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47
Solving A Polynomial Equation: Some History And Recent Progress
, 1997
"... The classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of presentday computing. We briefly recall the history of the algorithmic a ..."
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Cited by 85 (16 self)
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The classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of presentday computing. We briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms. We end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost.
A Gröbner free alternative for polynomial system solving
 Journal of Complexity
, 2001
"... Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic ..."
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Cited by 82 (16 self)
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Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation
Complexity of Bézout’s Theorem IV : Probability of Success, Extensions
 SIAM J. Numer. Anal
, 1996
"... � � � We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n +1 complex variables of fixed degrees will find all the roots of the system. We also extend the framework of our analysis to cover the ..."
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Cited by 60 (9 self)
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� � � We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n +1 complex variables of fixed degrees will find all the roots of the system. We also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context. Further complexity theory is developed. 1. Introduction. 1A. Bezout’s Theorem Revisited. Let f: � n+1 → � n be a system of homogeneous polynomials f =(f1,...,fn), deg fi = di, i=1,...,n. The linear space of such f is denoted by H (d) where d = (d1,...,dn). Consider the
Complexity of Bezout's theorem V: Polynomial time
 Theoretical Computer Science
, 1994
"... this paper is to show that the problem of finding approximately a zero of a polynomial system of equations can be solved in polynomial time, on the average. The number of arithmetic operations is bounded by cN ..."
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Cited by 52 (5 self)
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this paper is to show that the problem of finding approximately a zero of a polynomial system of equations can be solved in polynomial time, on the average. The number of arithmetic operations is bounded by cN
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 38 (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
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 30 (14 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)).
An Efficient Algorithm for the Complex Roots Problem
, 1996
"... Given a univariate polynomial f(z) of degree n with complex coefficients, whose norms are less than 2 m in magnitude, the root problem is to find all the roots of f(z) up to specified precision 2 \Gamma . Assuming the arithmetic model for computation, we provide an algorithm which has complexity ..."
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Cited by 24 (2 self)
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Given a univariate polynomial f(z) of degree n with complex coefficients, whose norms are less than 2 m in magnitude, the root problem is to find all the roots of f(z) up to specified precision 2 \Gamma . Assuming the arithmetic model for computation, we provide an algorithm which has complexity O(n log 5 n log b), where b = m + . This improves on the previous best known algorithm of Pan for the problem which has complexity O(n 2 log 2 n log b). A remarkable property of our algorithm is that it does not require any assumptions about the root separation of f , which were either explicitly, or implicitly, required by previous algorithms. Moreover it also has a work efficient parallel implementation. We also show that both the sequential and parallel implementations of the algorithm work without modification in the Boolean model of arithmetic. In this case, it follows from root perturbation estimates that we need only specify ` = dn(b + log n + 3)e bits of the binary representat...
Sequential and parallel complexity of approximate evaluation of polynomial zeros
 COMPUT. MATH. APPLIC
, 1987
"... Our new sequential and parallel algorithms establish new record upper bounds on both arithmetic and Boolean complexity of approximating to complex polynomial zeros. O(n 2 log b log n) arithmetic operations or O(n log n log (bn)) parallel steps and n log b/log (bn) processors suffice in order to appr ..."
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Cited by 18 (7 self)
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Our new sequential and parallel algorithms establish new record upper bounds on both arithmetic and Boolean complexity of approximating to complex polynomial zeros. O(n 2 log b log n) arithmetic operations or O(n log n log (bn)) parallel steps and n log b/log (bn) processors suffice in order to approximate with absolute errors ~< 2 mb to all the complex zeros of an nth degree polynomial p(x) whose coefficients have mod ~< 2 m. If we only need such an approximation to a single zero of p(x), then O(n log b log n) arithmetic operations or O(log z n log (bn)) steps and (n/log n)log b/log (bn) processors suffice (which places the latter problem in NC, that is, in the class of problems that can be solved using polylogarithmic parallel time and a polynomial number of processors). Those estimates are reached in computations with O(bn) binary bits where the polynomial has integer coefficients. We also reach the sequential Boolean time bounds O(bn31og (bn)log log(bn)) for approximating to all the zeros (very minor improvement of the bound announced in 1982 by Schrnhage) and O(bn21og log n Iog(bn)log log(bn)) for approximating to a single zero. Among further implications are the improvements of the known algorithm.q and complexity estimates for computing matrix eigenvalues, for polynomial factorization over the field of complex numbers and for solving systems of polynomial equations. The computations rely on recursive application of Turan's proximity test of 1968, on its more recent extensions to root radii computations, on contour integration via Fast Fourier transform (FFT) within geometric constructions for search and exclusion, and (for the final minor improvements ofthe complexity bounds) on the recursive factorization ofp(x) over discs on the complex plane via numerical integration and Newton's iterations.'