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Analysis of Asynchronous Polynomial Root Finding Methods on a Distributed Memory Multicomputer
June 1994 (vol. 5 no. 6)
pp. 639-648

We have studied various implementations of iterative polynomial root finding methods on a distributed memory multicomputer. These methods are based on the construction of a sequence of approximations that converge to the set of zeros. The synchronous version consists in sharing the computation of the next iterate among the processors and updating their data through a total exchange of their results. In order to decrease thecommunication cost, we introduce asynchronous versions. The computation of the nextiterate is still shared among the processor, but the updating is done by using only nearestneighbor communications. We prove that under weak conditions, these asynchronousversions are still locally convergent, even if their convergence orders are reduced. Weanalyze the behavior of the asynchronous methods in function of their delay, thetopology of the interconnection network, and the elementary computation andcommunication times. We have implemented and compared these methods on ahypercube multicomputer.

[1] O. Aberth, "Iteration methods for finding all zeros of a polynomial simultaneously,"Mathematics of Computation, vol. 27, no. 122, pp. 339-344, 1973.
[2] G. Alefeld and J. Herzberger, "On the convergence speed of some algorithms for the simultaneous approximation of polynomial roots,"SIAM J. Num. Anal.vol. 11, no. 2, pp. 237-243, 1974.
[3] E. D. Angelova and K. I. Semerdziev, "Methods for the simultaneous approximate derivation of the roots of algebraic, trigonometric, and exponential equations,"USSR Comput. Maths. Math. Phys., vol. 22, no. 1, pp. 226-232, 1982.
[4] G. M. Baudet, "Asynchronous iterative methods for multiprocessors,"J. ACM, vol. 25, no. 2, pp. 226-244, Apr. 1978.
[5] M. Ben-Or, E. Feig, D. Kozen, and P. Tiwari, "A fast parallel algorithm for determining all roots of a polynomial with real roots,"Proc. ACM, pp. 340-349, 1986.
[6] M. Ben-Or and P. Tiwari, "Simple algorithms for approximating all roots of a polynomial with real roots," Tech. Rep., Dept. of Comput. Sci., Hebrew Univ., Jerusalem, Israel, 1989.
[7] D. P. Bertsekas and J. N. Tsitsiklis,Parallel and Distributed Computations. Englewood Cliffs, NJ: Prentice-Hall, 1989.
[8] D. Bini, "Complexity of parallel polynomial computations," inParallel computing: methods, algorithms and applications, Proc. Int. Meeting Parallel Computing, 1988, pp. 115-127.
[9] D. Bini and L. Geminiani, "On the complexity of polynomial zeros," Tech. Rep., Dipartimento di Matematica II, universitàdi Roma, Tor Vergata (1989).
[10] D. Bini and V. Pan, "Numerical computations with polynomials and related matrix computations," Tech. Rep., Dipartimento di mathematica, Universita di Roma (1988).
[11] A. Bojanczyk, "Optimal asynchronous Newton method for the solution of nonlinear equations,"J. ACM, vol. 31, no. 4, pp. 792-803, 1984.
[12] L. Bomans and D. Roose, "Communication benchmarks for the iPSC/2," inHypercube and Distributed Computers. Amsterdam: North-Holland, 1989, pp. 311-328 (Proceedings of the 1st European Workshop on Hypercube and Distributed Computers, Rennes).
[13] D. Braess and K. P. Hadeler, "Simultaneous inclusion of the zeros of a polynomial,"Num. Math., pp. 161-165 (1973).
[14] P. Brand, E. Fleury, and P. Fraigniaud, "Algorithmes parallèles pour la recherche de racines de polynômes basée sur la méthode de Durand-Kerner," manuscript, LIP, ENS-Lyon, France, 1991.
[15] M. L. Lo Cascio, L. Pasquini, and D. Trigiante, "Simultaneous determination of polynomial complex roots and multyplicities: Algorithm and problems involved," Tech. Rep., Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universita di Roma, La Sapienza, 1985.
[16] M. Cosnard and P. Fraigniaud, "Finding the roots of a polynomial on an MIMD multicomputer,"J. Parallel Computing, vol. 15, pp. 75-85 (1990).
