An O(nlog/sup 3/ n) algorithm for the real root problem

Given a univariate complex polynomial f(x) of degree n with rational coefficients expressed as a ratio of two integers <2/sup m/, the root problem is to find all the roots of f(x) up to specified precision 2/sup -/spl mu//. In this paper we assume the arithmetic model for computation. We give an algorithm for the real root problem: where all the roots of the polynomial are real. Our real root algorithm has time cost of O(nlog/sup 2/ n(log n+log b)), where b=m+/spl mu/, thus has time bound O(nlog/sup 3/ n) even in the case of high precision m+/spl mu//spl les/n/sup O(1/). This is within a small polylog factor of optimality, thus (perhaps surprisingly) upper bounding the arithmetic complexity of our real root problem to nearly the same as basic arithmetic operations on polynomials. We require only /spl pi/=O(n(/spl mu/+m+n)) bits of precision to carry out our computations. The Boolean complexity of our algorithm is a multiplicative factor of M(/spl pi/)=O(/spl pi/(log /spl pi/)loglog /spl pi/) more.<<ETX>>

[1]  Victor Y. Pan,et al.  Polynomial division and its computational complexity , 1986, J. Complex..

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

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

[4]  Ephraim Feig,et al.  A Fast Parallel Algorithm for Determining all Roots of a Polynomial with Real Roots , 1988, SIAM J. Comput..

[5]  Victor Y. Pan,et al.  Some polynomial and Toeplitz matrix computations , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[6]  S. Smale.  On the efficiency of algorithms of analysis , 1985 .

[7]  George E. Collins.  Polynomial Remainder Sequences and Determinants , 1966 .

[8]  Stephen Smale,et al.  Computational Complexity: On the Geometry of Polynomials and a Theory of Cost: II , 1986, SIAM J. Comput..

[9]  David Y. Y. Yun,et al.  Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants , 1980, J. Algorithms.

[10]  D. Rose,et al.  Generalized nested dissection , 1977 .

[11]  Jacob T. Schwartz.  Probabilistic algorithms for verification of polynomial identities (invited) , 1979, EUROSAM.

[12]  P. Henrici,et al.  Circular arithmetic and the determination of polynomial zeros , 1971 .

[13]  V. Pan.  On computations with dense structured matrices , 1990 .

[14]  B. Anderson,et al.  Asymptotically fast solution of toeplitz and related systems of linear equations , 1980 .

[15]  Michael Ben-Or,et al.  Simple algorithms for approximating all roots of a polynomial with real roots , 1990, J. Complex..

[16]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

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

[18]  Victor Y. Pan,et al.  Fast and E cient Parallel Evaluation of the Zeros of a Polynomial Having Only Real Zeros , 1989 .

[19]  C. Andrew Neff,et al.  Specified precision polynomial root isolation is in NC , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[20]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[21]  M. Marden.  Geometry of Polynomials , 1970 .

[22]  V. Pan.  Sequential and parallel complexity of approximate evaluation of polynomial zeros , 1987 .

[23]  Victor Y. Pan,et al.  Fast and Efficient Parallel Solution of Sparse Linear Systems , 1993, SIAM J. Comput..

[24]  F. Gomes Teixeira.  Deuxième note sur le développement des fonctions satisfaisant à une équation différentielle , 1887 .