- Full text PDF available (16)
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)
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)
This paper is a review of results on computational methods of linear algebra over commutative domains. Methods for the following problems are examined: solution of systems of linear equations, computation of determinants, computation of adjoint and inverse matrices, computation of the characteristic polynomial of a matrix.
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.
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 , .