[17] J. Davenport,Computer Algebra for Cylindrical Algebraic Decomposition, TRITA-NA-8511.
[18] D. K. Dunaway, "Calculation of zeros of areal polynomial through factorization using Euclid's algorithm,"SIAM J. Num. Anal., vol. 11, no. 6, pp. 1087-1104 (1974).
[19] E. Durand,Solutions numériques deséquations algébriques, Tome 1: Equations du type F(x) = 0; Racines d'un polynôme, Masson, Paris (1960).
[20] L. W. Ehrlich, "A modified Newton method for polynomials,"Commun. ACM, vol. 10, no. 2, pp. 107-108 (1967).
[21] M. R. Farmer and G. Loizou, "An algorithm for the total, or partial, factorization of a polynomial,"Math. Proc. Camb. Phil. Soc., vol. 82, 427-437 (1977).
[22] P. Fraigniaud, "Performance analysis of broadcasting in hypercubes," inHypercube and Distributed Computers. Amsterdam: North-Holland, pp. 311-328, 1989 (Proceeding of the 1st European Workshop on Hypercube and Distributed Computers, Rennes).
[23] P. Fraigniaud, "The Durand-Kerner polynomials root finding method in case of multiple roots,"BIT, vol. 31, pp. 112-123 (1991).
[24] P. Fraigniaud and E. Lazard, "Methods and problems of communication in usual networks," Tech. Rep. LIP 91-33, ENS-Lyon, France, 1991.
[25] T. L. Freeman, "Calculating polynomial zeros on a local memory parallel computer,"J. Parallel Computing, vol. 12, pp. 351-358, 1989.
[26] T. L. Freeman and M. K. Bane, "Asynchronous polynomial zero-finding algorithms,"J. Parallel Computing, vol. 17, pp. 673-681, 1991.
[27] S. Gamboa, "Método de Graeffe para la solucion de ecuationes polinomicas de grado superior a dos," Tech. Rep., Dept. Math., Univ. of Santander, Colombia.
[28] M. W. Green, A. J. Korsak, and M. C. Pease, "Simultaneous iteration towards all roots of a complex polynomial,"SIAM Rev., vol. 18, pp. 501-502 (1976).
[29] H. Guggenheimer, "Initial approximations in Durand-Kerner's root finding method,"BIT, vol. 26, pp. 537-539, 1986.
[30] A. S. Householder,The Numerical Treatment of a Single Nonlinear Equation. New York: McGraw-Hill, (1970).
[31] A. S. Householder, "Generalization of an algorithm of Sebastao e Silva,"Numer. Math., vol. 16, pp. 375-382, 1971.
[32] F. Hoxha, "Calcul simultanédes racines d'un polynome complexe: contributionàl'algorithmique et mise en oeuvre sur un réseau de processeurs," These de l'Institut National Polytechnique de Toulouse, France, 1988.
[33] L. H. Jamieson and T. A. Rice, "A highly parallel algorithm for root extraction,"IEEE Trans. Comput., vol. 28, pp. 443-449 (1989).
[34] M. A. Jenkins and J. F. Traub, "A three stage variable shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration,"Num. Math., vol. 14, 252-263, 1970.
[35] C. T. Ho and S. L. Johnsson, "Optimum broadcasting and personalized communication in hypercubes,"IEEE Trans. Comput., vol. 38 (1989).
[36] I. O. Kerner, "Ein gesamtschrittverfahren zur berechnung der nullstellen von polynomen," Numer. Mathemat., vol. 8, pp. 290-294 (1966).
[37] Göran Kjellberg, "Two observations on Durand-Kerner root-finding method,"BIT, vol. 24, pp. 554-559, 1984.
[38] D. E. Knuth, "Big omicron and big omega and big theta,"SIGACT News, pp. 18-23, 1976.
[39] H. Maehly, "Zur iterativen auflösung algebraisher gleichungen,"Z. Angew. Math. Physik, vol. 5, pp. 260-263, 1954.
[40] A. J. Maeder and S. A. Wynton, "Some parallel methods for polynomial root finding,"J. Comput. Appl. Math., vol. 18, pp. 71-81, 1987.
