Root-Refining for a Polynomial Equation

Polynomial root-finding usually consists of two stages. At first a crude approximation to a root is slowly computed; then it is much faster refined by means of the same or distinct iterations. The efficiency of computing an initial approximation resists formal study, and the users employ empirical data. In contrast, the efficiency of refinement is formally measured by the classical concept q1/α where q is the convergence order and α is the number of function evaluations per iteration. To cover iterations not reduced to function evaluations alone, e.g., ones simultaneously refining n approximations to all n roots of a degree n polynomial, we let d denote the number of arithmetic operations involved in an iteration divided by 2n because we can evaluate such a polynomial at a point by using 2n operations. For this task we employ recursive polynomial factorization to yield refinement with the efficiency $2^{cn/\log^2 n}$ for a positive constant c. For large n this is a dramatic increase versus the record efficiency 2 of refining an approximation to a single root of a polynomial. The advance could motivate practical use of the proposed root-refiners.

[1]  Victor Y. Pan,et al.  Univariate polynomials: nearly optimal algorithms for factorization and rootfinding , 2001, ISSAC '01.

[2]  V. Pan The amended DSeSC power method for polynomial root-finding , 2005 .

[3]  V. Pan,et al.  Polynomial and Matrix Computations , 1994, Progress in Theoretical Computer Science.

[4]  Victor Y. Pan,et al.  Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding , 2002, J. Symb. Comput..

[5]  James H. Curry,et al.  On zero finding methods of higher order from data at one point , 1989, J. Complex..

[6]  Béla Bollobás,et al.  A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method , 2010, Math. Comput..

[7]  K. Mahler An inequality for the discriminant of a polynomial. , 1964 .

[8]  Victor Y. Pan,et al.  Solving a Polynomial Equation: Some History and Recent Progress , 1997, SIAM Rev..

[9]  Louis W. Ehrlich,et al.  A modified Newton method for polynomials , 1967, CACM.

[10]  Victor Y. Pan Solving Polynomials with Computers , 1998 .

[11]  A. Ostrowski Solution of equations and systems of equations , 1967 .

[12]  Victor Y. Pan,et al.  New progress in real and complex polynomial root-finding , 2011, Comput. Math. Appl..

[13]  Immo O. Kerner,et al.  Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen , 1966 .

[14]  Alexandre Ostrowski Recherches sur la méthode de graeffe et les zéros des polynomes et des séries de laurent , 1940 .

[15]  Oliver Aberth,et al.  Iteration methods for finding all zeros of a polynomial simultaneously , 1973 .

[16]  A. Householder Dandelin, Lobacevskii, or Graeffe , 1959 .

[17]  M. Petkovic,et al.  Point estimation of simultaneous methods for solving polynomial equations , 2007 .

[18]  Victor Y. Pan,et al.  Multivariate Polynomials, Duality, and Structured Matrices , 2000, J. Complex..

[19]  Peter Kirrinnis,et al.  Partial Fraction Decomposition in (z) and Simultaneous Newton Iteration for Factorization in C[z] , 1998, J. Complex..

[20]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[21]  B. Wyman,et al.  Linear algebra for control theory , 1994 .

[22]  S. Smale Newton’s Method Estimates from Data at One Point , 1986 .

[23]  John H. Reif,et al.  An O(n/sup 1+/spl epsiv// log b) algorithm for the complex roots problem , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[24]  E. Bell The development of mathematics , 1941 .

[25]  Myong-Hi Kim Computation complexity of the euler algorithms for the roots of complex polynomials , 1986 .

[26]  A. C. Aitken XXV.—On Bernoulli's Numerical Solution of Algebraic Equations , 1927 .

[27]  Jean Charles Faugère,et al.  A new efficient algorithm for computing Gröbner bases without reduction to zero (F5) , 2002, ISSAC '02.

[28]  S. Smale The fundamental theorem of algebra and complexity theory , 1981 .

[29]  V. Pan Optimal and nearly optimal algorithms for approximating polynomial zeros , 1996 .

[30]  D. E. Muller A method for solving algebraic equations using an automatic computer , 1956 .

[31]  Victor Y. Pan,et al.  Efficient polynomial root-refiners: A survey and new record efficiency estimates , 2012, Comput. Math. Appl..

[32]  Victor Y. Pan,et al.  Numerical methods for roots of polynomials , 2007 .

[33]  James Renegar,et al.  On the worst-case arithmetic complexity of approximating zeros of polynomials , 1987, J. Complex..

[34]  Clifford T. Mullis,et al.  A Newton-Raphson method for moving-average spectral factorization using the Euclid algorithm , 1990, IEEE Trans. Acoust. Speech Signal Process..

[35]  S. Barnett Polynomials and linear control systems , 1983 .

[36]  K. Weierstrass Neuer Beweis des Fundamentalsatzes der Algebra , 2013 .

[37]  J. McNamee A 2002 update of the supplementary bibliography on roots of polynomials , 2002 .

[38]  Carl B. Boyer,et al.  A History of Mathematics. , 1993 .

[39]  G. Wilson Factorization of the Covariance Generating Function of a Pure Moving Average Process , 1969 .

[40]  Victor Y. Pan,et al.  Root-finding by expansion with independent constraints , 2011, Comput. Math. Appl..

[41]  P. Dooren Some numerical challenges in control theory , 1994 .

[42]  Victor Y. Pan,et al.  Optimal (up to polylog factors) sequential and parallel algorithms for approximating complex polynomial zeros , 1995, STOC '95.