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If A_1,...,A_N are real square matrices then the p-radius, generalisedLyapunov exponent or matrix pressure is defined to be the asymptoticexponential growth rate of the sum $\sum_{i_1,\ldots,i_n=1}^N \|A_{i_n}\cdotsA_{i_1}\|^p$, where p is a real parameter. Under its various names thisquantity has been investigated for its applications to topics including waveletregularity and refinement equations, fractal geometry and the large deviationstheory of random matrix products. In this article we present a new algorithmfor computing the p-radius under the hypothesis that the matrices are allpositive, or more generally under the hypothesis that they satisfy a weakercondition called domination. This algorithm is based on interpreting thep-radius as the leading eigenvalue of a trace-class operator on a Hilbert spaceand estimating that eigenvalue via approximations to the Fredholm determinantof the operator. In this respect our method is closely related to the work ofZ.-Q. Bai and M. Pollicott on computing the top Lyapunov exponent of a randommatrix product. For pairs of positive matrices of low dimension our methodyields substantial improvements over existing methods.

Fast approximation of the $p$-radius, matrix pressure or generalised Lyapunov exponent for positive and dominated matrices