Monte Carlo estimates of the log determinant of large sparse matrices
نویسنده
چکیده
Maximum likelihood estimates of parameters of some spatial models require the computation of the log-determinant of positive-de®nite matrices of the form Iÿ aD, where D is a large, sparse matrix with eigenvalues in ÿ1; 1 and where 0 < a < 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 Iÿ aD for many values of a simultaneously. Using this estimator, we estimate the log-determinant for a 1,000,000 1,000,000 matrix D, for 100 values of a, in 23.1 min on a 133 MHz pentium with 64 MB of memory using Matlab. Ó 1999 Published by Elsevier Science Inc. All rights reserved.
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