On the worst-case arithmetic complexity of approximating zeros of polynomials
Journal of Complexity
ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
A bibliography on roots of polynomials
Journal of Computational and Applied Mathematics
Specified precision polynomial root isolation is in NC
Journal of Computer and System Sciences - Special issue: 31st IEEE conference on foundations of computer science, Oct. 22–24, 1990
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
An efficient algorithm for the complex roots problem
Journal of Complexity
Complexity and real computation
Complexity and real computation
Asymptotic acceleration of solving multivariate polynomial systems of equations
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
The complexity of the matrix eigenproblem
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Roots of a polynomial and its derivatives
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Approximate polynomial Gcds, Padé approximation, polynomial zeros and bipartite graphs
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Multivariate polynomials, duality, and structured matrices
Journal of Complexity
Computation of approximate polynomial GCDs and an extension
Information and Computation
Structured matrices and polynomials: unified superfast algorithms
Structured matrices and polynomials: unified superfast algorithms
EUROCAM '82 Proceedings of the European Computer Algebra Conference on Computer Algebra
Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Randomized Acceleration of Fundamental Matrix Computations
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
Parametrization of approximate algebraic curves by lines
Theoretical Computer Science - Algebraic and numerical algorithm
Parametrization of approximate algebraic surfaces by lines
Computer Aided Geometric Design
Distance bounds of ε-points on hypersurfaces
Theoretical Computer Science
Computing Approximate GCDs in Ill-conditioned Cases
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Computing clustered close-roots of univariate polynomials
Proceedings of the 2009 conference on Symbolic numeric computation
Parametrization of approximate algebraic surfaces by lines
Computer Aided Geometric Design
Approximate parametrization of plane algebraic curves by linear systems of curves
Computer Aided Geometric Design
Algebraic and numerical algorithms
Algorithms and theory of computation handbook
Root-finding by expansion with independent constraints
Computers & Mathematics with Applications
Efficient polynomial root-refiners: A survey and new record efficiency estimates
Computers & Mathematics with Applications
Univariate polynomial root-finding by arming with constraints
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
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To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zero-free annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the n-th degree into two factors balanced in the degrees and with the zero sets separated by the basic annulus. Recursive combination of the two algorithms leads to recursive computation of the complete numerical factorization of a polynomial into the product of linear factors and further to the approximation of the roots. The new rootfinder incorporates the earlier techniques of Schönhage and Kirrinnis and our old and new techniques and yields nearly optimal (up to polylogarithmic factors) arithmetic and Boolean cost estimates for the complexity of both complete factorization and rootfinding. The improvement over our previous record Boolean complexity estimates is by roughly the factor of n for complete factorization and also for the approximation of well-conditioned (well isolated) roots, whereas the same algorithm is also optimal (under both arithmetic and Boolean models of computing) for the worst case input polynomial, where the roots can be ill-conditioned, forming clusters. (The worst case bounds are supported by our previous algorithms as well.) Al our algorithms allow processor efficient acceleration to achieve solution in polygarithmic parallel time.