Monte Carlo estimates of the log determinant of large sparse matrices

A Monte Carlo estimate of the log-determinant of positive-definite matrices of the formI —αD is proposed that is simple to program, very sparing in its use of memory, easily computed in parallel and can estimate log det(I-αD) for many values ofα simultaneously.

Abstract

Maximum likelihood estimates of parameters of some spatial models require the computation of the log-determinant of positive-definite matrices of the formI —αD. whereD is a large, sparse matrix with eigenvalues in [−1, 1] and where 0<α<1, with extremely large matrices the usual direct methods of obtaining the log-determinant require too much time and memory. We propose a Monte Carlo estimate of the log-determinant. This estimate is simple to program, very sparing in its use of memory, easily computed in parallel and can estimate log det(I-αD) for many values ofα simultaneously Using this estimator, we estimate the log-determinant for a 1,000,000 × 1,000,000 matrixD, for 100 values ofα, in 23.1 min on 133 MHz pentium with 64 MB of memory using Matlab.