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In [14], the authors developed a new approach to the computation of theHausdorff dimension of the invariant set of an iterated function system or IFS.In this paper, we extend this approach to incorporate high order approximationmethods. We again rely on the fact that we can associate to the IFS aparametrized family of positive, linear, Perron-Frobenius operators $L_s$, anidea known in varying degrees of generality for many years. Although $L_s$ isnot compact in the setting we consider, it possesses a strictly positive $C^m$eigenfunction $v_s$ with eigenvalue $R(L_s)$ for arbitrary $m$ and all otherpoints $z$ in the spectrum of $L_s$ satisfy $|z| \le b$ for some constant $b <R(L_s)$. Under appropriate assumptions on the IFS, the Hausdorff dimension ofthe invariant set of the IFS is the value $s=s_*$ for which $R(L_s) =1$. Thiseigenvalue problem is then approximated by a collocation method at the extendedChebyshev points of each subinterval using continuous piecewise polynomials ofarbitrary degree $r$. Using an extension of the Perron theory of positivematrices to matrices that map a cone $K$ to its interior and explicit a prioribounds on the derivatives of the strictly positive eigenfunction $v_s$, we giverigorous upper and lower bounds for the Hausdorff dimension $s_*$, and thesebounds converge rapidly to $s_*$ as the mesh size decreases and/or thepolynomial degree increases.

Hidden Positivity and a New Approach to Numerical Computation of Hausdorff Dimension: Higher Order Methods