Numerical computation of the roots of Mandelbrot polynomials: an experimental analysis
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This paper deals with the problem of numerically computing the roots of polynomials p_k(x), k=1,2,…, of degree n=2^k-1 recursively defined by p_1(x)=x+1, p_k(x)=xp_k-1(x)^2+1. An algorithm based on the Ehrlich-Aberth simultaneous iterations complemented by the Fast Multipole Method, and by the fast search of near neighbors of a set of complex numbers, is provided. The algorithm has a cost of O(nlog n) arithmetic operations per step. A Fortran 95 implementation is given and numerical experiments are carried out. Experimentally, it turns out that the number of iterations needed to arrive at numerical convergence is O(log n). This allows us to compute the roots of p_k(x) up to degree n=2^24-1 in about 16 minutes on a laptop with 16 GB RAM, and up to degree n=2^28-1 in about one hour on a machine with 256 GB RAM. The case of degree n=2^30-1 would require higher memory and higher precision to separate the roots. With a suitable adaptation of FMM to the limit of 256 GB RAM and by performing the computation in extended precision (i.e. with 10-byte floating point representation) we were able to compute all the roots in about two weeks of CPU time for n=2^30-1. From the experimental analysis, explicit asymptotic expressions of the real roots of p_k(x) and an explicit expression of min_i j|ξ_i^(k)-ξ_j^(k)| for the roots ξ_i^(k) of p_k(x) are deduced. The approach is extended to classes of polynomials defined by a doubling recurrence.
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