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Efficiency of Root-Finding Iterations
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Mathematics

Root-Finding with Implicit Deflation

Profile image of Vitaly ZadermanVitaly Zaderman

2019, Lecture Notes in Computer Science

Abstract

Functional iterations such as Newton's are a popular tool for polynomial root-finding. We consider a realistic situation where some roots have already been approximated (we say tamed), and one would like to restrict further root-finding to the approximation of the remaining (wild) roots. A natural approach of applying explicit deflation has been much studied and recently advanced by one of the authors of this paper, but presently we consider the alternative of implicit deflation combined with mapping of the variable and reversion of an input polynomial. The hope is that the union of the sets of tame roots approximated in a number of such transformations can cover all roots of a polynomial. We also show another direction to substantial further progress in this 1 long and extensively studied area. Namely we dramatically increase the local efficiency of root-finding by means of the incorporation of fast algorithms for multipoint polynomial evaluation and the Fast Multipole Method.

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References (71)

  1. Aberth, O.: Iteration Methods for Finding All Zeros of a Polynomial Simultaneously, Mathematics of Computation 27, 122, 339-344 (1973) doi: 10.1090/S0025-5718-1973-0329236-7
  2. Bell, E.T.: The Development of Mathematics. McGraw-Hill, New York (1940) doi: 10.2307/2268176
  3. Boyer, C.A.: A History of Mathematics. Wiley, New York (1991) doi: 10.1177/027046769201200316
  4. Barnett, S.: Polynomial and Linear Control Systems. Marcel Dekker, New York (1983) doi: 10.1112/blms/16.3.331
  5. Bini, D.A.: Parallel Solution of Certain Toeplitz Linear Systems. SIAM J. Comput. 13, 2, 268-279 (1984) doi: 10.1137/0213019
  6. Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Com- putation. Springer (1998). doi: 10.1007/978-1-4612-0701-6
  7. Bini, D.A., Fiorentino, G.: Design, Analysis, and Implementation of a Multiprecision Polynomial Rootfinder. Numer. Algorithms 23, 127-173 (2000) doi: 10.1023/A:1019199917103
  8. Bini, D.A., Gemignani, L., Pan, V.Y.: Inverse Power and Du- rand/Kerner Iteration for Univariate Polynomial Root-finding. Com- put. Math. Appl. 47, 2/3, 447-459 (2004) doi: 10.1016/S0898- 1221(04)90037-5
  9. Box, G.E.P., Jenkins, G.M.: Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco, California (2015, fifth edition) doi: 10.1111/jtsa.12194
  10. Bini, D., Pan, V.Y.: Computing Matrix Eigenvalues and Polynomial Zeros Where the Output Is Real. SIAM J. Comput. 27, 4, 1099- 1115 (1998). Proc. version in SODA'91, 384-393. ACM Press, NY, and SIAM Publ., Philadelphia (1991) doi: 10.1137/S0097539790182482
  11. Bini, D., Pan, V.Y.: Graeffe's, Chebyshev, and Cardinal's Processes for Splitting a Polynomial into Factors. J. Complex. 12, 492-511 (1996) doi: 10.1006/jcom.1996.0030
  12. Bini, D.A., Robol, L.: Solving Secular and Polynomial Equations: a Multiprecision Algorithm. J. Comput Appl Math 272, 276-292 (2014) doi: 10.1016/j.cam.2013.04.037
  13. Becker, R., Sagraloff, M., Sharma, V., Xu, J., Yap, C.: Complexity Analysis of Root Clustering for a Complex Polynomial. In: Interna- tional Symposium on Symbolic and Algebraic Computation (ISSAC 2016), 71-78 (2016) doi: 10.1145/2930889.2930939
