Monte Carlo estimates of the log determinant of large sparse matrices

Document Type

Article

Publication Date

3-1-1999

Abstract

Maximum likelihood estimates of parameters of some spatial models require the computation of the log-determinant of positive-definite matrices of the form I - αD. where D 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 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 α, in 23.1 min on a 133 MHz pentium with 64 MB of memory using Matlab. © 1999 Published by Elsevier Science Inc. All rights reserved.

Publication Source (Journal or Book title)

Linear Algebra and Its Applications

First Page

41

Last Page

54

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