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Abstract: In this paper we describe how an idea centered on the concept of self-saturation allows several improvements in the computation of Gröbner bases via Buchberger’s Algorithm. In a nutshell, the idea is to extend the advantages of computing with homogeneous polynomials or vectors to the general case. When the input data are not homogeneous, we use as a main tool the procedure of a self-saturating Buchberger’s Algorithm. Another strictly related topic is treated later when a mathematical foundation is given to the sugar trick which is nowadays widely used in most of the implementations of Buchberger’s Algorithm. A special emphasis is also given to the case of a single grading, and subsequently some timings and indicators showing the practical merits of our approach. [Copyright &y& Elsevier]

Abstract: In this paper we present an algorithm for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Gröbner bases. We propose a general framework of Ore localizations of non-commutative -algebras and show its merits and constructiveness. It allows us to handle, among others, common operator algebras with rational coefficients. We introduce the splitting of the computation of a normal form (like the Jacobson form over simple domain) for matrices over Ore localizations into the diagonalization (the computation of a diagonal form of a matrix) and the normalization (the computation of the normal form of a diagonal matrix). These ideas are also used for the computation of the Smith normal form in the commutative case. We give a special algorithm for the normalization of a diagonal matrix over the rational Weyl algebra and present counterexamples to its idea over rational shift and -Weyl algebras. Our implementation of the algorithm in Singular:Plural relies on the fraction-free polynomial strategy, details of which will be described in the forthcoming article. It shows quite an impressive performance, compared with methods which directly use fractions. In particular, we experience quite a moderate swell of coefficients and obtain uncomplicated transformation matrices. We leave questions on the algorithmic complexity of this algorithm open, but we stress the practical applicability of the proposed method to a large class of non-commutative algebras. [Copyright &y& Elsevier]

Abstract: Proximal point algorithms (PPA) are attractive methods for monotone variational inequalities. The approximate versions of PPA are more applicable in practice. A modified approximate proximal point algorithm (APPA) presented by Solodov and Svaiter [Math. Programming, Ser. B 88 (2000) 371–389] relaxes the inexactness criterion significantly. This paper presents an extended version of Solodov–Svaiter''s APPA. Building the direction from current iterate to the new iterate obtained by Solodov–Svaiter''s APPA, the proposed method improves the profit at each iteration by choosing the optimal step length along this direction. In addition, the inexactness restriction is relaxed further. Numerical example indicates the improvement of the proposed method. [Copyright &y& Elsevier]