  14. Becker, R., Sagraloff, M., Sharma, V., Yap, C.: A Near-Optimal Sub- division Algorithm for Complex Root Isolation based on the Pellet Test and Newton Iteration. J. Symb Comput 86, 51-96 (2018) doi: 10.1016/j.jsc.2017.03.009
  15. Ben-Or, M., Tiwari, P.: Simple algorithms for approximating all roots of a polynomial with real roots. J. Complexity 6(4), 417-442 (1990) doi: 10.1016/0885-064X(90)90032-9
  16. Barba, L. A., Yokota, R.: How Will the Fast Multipole Method Fare in Exascale Era? SIAM News, 46, 6, 1-3, July/August (2013)
  17. Durand, E.: Solutions numériques des équations algébriques, Tome 1: Equations du type F(X)=0; Racines d'un polynôme. Masson, Paris (1960)
  18. Du, Q., Jin, M., Li, T.Y., Zeng, Z.: The quasi-Laguerre iteration. Math. Comput. 66(217), 345-361 (1997) doi: 10.1090/S0025-5718-97- 00786-2
  19. Delves, L.M., Lyness, J.N.: A Numerical Method for Locating the Zeros of an Analytic Function. Math. Comput. 21, 543-560 (1967). doi: 10.1090/S0025-5718-1967-0228165-4
  20. Demeure, C.J., Mullis, C.T.: The Euclid Algorithm and the Fast Com- putation of Cross-covariance and Autocovariance Sequences. IEEE Trans. Acoust, Speech, Signal Process. 37, 545-552 (1989) doi: 10.1109/29.17535
  21. Demeure, C.J., Mullis, C.T.: A Newton-Raphson Method for Moving- average Spectral Factorization Using the Euclid Algorithm. IEEE Trans. Acoust, Speech, Signal Processing 38, 1697-1709 (1990) doi: 10.1109/29.60101
  22. Ehrlich, L.W.: A Modified Newton Method for Polynomials. Commun ACM 10, 107-108 (1967) doi: 10.1145/363067.363115
  23. Encyclopedia of Mathematics. Newton method (Hazewinkel, Michiel, ed.). Springer Science+Business Media B.V. Kluwer Academic Publish- ers (1994, first edition) doi: 10.1007/978-94-009-5991-0 , (2000, second edition) doi: 10.1007/978-94-015-1279-4
  24. Emiris, I.Z., Pan, V.Y., Tsigaridas, E.: Algebraic Algorithms. In: Tucker, A.B., Gonzales, T., Diaz-Herrera, J.L. (eds) Computing Hand- book (3rd edition), Vol. I: Computer Science and Software Engineering, Ch. 10, pp. 10-1 -10-40. Taylor and Francis Group (2014)
  25. Gerasoulis, A., Grigoriadis, M. D., Sun, L.: A Fast Algorithm for Trummer's Problem. SIAM Journal on Scientific and Statistical Com- puting 8, 1, 135-138 (1987) doi: 10.1137/0908017
  26. Greengard, L., Rokhlin, V.: A Fast Algorithm for Particle Simu- lation. Journal of Computational Physics 73, 325-348 (1987) doi: 10.1016/0021-9991(87)90140-9
  27. Householder, A.S.: Dandelin, Lobachevskii, or Graeffe? Amer. Math. Monthly 66, 464-466. (1959) doi: 10.2307/2310626
  28. Henrici, P.: Applied and Computational Complex Analysis. Vol. 1: Power Series, Integration, Conformal Mapping, Location of Zeros. Wi- ley (1974)
  29. Henrici, P., Gargantini, I.: Uniformly Convergent Algorithms for the Simultaneous Approximation of All Zeros of a Polynomial. In: Dejon, B., Henrici, P. (eds) Constructive Aspects of the Fundamental Theorem of Algebra. Wiley (1969)
  30. Habbard, J., Schleicher, D., Sutherland, S.: How to Find All Roots of Complex Polynomials by Newton's Method. Invent. Math. 146, 1-33 (2001) doi: 10.1007/s002220100149
  31. Imbach, R., Pan, V.Y., Yap, C.: Implementation of a Near- Optimal Complex Root Clustering Algorithm. In: Proc. of Interna- tional Congress on Math Software (ICMS 2018), 235-244 (2018) doi: 10.1007/978-3-319-96418-8 28
  32. Imbach, R., Pan, V.Y., Yap, C., Kotsireas, I.S., Zaderman, V.: Root- finding with Implicit Deflation. In: Proc. of CASC 2019 (to appear). arXiv:1606.01396, submitted on 21 May (2019)
