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-I and applied to several classes of polynomials with integer coefficients, Uspensky's method proves to be a strong competitor of the recently discovered algorithm of Collins and Loos. It is shown, however, that it's maximum computing time is exponential in the coefficient length. This motivates a modification of the Uspensky algorithm which is quadratic in the coefficient length and which also performs well in the practical test cases.
Recent progress in polynomial elimination has rendered the computation of the real roots of ill-conditioned polynomials of high degree (over 1000) with huge coefficients (several thousand digits) a critical operation in computer algebra.
To rise to the occasion, the only method-candidate that has been considered by various authors for modification and improvement has been the Collins-Akritas bisection method [1], which is a based on a variation of Vincent’s theorem [2].
The most recent example is the paper by Rouillier and Zimmermann [3], where the authors present “... a new algorithm, which is optimal in terms of memory usage and as fast as both Collins and Akritas’ algorithm and Krandick variant ...” [3]
In this paper we compare our own continued fractions method CF [4] (which is directly based on Vincent’s theorem) with the best bisection method REL described in [3]. Experimentation with the data presented in [3] showed that, with respect to time, our continued fractions method CF is by far superior to REL, whereas the two are about equal with respect to space.
Summary. A new method is presented for the isolation of the real roots of a polynomial equation; it is based on Vincenrs forgotten theorem of 1836 and has been implemented using exact (infinite precision) integer arithmetic algorithms. A theoretical analysis of the computing time of this method is given along with some empirical results.
In 1971 using pseudo-divisions - that is, by working in Z[x] - Brown and Traub computed Euclid’s polynomial remainder sequences (prs’s) and (proper) subresultant prs’s using sylvester1, the most widely known form of Sylvester’s matrix, whose determinant defines the resultant of two polynomials. In this paper we use, for the first time in the literature, the Pell-Gordon Theorem of 1917, and sylvester2, a little known form of Sylvester’s matrix of 1853 to initially compute Sturm sequences in Z[x] without pseudodivisions - that is, by working in Q[x]. We then extend our work in Q[x] and, despite the fact that the absolute value of the determinant of sylvester2 equals the absolute value of the resultant, we construct modified subresultant prs’s, which may differ from the proper ones only in sign.
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