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Parallel algorithms for finding polynomial Roots on OTIS-torus

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Abstract

We present two parallel algorithms for finding all the roots of an N-degree polynomial equation on an efficient model of Optoelectronic Transpose Interconnection System (OTIS), called OTIS-2D torus. The parallel algorithms are based on the iterative schemes of Durand–Kerner and Ehrlich methods. We show that the algorithm for the Durand–Kerner method requires (N 0.75+0.5N 0.25−1) electronic moves + 2(N 0.5−1) OTIS moves using N processors. The parallel algorithm for Ehrlich method is shown to run in (N 0.75+0.5N 0.25−1) electronic moves + 2(N 0.5−1) OTIS moves with the same number of processors. The algorithms have lower AT cost than the algorithms presented in Jana (Parallel Comput 32:301–312, 2006). The scalability of the algorithms is also discussed.

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References

  1. Aberth O (1973) Iteration method for finding zeros of a polynomial simultaneously. Math Comput 27:339–344

    MATH  MathSciNet  Google Scholar 

  2. Akl SG (1985) Parallel sorting algorithm. Academic Press, Orlando

    Google Scholar 

  3. Anderson E, Brooks J, Gassl C, Scott S (1997) Performance of the Cray T3E. In: Proc of supercomputing conference, San Jose, California, USA

  4. Arabnia HR (1993) In: Atkins S, Wagner AS (eds) A transputer-based reconfigurable parallel system, transputer research and applications (NATUG 6). IOS Press, Vancouver, pp 153–169

    Google Scholar 

  5. Arif Wani M, Arabnia HR (2003) Parallel edge-region-based segmentation algorithm targeted at reconfigurable multi-ring network. J Supercomput 25(1):43–63

    Article  Google Scholar 

  6. Azad HS (2007) The performance of synchronous parallel polynomial root extraction on a ring multicomputer. Clust Comput 10(2):167–174

    Article  MathSciNet  Google Scholar 

  7. Ben-Or M, Feig E, Kozzen D, Tiwary P (1968) A fast parallel algorithm for determining all roots of a polynomial with real roots. In: Proc of ACM, pp 340–349

  8. Bhandarkar SM, Arabnia HR (1995) The REFINE multiprocessor theoretical properties and algorithms. Parallel Comput 21(11):1783–1806

    Article  Google Scholar 

  9. Bini DA, Gemignani L (2004) Inverse power and Durand–Kerner iterations for univariate polynomial root-finding. Comput Math Appl 47:447–459

    Article  MATH  MathSciNet  Google Scholar 

  10. Bose B, Broeg B, Kwon Y, Ashir Y (1995) Lee distance and topological properties of k- ary n-cubes. IEEE Trans Comput 44:1021–1030

    Article  MATH  MathSciNet  Google Scholar 

  11. Cosnard M, Fraigniaud P (1990) Finding the roots of a polynomial on an MIMD multicomputer. Parallel Comput 15:75–85

    Article  MATH  MathSciNet  Google Scholar 

  12. Day K (2002) Topological properties of OTIS-Networks. IEEE Trans Parallel Distr Syst 13:359–366

    Article  Google Scholar 

  13. Durand E (1960) Solutions numeriques des equations algebriques, vol I. Mason, Paris

    MATH  Google Scholar 

  14. Ehrlich LW (1967) A modified Newton method for polynomials. Commun ACM 10:107–108

    Article  MATH  Google Scholar 

  15. Freeman TL (1989) Calculating polynomial zeros on local memory parallel computer. Parallel Comput 12:351–358

    Article  MATH  Google Scholar 

  16. Freeman TL, Bane MK (1991) Asynchronous polynomial zero-finding algorithms. Parallel Comput 17:673–681

    Article  MATH  MathSciNet  Google Scholar 

  17. Gemignani L (2007) Structured matrix methods for polynomial root finding. In: Proc of the 2007 Intl symposium on symbolic and algebraic computation, pp 175–180

  18. Guggenheimer H (1986) Initial approximation in Durand–Kerner’s root finding method. BIT 26:537–539

    Article  MATH  Google Scholar 

  19. Hildebrand FB (1956) Introduction to numerical analysis. McGraw-Hill, New York

    MATH  Google Scholar 

  20. Jana PK (2006) Polynomial interpolation and polynomial root finding on OTIS-Mesh. Parallel Comput 32:301–312

    Article  MathSciNet  Google Scholar 

  21. Jana PK (1999) Finding polynomial zeroes on a Multi-mesh of trees (MMT). In: Proc of the 2nd int conference on information technology, Bhubaneswar, December 20–22, pp 202–206

