Alkiviadis G. Akritas

Various proofs of Sylvester's (determinant) identity

Despite the fact that the importance of Sylvesters determinant identity has been recognized in the past, we were able to find only one proof of it in English (Bareiss, 1968), with reference to some others. (Recall that Sylvester (1857) stated this theorem without proof.) Having used this identity, recently, in the validity proof of our new, improved, matrix-triangularization subresultant polynomial remainder sequence method (Akritas et al., 1995), we decided to collect all the proofs we found of this identity-one in English, four in German and two in Russian, in that order-in a single paper (Akritas et al., 1992). It turns out that the proof in English is identical to an earlier one in German. Due to space limitations two proofs are omitted.

An implementation of Vincent's theorem

SummaryA new method is presented for the isolation of the real roots of a polynomial equation; it is based on Vincents 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.

Applications of singular-value decomposition (SVD)

Let A be an m×n matrix with m≥n. Then one form of the singular-value decomposition of A is A=UTΣV,where U and V are orthogonal and Σ is square diagonal. That is, UUT=Irank(A), VVT=Irank(A), U is rank(A)×m, V is rank(A)×n and Σ=σ10⋯000σ2⋯00⋮⋮⋱⋮⋮00⋯σrank(A)−1000⋯0σrank(A)is a rank(A)×rank(A) diagonal matrix. In addition σ1≥σ2≥⋯≥σrank(A)>0. The σi’s are called the singular values of A and their number is equal to the rank of A. The ratio σ1/σrank(A) can be regarded as a condition number of the matrix A.

A new method for computing polynomial greatest common divisors and polynomial remainder sequences

SummaryA new method is presented for the computation of a greatest common divisor (gcd) of two polynomials, along with their polynomial remainder sequence (prs). This method is based on our generalization of a theorem by Van Vleck [12] and uniformly treats both normal and abnormal prss, making use of Bareisss [3] integer-preserving transformation algorithm for Gaussian elimination. Moreover, for the polynomials of the prss, this method provides the smallest coefficients that can be expected without coefficient ged computations (as in Bareiss [3]) and it clearly demonstrates the divisibility properties; hence, it combines the best of both the reduced and the subresultant prs algorithms.

There is no “Uspensky's method.”

In this paper an attempt is made to correct the misconception of several authors [1] that there exists a method by Upensky (based on Vincents theorem) for the isolation of the real roots of a polynomial equation with rational coefficients. Despite Uspenskys claim, in the preface of his book [2], that he invented this method, we show that what Upensky actually did was to take Vincents method and double its computing time. Upensky must not have understood Vincents method probably because he was not aware of Budans theorem [3]. In view of the above, it is historically incorrect to attribute Vincents method to Upensky.

Advances on the continued fractions method using better estimations of positive root bounds

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, or any of its variants.

Exact algorithms for the implementation of cauchy's rule

Cauchys little known rule for computing a lower (or upper) bound on the values of the positive roots of a polynomial equation has proven to be of great importance; namely it constitutes an indispensable and crucial part of the fastest method existing for the isolation of the real roots of an equation, a method which was recently developed by the author of this article. In this paper efficient, exact (infinite precision) algorithms, along with their computing time analysis, are presented for the implementation of this important rule.

Matrix computation of subresultant polynomial remainder sequences in integral domains

We present an improved variant of the matrix-triangularization subresultant prs method [1] for the computation of a greatest common divisor of two polynomialsA andB (of degreesm andn, respectively) along with their polynomial remainder sequence. It is improved in the sense that we obtain complete theoretical results, independent of Van Vleck’s theorem [13] (which is not always true [2, 6]), and, instead of transforming a matrix of order 2·max(m, n) [1], we are now transforming a matrix of orderm+n. An example is also included to clarify the concepts.AbstractПредставлен улучшенный вариант матрично-грианіулярннационноіо субрезультантного метода пояиномиальных носледовательностей остатков (ППО) [1] для вычисления наибольшего общеіо дедителя двух мноточленовA иB (стеиенейm иn соответственно) с одновременным нахождением нх ПОП. Улучшение заключается в том, что нолучены законченные теоретические результаты, независимые от теоремы Ван Влека [13] (которая нэ всегда снраведлива, см [2, 6]). Кроме того, вместо преобразования матрицы норядка 2 · мах (m, n) [1] тенерь нреобразуется матрица норядкаm+n. Представлен численный нример для иллюстрации этих ноложений.

Exact algorithms for the matrix-triangularization subresultant PRS method

In [2] a new method is presented for the computation of a greatest common divisor (gcd) of two polynomials, along with their polynomial remainder sequence (prs). This method is based on our generalization of a theorem by Van Vleck (1899)[12] and uniformly treats both normal and abnormal prs’s, making use of Bareiss’s (1968)[4] integer-preserving transformation algorithm for Gaussian elimination; moreover, for the polynomials of the prs’s, this method provides the smallest coefficients that can be expected without coefficient gcd computations. In this paper we present efficient, exact algorithms for the implementation of this new method, along with an example where bubble pivot is needed.

A simple proof of tha validity of the reduced prs algorithm

Given two univariate polynomials with integer coefficients, it has beenrediscovered [2] that the reduced polynomial remainder sequence (prs) algorithm can be used mainly to compute over the integers the members of anormal prs, keeping under control the coefficient growth and avoiding greatest common divisor (ged) computations of the coefficients. The validity proof of this algorithm as presented in the current literature [2] is very involved and has obscured simple divisibility properties. In this note, we present Sylvesters theorem of 1853 [4] which makes these simple divisibility properties clear for normal prss. The proof presented here is a modification of Sylvesters original proof.ZusammenfassungFür zwei gegebene Polynome in einer Variablen und mit ganzzahligen Koeffizienten wurdewiederentdeckt [2], baß der reduzierte prs-Algorithmus hauptsächlich verwendet werden kann, um die Elemente einernormalen prs mit ganzzahligen Operationen zu berechnen, wobei das Anwachsen der Koeffizienten unter Kontrolle gehalten und vermieden wird, Berechnungen vom größten gemeinsamen Teiler der Koeffizienten durchzuführen. Der Beweis für diesen Algorithmus, wie er in der heutigen Literatur [2] präsentiert wird, ist sehr kompliziert und hat einfache Divisionseigenschaften verborgen. In dieser Mitteilung wird das Sylvestertheorem von 1853, welches diese einfache Divisionseigenschaften für normale prs klar macht, dargestellt. Der Beweis, der hier präsentiert wird, ist eine Modifikation von Sylvesters ursprünglichem Beweis.