Matrix Computations Using Quasirandom Sequences

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 Soból, Halton, and Faure QRNs.