  33. Kerner, I. O.: Ein Gesamtschrittverfahren zur Berechung der Null- stellen von Polynomen. Numerische Math. 8, 290-294 (1966) doi: 10.1007/BF0216256
  34. Kirrinnis, P.: Polynomial Factorization and Partial Fraction Decom- position by Simultaneous Newton's Iteration. In: J. Complex. 14, 378- 444 (1998) doi: 10.1006/jcom.1998.0481
  35. Kim, M.-H., Sutherland, S.: Polynomial Root-Finding Algorithms and Branched Covers. SIAM Journal on Computing 23, 2, 415-436 (1994) doi: 10.1137/S0097539791201587
  36. Kobel, A., Rouillier, F., Sagraloff, M.: Computing Real Roots of Real Polynomials ... and Now for Real! In: Intern. Symp. Symb. Algebraic Computation (ISSAC 2016), 301 -310. ACM Press, New York (2016) doi: 10.1145/2930889.2930937
  37. Mahley, H.: Zur Auflösung Algebraisher Gleichngen, Z. Andew. Math. Physik. 5, 260 -263 (1954)
  38. McNamee, J.M.: Numerical Methods for Roots of Polynomials, Part I, XIX+354 pages. Elsevier (2007)
  39. Moenck, R., Borodin, A.: Fast Modular Transforms via Division., In: Proc. 13th Annual Symposium on Switching and Automata The- ory (SWAT 1972), 90-96, IEEE Computer Society Press (1972) doi: 10.1109/SWAT.1972.5.
  40. Mourrain, B., Pan, V.Y.: Lifting/Descending Processes for Polynomial Zeros and Applications. J. of Complexity 16, 1, 265 -273 (2000) doi: 10.1006/jcom.1999.0533
  41. McNamee, J.M., Pan, V.Y.: Numerical Methods for Roots of Polyno- mials, Part II, XXI+728 pages. Elsevier (2013)
  42. Neff, C.A., Reif, J.H.: An o(n 1+ǫ ) Algorithm for the Complex Root Problem. In: Proc. of 35th Ann. IEEE Symp. on Foundations of Com- puter Science (FOCS '94), 540-547. IEEE Computer Society Press (1994) doi: 10.1109/SFCS.1994.365737
  43. Ostrowski, A. M.: Solution of Equations and Systems of Equations. Academic Press, New York (1966) doi: 10.1017/S0008439500029805
  44. Pan, V.Y.: Optimal (up to Polylog Factors) Sequential and Parallel Algorithms for Approximating Complex Polynomial Zeros. In: Proc. 27th Ann. ACM Symp. on Theory of Computing (STOC'95), 741-750. ACM Press, New York (1995) doi: 10.1145/225058.225292
  45. Pan, V.Y.: Solving a Polynomial Equation: Some History and Recent Progress. SIAM Rev 39, 2, 187-220 (1997) doi: 10.1137/S0036144595288554
  46. Pan, V.Y.: Solving Polynomials with Computers. Am Sci 86, 62-69. January-February (1998) doi: 10.1511/1998.1.62
  47. Pan, V.Y.: Approximation of Complex Polynomial Zeros: Modified Quadtree (Weyl's) Construction and Improved Newton's Iteration. J.Complex. 16, 1, 213-264 (2000) doi: 10.1006/jcom.1999.0532
  48. Pan, V.Y.: Structured Matrices and Polynomials: Unified Super- fast Algorithms. Birkhäuser/Springer, Boston/New York, (2001) doi: 10.1007/978-1-4612-0129-8
  49. Pan, V.Y.: Univariate Polynomials: Nearly Optimal Algorithms for Factorization and Rootfinding. J. Symb Comput 33, 5, 701-733 (2002) doi: 10.1006/jsco.2002.0531
  50. Pan, V.Y.: Transformations of Matrix Structures Work Again. Linear Algebra and Its Applications 465, 1 -32 (2015) doi: 10.1016/j.laa.2014.09.004
  51. Pan, V.Y.: Root-finding with Implicit Deflation. arXiv:1606.01396, submitted on 4 June (2016)
  52. Pan, V.Y.: Simple and Nearly Optimal Polynomial Root-Finding by Means of Root Radii Approximation. In: Kotsireas I.S., Martinez- Moro, E. (eds) Springer Proceedings in Mathematics and Statistics, Ch. 23: Applications of Computer Algebra 198 AG. Springer Interna- tional Publishing (2017).
  53. Chapter DOI.10.1007/978-3-319-56932-1 23
