A parallel quasi-Monte Carlo method for computing extremal eigenvalues

The convergence of Monte Carlo methods for numerical integration can often be improved by replacing pseudorandom numbers (PRNs) with more uniformly distributed numbers known as quasirandom numbers (QRNs). In this paper the convergence of a Monte Carlo method for evaluating the extremal eigenvalues of a given matrix is studied when quasirandom sequences axe used. An error bound is established and numerical experiments with large sparse matrices are performed using three different QRN sequences: Sobol', Halton and Faure. The results indicate: An improvement in both the magnitude of the error and in tile convergence rate that can be achieved when using QRNs in place of PRNs. The high parallel efficiency established for Monte Carlo methods is preserved for quasi-Monte Carlo methods in this case. The execution time for computing an extremal cigenvalue of a large, sparse matrix on p processors is bounded by O(lN/p), where l is the length of the Markov chain in the stochastic process and N is the number of chains, both of which are independent of the matrix size.