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We consider summation of consecutive values ϕ (v), ϕ ( v  + 1), ..., ϕ (w) of a meromorphic function ϕ (z) where . We assume that ϕ ( z ) satisfies a linear difference equation L ( y ) = 0 with polynomial coefficients, and that a summing operator for L exists (such an operator can be found – if it exists – by the Accurate Summation algorithm, or alternatively, by Gosper’s algorithm when ord L  = 1). The notion of bottom summation which covers the case where ϕ (z) has poles in is introduced.

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.

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.

In this paper we introduce an improved variant of the LLL algorithm. Using the Gram matrix to avoid expensive correction steps necessary in the Schnorr-Euchner algorithm and introducing the use of buffered transformations allows us to obtain a major improvement in reduction time. Unlike previous work, we are able to achieve the improvement while obtaining a strong reduction result and maintaining the stability of the reduction algorithm.

Starting from a chain contraction (a special chain homotopy equivalence) connecting a differential graded algebra A with a differential graded module M , the so-called homological perturbation technique “tensor trick” [8] provides a family of maps, { m _ i }_ i  ≥ 1, describing an A _ ∞ -algebra structure on M derived from the one of algebra on A . In this paper, taking advantage of some annihilation properties of the component morphisms of the chain contraction, we obtain a simplified version of the existing formulas of the mentioned A _ ∞ -maps, reducing the computational cost of computing m _ n from O ( n !^2) to O ( n !).

For graphs there exist highly elaborated drawing algorithms. We concentrate here in an analogous way on visualizing relations represented as Boolean matrices as, e.g., in RelView . This means rearranging the matrix appropriately, permuting rows and columns simultaneously or independently as required. In this way, many complex situations may successfully be handled in various application fields. We show how relation algebra and RelView can be combined to solve such tasks.

We introduce the concept of comprehensive triangular decomposition (CTD) for a parametric polynomial system F with coefficients in a field. In broad words, this is a finite partition of the the parameter space into regions, so that within each region the “geometry” (number of irreducible components together with their dimensions and degrees) of the algebraic variety of the specialized system F ( u ) is the same for all values u of the parameters. We propose an algorithm for computing the CTD of F . It relies on a procedure for solving the following set theoretical instance of the coprime factorization problem. Given a family of constructible sets A _1, ..., A _ s , compute a family B _1, ..., B _ t of pairwise disjoint constructible sets, such that for all 1 ≤  i  ≤  s the set A _ i writes as a union of some of the B _1, ..., B _ t . We report on an implementation of our algorithm computing CTDs, based on the RegularChains library in maple . We provide comparative benchmarks with maple implementations of related methods for solving parametric polynomial systems. Our results illustrate the good performances of our CTD code.

We investigate the stability of the modified difference scheme of Kim and Moin for numerical integration of two-dimensional incompressible Navier–Stokes equations by the Fourier method and by the method of discrete perturbations. The obtained analytic-form stability condition gives the maximum time steps allowed by stability, which are by factors from 2 to 58 higher than the steps obtained from previous empirical stability conditions. The stability criteria derived with the aid of CAS Mathematica are verified by numerical solution of two test problems one of which has a closed-form analytic solution.

The boundary problem in cylindrical coordinates for the Schrödinger equation describing a hydrogen-like atom in a strong homogeneous magnetic field is reduced to the problem for a set of the longitudinal equations in the framework of the Kantorovich method. The effective potentials of these equations are given by integrals over transversal variable of a product of transverse basis functions depending on the longitudinal variable as a parameter and their first derivatives with respect to the parameter. A symbolic-numerical algorithm for evaluating the transverse basis functions and corresponding eigenvalues which depend on the parameter, their derivatives with respect to the parameter and corresponded effective potentials is presented. The efficiency and accuracy of the algorithm and of the numerical scheme derived are confirmed by computations of eigenenergies and eigenfunctions for the low-excited states of a hydrogen atom in the strong homogeneous magnetic field.

The normal form method is widely used in the theory of nonlinear ordinary differential equations (ODEs). But in practice it is impossible to evaluate the corresponding transformations without computer algebra packages. Here we describe an algorithm for normalization of nonlinear autonomous ODEs. Some implementations of these algorithms are also discussed.