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66
Factoring Multivariate Polynomials via Partial Differential Equations
 Math. Comput
, 2000
"... A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms for factorin ..."
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Cited by 50 (9 self)
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A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms for factoring univariate polynomials, the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd's and by factoring univariate polynomials over the ground field. The new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields, local fields, number fields, and the complex number field. The theory of the new method allows an effective Hilbert irreducibility theorem, thus an efficient reduction of polynomials from multivariate to bivariate.
Univariate polynomials: nearly optimal algorithms for factorization and rootfinding
 In Proceedings of the International Symposium on Symbolic and Algorithmic Computation
, 2001
"... 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 zerofree annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of t ..."
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Cited by 37 (11 self)
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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 zerofree annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the nth 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 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, Neff/Reif, and Kirrinnis and our old and new techniques and yields nearly optimal (up to polylogarithmic factors) arithmetic and Boolean cost estimates for the computational 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 wellconditioned (well isolated) roots, whereas the same algorithm is also optimal (under both arithmetic and Boolean models of computing) for the worst case input polynomial, whose roots can be illconditioned, forming
Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems
 SIAM J. Numer. Anal
, 2001
"... Many polynomial systems have solution sets comprising multiple irreducible components, possibly of dierent dimensions. A fundamental problem of numerical algebraic geometry is to decompose such a solution set, using oatingpoint numerical processes, into its components. ..."
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Cited by 35 (21 self)
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Many polynomial systems have solution sets comprising multiple irreducible components, possibly of dierent dimensions. A fundamental problem of numerical algebraic geometry is to decompose such a solution set, using oatingpoint numerical processes, into its components.
A Descartes algorithm for polynomials with bitstream coefficients
 CASC, VOLUME 3718 OF LNCS
, 2005
"... The Descartes method is an algorithm for isolating the real roots of squarefree polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bitstreams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We s ..."
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Cited by 31 (3 self)
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The Descartes method is an algorithm for isolating the real roots of squarefree polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bitstreams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bitstream coefficients. To isolate the real roots of a squarefree real polynomial q(x) = qnx n +...+q0 with root separation ρ, coefficients qn  ≥ 1 and qi  ≤ 2 τ, it needs coefficient approximations to O(n(log(1/ρ)+τ)) bits after the binary point and has an expected cost of O(n 4 (log(1/ρ)+τ) 2) bit operations.
Optimal and nearly optimal algorithms for approximating polynomial zeros
 Comput. Math. Appl
, 1996
"... AbstractWe substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p(x). Our new algorithms save both Boolean and arithmetic sequential time, versus the previous best algorithms of SchSnhage [1], Pan [2], and Neff and Reif [3]. In parallel (N ..."
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Cited by 29 (13 self)
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AbstractWe substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p(x). Our new algorithms save both Boolean and arithmetic sequential time, versus the previous best algorithms of SchSnhage [1], Pan [2], and Neff and Reif [3]. In parallel (NC) implementation, we dramatically decrease the number of processors, versus the parallel algorithm of Neff [4], which was the only NC algorithm known for this problem so far. Specifically, under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p(x) to the unit disc {x: Ix [ < 1}, our algorithms (which promise to be practically effective) approximate all the zeros of p(x) within the absolute error bound 2b, by using order of n arithmetic operations and order of (b + n)n 2 Boolean (bitwise) operations (in both cases up to within polylogarithmic factors). The algorithms allow their optimal (work preserving) NC parallelization, so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or (b + n)n 2 Boolean processors. All the cited bounds on the computational complexity are within polylogarithmic factors from the optimum (in terms of n and b) under both arithmetic and Boolean models of computation (in the Boolean case, under the additional (realistic) assumption that n = O(b)).
Towards Factoring Bivariate Approximate Polynomials
"... A new algorithm is presented for factoring bivariate approximate polynomials over C [x, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the de ..."
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Cited by 20 (0 self)
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A new algorithm is presented for factoring bivariate approximate polynomials over C [x, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the degree of the problem. This method has been implemented in Maple, and has been demonstrated to be efficient and numerically robust.
An Iterated Eigenvalue Algorithm for Approximating Roots of Univariate Polynomials
 J. Symbolic Comput
, 2001
"... We present an iterative algorithm that approximates all roots of a univariate polynomial. The iteration uses floatingpoint eigenvalue computation of a generalized companion matrix. With some assumptions, we show that the algorithm approximates the roots within about log ae=ffl (P ) iterations, w ..."
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Cited by 20 (0 self)
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We present an iterative algorithm that approximates all roots of a univariate polynomial. The iteration uses floatingpoint eigenvalue computation of a generalized companion matrix. With some assumptions, we show that the algorithm approximates the roots within about log ae=ffl (P ) iterations, where ffl is the relative error of floatingpoint arithmetic, ae is the relative separation of the roots, and (P ) is the condition number of the polynomial. Each iteration requires an n\Thetan floatingpoint eigenvalue computation, n the polynomial degree, and evaluation of the polynomial to floatingpoint accuracy at up to n points. We describe a careful implementation of the algorithm, including many techniques that contribute to the practical efficiency of the algorithm. On some hard examples of illconditioned polynomials, e.g. highdegree Wilkinson polynomials, the implementation is an order of magnitude faster than the BiniFiorentino implementation mpsolve. 1
Polynomial Root Finding Using Iterated Eigenvalue Computation
 in « Proc. ISSAC », NewYork, ACM
, 2001
"... We present a novel iterative algorithm that approximates all roots of a univariate polynomial. The iteration uses floatingpoint eigenvalue computation of a generalized companion matrix. With some assumptions, we show that the algorithm approximates the roots to floatingpoint accuracy within about ..."
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Cited by 15 (1 self)
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We present a novel iterative algorithm that approximates all roots of a univariate polynomial. The iteration uses floatingpoint eigenvalue computation of a generalized companion matrix. With some assumptions, we show that the algorithm approximates the roots to floatingpoint accuracy within about log ae=ffl (P ) iterations, where ffl is the relative error of floatingpoint arithmetic, ae is the relative separation of the roots, and (P ) is the condition number of the polynomial. Each iteration requires an n\Thetan floatingpoint eigenvalue computation, n the polynomial degree, and evaluation of the polynomial to floatingpoint accuracy at n points. On some hard examples of illconditioned polynomials, e.g. highdegree Wilkinson polynomials, a careful implementation of the algorithm is an order of magnitude faster than the best alternative. 1 Introduction The algorithmic problem of approximating the roots of a univariate polynomial, presented by its coefficients, is classic in numeri...