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Uspensky's 1948 book on the theory of equations presents an algorithm, based on Descartes' rule of signs, for isolating the real roots of a squarefree polynomial with real coefficients. Programmed in SAC-1 and applied to several classes of polynomials with integer coefficients, Uspensky's method proves to be a strong competitor of the recently discovered… (More)
Finding an upper bound for the positive roots of univariate polynomials is an important step of the continued fractions real root isolation algorithm. The revived interest in this algorithm has highlighted the need for better estimations of upper bounds of positive roots. In this paper we present a new theorem, based on a generalization of a theorem by D.… (More)
This paper discusses a set of algorithms which, given a polynomial equation with integer coefficients and without any multiple roots, uses exact (infinite precision) integer arithmetic and the Vincent-Uspensky-Akritas theorem to compute intervals containing the real roots of the polynomial equation. Theoretical computing time bounds are developed for these… (More)
Abstract. In this paper we compare four implementations of the Vincent-AkritasStrzeboński Continued Fractions (VAS-CF) real root isolation method using four different (two linear and two quadratic complexity) bounds on the values of the positive roots of polynomials. The quadratic complexity bounds were included to see if the quality of their estimates… (More)
Given an m×n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see ). Fig. 1. The row spaces and the nullspaces of A and A T ; a 1 through a n and h 1 through h m are abbreviations of the alignerframe and hangerframe vectors respectively (see ). The Fundamental Theorem of Linear Algebra tells us that N (A) is the… (More)
In this paper an attempt is made to correct the misconception of several authors  that there exists a method by Upensky (based on Vincent's theorem) for the isolation of the real roots of a polynomial equation with rational coefficients. Despite Uspensky's claim, in the preface of his book , that he invented this method, we show that what Upensky… (More)
Let A be an m × n matrix with m ≥ n. Then one form of the singular-value decomposition of A is A = UΣV, where U and V are orthogonal and Σ is square diagonal. That is, UUT = Irank(A), V V T = Irank(A), U is rank(A)×m, V is rank(A)× n and Σ = σ1 0 · · · 0 0 0 σ2 · · · 0 0 .. .. . . . .. .. 0 0 · · · σrank(A)−1 0 0 0 · · · 0 σrank(A) is a… (More)
In this paper we compare two real root isolation methods using Descartes’ Rule of Signs: the Interval Bisection method, and the Continued Fractions method. We present some time-saving improvements to both methods. Comparing computation times we conclude that the Continued Fractions method works much faster save for the case of very many very large roots.
It is well known that Euclid's algorithm for computing the greatest common divisor (gcd) of two integer numbers is more than two thousand years old and, as it turns out, it is the oldest known algorithm. Interest in computing a gcd of two polynomials first appeared only in the sixteenth century and the problem was solved by Simon Stevin  simply by… (More)
The purpose of this short note is to make known to our community some of the results of my Ph.D. Dissertation . More details can also be found elsewhere , .