretically and through numerical tests, that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the corresponding Monte Carlo methods. Numerical tests are performed on sparse matrices using PRNs and Sobol, Halton, and Faure QRNs.