  22. Jana PK, Sinha BP (2006) An improved parallel prefix algorithm on OTIS-Mesh. Parallel Proc Lett 16:429–440

    Article  MathSciNet  Google Scholar 

  23. Jana PK, Sinha BP (1998) Fast parallel algorithm for Graeffe’s root squaring technique. Comput Math Appl 35:71–80

    Article  MATH  MathSciNet  Google Scholar 

  24. Jana PK, Sinha BP, Datta Gupta R (1999) Efficient parallel algorithms for finding polynomial zeroes. In: Proc of the 6th int conference on advance computing, CDAC, Pune University Campus, India, December 14–16, pp 189–196

  25. Kalantari B (2008) Polynomial root finding and polynomiography. World Scientific, New Jersey

    Book  Google Scholar 

  26. Kerner IO (1966) Ein Gesamtschrittverfahren Zur Berechnung der Nullstetten on polynomen. Numer Math 8:290–294

    Article  MATH  MathSciNet  Google Scholar 

  27. Kessler RE, Schwarzmeier JL (1993) CRAY T3D: A new dimension for Cray research. CompCon, Houston, pp 176-182

    Google Scholar 

  28. Marsden G, Marchand P, Harvey P, Esener S (1993) Optical transpose interconnection system architectures. Optic Lett 18(13):1083–1085

    Article  Google Scholar 

  29. Mirankar WL (1968) Parallel methods for approximating the roots of a function. IBM Res Dev 297–301

  30. Mirankar WL (1971) A survey of parallelism in numerical analysis. SIAM Rev 524–547

  31. Noakes M et al (1993) The J-machine multicomputer: an architectural evaluation. In: Proc of the 20th int symposium on computer architecture

  32. Rajasekaran S, Sahni S (1998) Randomized routing, selection and sorting on the OTIS-Mesh. IEEE Trans Parallel Distr Syst 9:833–840

    Article  Google Scholar 

  33. Rice TA, Jamieson LH (1989) A highly parallel algorithm for root extraction. IEEE Trans Comp 38(3):443–449

    Article  MathSciNet  Google Scholar 

  34. Sahni S (2001) Models and algorithms for optical and optoelectronic parallel computers. Int J Found Comput Sci 12(13):249–264

    Article  Google Scholar 

  35. Sahni S, Wang CF (1998) BPC permutations on the OTIS hypercube optoelctronic computer. Informatica 263–269

  36. Schedler GS (1967) Parallel numerical methods for the solution of equations. Commun ACM 286–290

  37. Skachek V, Roth RM (2008) Probabilistic algorithm for finding roots of linearized polynomials. Design, codes and cryptography. Kluwer, Norwell

    Google Scholar 

  38. Wang CF, Sahni S (2001) Matrix Multiplication on the OTIS-Mesh optoelectronic computer. IEEE Trans Comput 50(7):635–646

    Article  Google Scholar 

  39. Winogard S (1972) Parallel iteration methods in complexity of computer communications. Plenum, New York

    Google Scholar 

  40. Zane F, Marchand P, Paturi R, Esener S (2000) Scalable network architectures using the optical transpose interconnection system (OTIS). J Parallel Distr Comput 60:521–538

    Article  MATH  Google Scholar 

  41. Zhanc X, Wan M, Yi Z (2008) A constrained learning algorithm for finding multiple real roots of polynomial. In: Proc of the 2008 intl symposium on computational intelligence and design, pp 38–41

  42. Zhu W, Zeng Z, Lin D (2008) An adaptive algorithm finding multiple roots of polynomials. Lect Notes Comput Sci 5262:674–681

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Information Management, Xavier Institute of Social Service, Ranchi, 834001, India

    Keny T. Lucas

  2. Department of Computer Science and Engineering, Indian School of Mines University, Dhanbad, 826 004, India

    Prasanta K. Jana

Authors
  1. Keny T. Lucas
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  2. Prasanta K. Jana
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Correspondence to Keny T. Lucas.

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Lucas, K.T., Jana, P.K. Parallel algorithms for finding polynomial Roots on OTIS-torus. J Supercomput 54, 139–153 (2010). https://doi.org/10.1007/s11227-009-0312-7

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  • DOI: https://doi.org/10.1007/s11227-009-0312-7

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