  54. Pan, V.Y.: Old and New Nearly Optimal Polynomial Root-finders. In: CASC (2019) (to appear) Also arxiv: 1805.12042 [cs.NA] May (2019)
  55. Pan, V.Y., Sadikou, A., Landowne, E.: Univariate Polynomial Division with a Remainder by Means of Evaluation and Interpolation. In: Proc. of 3rd IEEE Symposium on Parallel and Distributed Processing, 212- 217. IEEE Computer Society Press, Los Alamitos, California (1991) doi: 10.1109/SPDP.1991.218277
  56. Pan, V.Y., Sadikou, A., Landowne, E.: Polynomial Division with a Remainder by Means of Evaluation and Interpolation. Inform Process Lett 44, 149-153 (1992) doi: 10.1016/0020-0190(92)90055-Z
  57. Pan, V.Y., Tsigaridas, E.P.: Nearly Optimal Refinement of Real Roots of a Univariate Polynomial. J. Symb Comput 74, 181-204 (2016) doi: 10.1016/j.jsc.2015.06.009. Also in: Kauers, M. (ed) Proc. of ISSAC 2013, pp. 299-306. ACM Press, New York (2013)
  58. Pan, V.Y., Tsigaridas, E.P.: Nearly Optimal Computations with Struc- tured Matrices. In: Theor. Comput. Sci., Watt, S., Verschelde, J., Zhi, L. (eds) Special Issue on Symbolic-Numerical Algorithms 681, 117- 137 (2017). doi: 10.1016/j.tcs.2017.03.030.
  59. Pan, V.Y., Zhao, L.: Real Polynomial Root-finding by Means of Ma- trix and Polynomial Iterations. In: Theor. Comput. Sci., Watt, S., Verschelde, J., Zhi, L. (eds) Special Issue on Symbolic-Numerical Al- gorithms 681, 101-116 (2017). doi: 10.1016/j.tcs.2017.03.032.
  60. Renegar, J.: On the Worst-case Arithmetic Complexity of Approxi- mating Zeros of Polynomials. J. Complex. 3, 2, 90-113 (1987) doi: 10.1016/0885-064X(87)90022-7
  61. A. Schönhage. The Fundamental Theorem of Algebra in Terms of Computational Complexity. Tech. Report, Math. Dept., University of Tübingen, Tübingen, Germany, (1982)
  62. Schönhage, A.: Asymptotically Fast Algorithms for the Numerical Muitiplication and Division of Polynomials with Complex Coefficients. In: Proc. of European Computer Algebra Conference (EUROCAM 1982), 3-15. Computer Algebra, (1982) doi: 10.1007/3-540-11607-9 1
  63. Schönhage, A.: Quasi GCD Computations. J. Complex. 1, 118-137 (1985) doi: 10.1016/0885-064X(85)90024-X
  64. Schleicher, D.: private communication.
  65. Schleicher, D., Stoll, R.: Newton's Method in Practice: Finding All Roots of Polynomials of Degree One Million Efficiently. Theor. Com- put. Sci. 681, 146-166 (2017) doi: 10.1016/j.tcs.2017.03.025
  66. Tilli, P.: Convergence Conditions of Some Methods for the Simulta- neous Computation of Polynomial Zeros. Calcolo 35, 3-15 (1998) doi: 10.1007/s100920050005
  67. Van Dooren, P..M.: Some Numerical Challenges in Control Theory. Linear Algebra for Control Theory, IMA Vol. Math. Appl. 62, 177 - 189 (1994) doi: 10.1007/978-1-4613-8419-9 12
  68. Weierstrass, K.: Neuer Beweis des Fundamentalsatzes der Algebra. Mathematische Werker, Tome III, 251-269. Mayer und Mueller, Berlin (1903)
  69. Weyl, H.: Randbemerkungen zu Hauptproblemen der Mathematik. II. Fundamentalsatz der Algebra and Grundlagen der Mathematik. Math- ematische Zeitschrift, 20, 131-151 (1924)
  70. Wilson, G.T.: Factorization of the Covariance Generating Function of a Pure Moving-average. SIAM J. Numer Anal 6, 1-7 (1969) doi: 10.1137/0706001
  71. Werner, W.: Some Improvements of Classical Iterative Methods for the Solution of Nonlinear Equations. In: Allgower, E.L. et al (eds) Nu- merical Solution of Nonlinear Equations, Proc. Bremen 1980 (L.N.M. 878), 427 -440. Springer, Berlin (1982) doi: 10.1007/BFb0090691

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