Monte Carlo estimates of the log determinant of large sparse matrices

Maximum likelihood estimates of parameters of some spatial models require t he computation of the log-determinant of positive-definite matrices of the form I - alpha D, where D is a large, sparse matrix with eigenvalues in [-1 , 1] and where 0 < alpha < 1, With extremely large matrices the usual direc t methods of obtaining the log-determinant require too much time and memory , We propose a Monte Carlo estimate of the log-determinant. This estimate i s simple to program, very sparing in its use of memory, easily computed in parallel and can estimate log det(I - alpha D) for many values of alpha sim ultaneously. Using this estimator, we estimate the log-determinant for a 1, 000,000 x 1,000,000 matrix D, for 100 values of alpha, in 23.1 min on a 133 MHz pentium with 64 MB of memory using Matlab. (C) 1999 Published by Elsev ier Science Inc. All rights reserved.