[41] I. L. Makrelov and K. I. Semerdziev, "Methods of finding simultaneously all the roots of algebraic, trigonometric, and exponential equations,"USSR Comput. Maths. Math. Phys., vol. 24, no. 5, pp. 9-105 (1984).
[42] C. A. Neff, "Specified precision polynomial root isolation is in NC," Proc. 31st Ann. Symp. Found. Comput. Sci., 1990, to appear inJ. Comput. Syst. Sci., special issue for FOCS'90.
[43] J. L. Nicolas and A. Schinzel, "Localisation des zéros de polynomes intervenant en théorie du signal," Tech. Rep., Univ. of Lyon 1, France, 1988.
[44] V. Pan, "Sequential and parallel complexity of approximate evaluation of polynomial zeros,"Comput. Math. Applic., vol. 14, no. 8, pp. 591-622, 1987.
[45] V. Pan, "Algebraic complexity of computing polynomial zeros,"Comput. Math. Applic., vol. 14, no. 4, pp. 285-304, 1987.
[46] B. Park and S. Hitotumatu, "A study on new Muller's method,"Publ. RIMS Kyoto Univ., vol. 23, pp. 667-672, 1987.
[47] L. Pasquini and D. Trigiante, "A globally convergent method for simultaneously finding polynomial roots,"Mathemat. Computation, vol. 44, no. 169, pp. 135-149, 1985.
[48] M. L. Patrick, "A highly parallel algorithm for approximating all zeros of a polynomial with only real zero,"Commun. ACM, vol. 15, no. 11, pp. 952-955, 1972.
[49] M. S. Petkovic and L. V. Stefanovic, "On some iteration functions for the simultaneous computation of multiply complex polynomial zeros,"BIT, vol. 27, 11-122, 1987.
[50] L. B. Rail, "Convergence of the Newton process to multiple solutions,"Numer. Mathemat., vol. 9, pp. 23-37, 1966.
[51] H. Rutishauser, "On a modification of the QD-algorithm with Graeffetypeconvergence,"Z. Angew. Math. Phys., vol. 13, pp. 493-496, 1962.
[52] Y. Saad and M. H. Schultz, "Data comunication in parallel architectures,"J. Parallel Computing, vol. 11, pp. 131-150, 1989.
[53] A. Schönhagie, "The fundamental theorem of algebra in term of computational complexity," preliminary rep., Mathematisches Institut der Universität Tübingen, Germany, 1982.
[54] S. Smale, "On the efficiency of algorithms of analysis," Tech. Rep., Math. Dept., Univ. of California, Berkeley.
[55] S. Smale, "The fundamental theorem of algebra and complexity theory,"Bull. Am. Math. Soc., vol. 4, no. 1, pp. 1-36 (1981).
[56] J. Stoer and R. Bulirsch,Introduction to Numerical Analysis. New York: Springer Verlag (1980).
[57] Q. F. Stout and B. Wager, "Intensive hypercube communication I," Univ. Michigan, Computing Res. Lab., CRL-TR-9-87, 1987.
[58] W. Xing-Hua and Z. Shi-ming, "The quasi-Newton method in parallel circular iteration,"J. Computational Mathemat., vol. 2, no. 4, pp. 305-309 (1984).
[59] P. Weidner, "The Durant-Kerner methods for trigonometric and exponential polynomials,"Computing, vol. 40, pp. 175-179, 1988.
[60] W. Werner, "On the simultaneous determination of polynomial roots,"Lecture Notes in Matematics, vol. 953, pp. 188-202, 1982.
[61] J. H. Wilkinson, "The evaluation of the zeros of ill-conditioned polynomials: Part I and II,"Numer. Mathemat., vol. 1, pp. 150-166, 1959.

Index Terms:
Index Termsconvergence of numerical methods; poles and zeros; polynomials; distributed memorysystems; asynchronous polynomial root finding; distributed memory multicomputer;iterative polynomial root finding; synchronous; locally convergent; convergence;hypercube multicomputer; polynomial zeros; asynchronous methods
Citation:
M. Cosnard, P. Fraigniaud, "Analysis of Asynchronous Polynomial Root Finding Methods on a Distributed Memory Multicomputer," IEEE Transactions on Parallel and Distributed Systems, vol. 5, no. 6, pp. 639-648, June 1994, doi:10.1109/71.285609
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