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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)
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)
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)
Let A be an m × n matrix with m ≥ n. Then one form of the singular-value decomposition of A is A = U T ΣV, where U and V are orthogonal and Σ is square diagonal. That is, UU T = I rank(A) , V V T = I rank(A) , U is rank(A) × m, V is rank(A) × n and Σ = σ 1 0 · · · 0 0 0 σ 2 · · · 0 0. .. 0 0 · · · σ rank(A)−1 0 0 0 · · · 0 σ rank(A) … (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)
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 , .
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.
Three algorithms for computing the coefficients of translated polynomials are discussed and compared from the point of view of complexity. The two classical translation algorithms based on explicit application of the Taylor expansion theorem and the Ruffini-Horner method, respectively, have complexityO (n 2). A third algorithm based on the fast Fourier… (More)
We present an implementation of the Continued Fractions (CF) real root isolation method using a recently developed upper bound on the positive values of the roots of polynomials. Empirical results presented in this paper verify that this implementation makes the CF method always faster than the Vincent-Collins-Akritas bisection method 3 , or any of its… (More)