ETNA  Electronic Transactions on Numerical Analysis

Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A1011 Wien, Dr. Ignaz SeipelPlatz 2
Tel. +431515 81/DW 3420, Fax +431515 81/DW 3400 https://verlag.oeaw.ac.at, email: verlag@oeaw.ac.at 

DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)

ETNA  Electronic Transactions on Numerical Analysis ISBN 9783700182580 Online Edition Research Article
Dario A. Bini
S. 1  27 doi:10.1553/etna_vol61s1 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol61s1
Abstract: This paper deals with the problem of numerically computing the roots of polynomials $p_k(x)$, $k=1,2,\ldots$, of degree $n=2^k1$recursively defined by $p_1(x)=x+1$, $p_k(x)=xp_{k1}(x)^2+1$. An algorithm based on the EhrlichAberth simultaneous iterations complemented by the Fast Multipole Method (FMM) and the fast search of near neighbors of a set of complex numbers is provided. The algorithm, which relies on a specific strategy of selecting initial approximations, costs $O(n\log n)$ 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 69 minutes on a machine with 256 GB RAM. The case of degree $n=2^{30}1$ would require more memory and higher precision to separate the roots. With a suitable adaptation of the FMM to the limit of 256 GB RAM and by performing the computation in extended precision (i.e. with 10byte 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 $\min_{i\ne j}\xi_i^{(k)}\xi_j^{(k)}$ for the roots $\xi_i^{(k)}$ of $p_k(x)$ are deduced. The approach is effectively applied to general classes of polynomials defined by a doubling recurrence. Keywords: Mandelbrot polynomials, polynomial roots, EhrlichAberth iteration, fast multipole method Published Online: 2024/03/06 10:49:49 Object Identifier: 0xc1aa5572 0x003edfa8 Rights: . Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 10689613. …

Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A1011 Wien, Dr. Ignaz SeipelPlatz 2
Tel. +431515 81/DW 3420, Fax +431515 81/DW 3400 https://verlag.oeaw.ac.at, email: verlag@oeaw.ac.at 