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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)
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 [1] 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 [2], that he invented this method, we show that what Upensky(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.