On the worst-case arithmetic complexity of approximating zeros of polynomials
Journal of Complexity
On zero finding methods of higher order from data at one point
Journal of Complexity
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
Multivariate polynomials, duality, and structured matrices
Journal of Complexity
A modified Newton method for polynomials
Communications of the ACM
Polynomials and Linear Control Systems
Polynomials and Linear Control Systems
Point estimation of simultaneous methods for solving polynomial equations: a survey
Journal of Computational and Applied Mathematics
Time Series Analysis, Forecasting and Control
Time Series Analysis, Forecasting and Control
Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
A 2002 update of the supplementary bibliography on roots of polynomials
Journal of Computational and Applied Mathematics
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Computation complexity of the euler algorithms for the roots of complex polynomials
Computation complexity of the euler algorithms for the roots of complex polynomials
An O(n/sup 1+/spl epsiv// log b) algorithm for the complex roots problem
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
The amended DSeSC power method for polynomial root-finding
Computers & Mathematics with Applications
New progress in real and complex polynomial root-finding
Computers & Mathematics with Applications
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
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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.