Results 1  10
of
17
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 ..."
Abstract

Cited by 85 (16 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 38 (11 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 32 (3 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 30 (14 self)
 Add to MetaCart
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)).
On Approximating Complex Polynomial Zeros: Modified Quadtree (Weyl's) Construction and Improved Newton's Iteration
, 1996
"... The known record complexity estimates for approximating polynomial zeros rely on geometric constructions on the complex plane, which achieve initial approximation to the zeros and/or their clusters as well as their isolation from each other, and on the subsequent fast analytic refinement of the init ..."
Abstract

Cited by 17 (9 self)
 Add to MetaCart
The known record complexity estimates for approximating polynomial zeros rely on geometric constructions on the complex plane, which achieve initial approximation to the zeros and/or their clusters as well as their isolation from each other, and on the subsequent fast analytic refinement of the initial approximations. We modify Weyl's classical geometric construction for approximating all the n polynomial zeros in order to more rapidly achieve their strong isolation. For approximating the isolated zeros or clusters of zeros, we propose a new extension of Newton's iteration to yield quadratic global convergence (right from the start), under substantially weaker requirements to their initial isolation than one needs in the known algorithms.
Filled Julia sets with empty interior are computable. eprint, math.DS/0410580
"... Abstract. We show that if a polynomial filled Julia set has empty interior, then it is computable. 1. ..."
Abstract

Cited by 14 (8 self)
 Add to MetaCart
Abstract. We show that if a polynomial filled Julia set has empty interior, then it is computable. 1.
Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
Approximation of Polynomial Root Using a Single Input and the Corresponding Derivative Values
, 1998
"... A new formula for the approximation of root of polynomials with complex coe#cients is presented. For each simple root there exists a neighborhood such that given any input within this neighborhood, the formula generates a convergent sequence, computed via elementary operations on the input and the c ..."
Abstract

Cited by 8 (7 self)
 Add to MetaCart
A new formula for the approximation of root of polynomials with complex coe#cients is presented. For each simple root there exists a neighborhood such that given any input within this neighborhood, the formula generates a convergent sequence, computed via elementary operations on the input and the corresponding derivative values. Each element of the sequence is defined in terms of the quotient of two determinants, computable via a recursive formula. Convergence is proved by deriving an explicit error estimate. For special polynomials explicit neighborhoods and error estimates are derived that depend only on the initial error. In particular, the latter applies to the approximation of root of numbers. The proof of convergence utilizes a family of iteration functions, called the Basic Family; a nontrivial determinantal generalization of Taylor's theorem; a lower bound on determinants; Gerschgorin's theorem and Hadamard's inequality; as well as several new key results. The convergence resu...
Between Russell And Hilbert: Behmann On The Foundations Of Mathematics
 Bulletin of Symbolic Logic
, 1999
"... . After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in G ottingen in the period 19141921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfini ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
. After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in G ottingen in the period 19141921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflosung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann's work and Hilbert's foundational thought. 1. Logic and Foundations of Mathematics in G ...