Algpath is a certified homotopy continuation software. We upgrade the previous fixed-precision Rust implementation by incorporating mixed, adaptive precision with minimal overhead. This allows us to tackle problems on which the initial implementation fails due to the inability to increase precision, and where uncertified methods may fail or path jump.
When solving systems of polynomial equations and inequalities, the task of computing their solutions with integer coordinates is a much harder problem than that of computing their real solutions or that of computing all their solutions. In fact, in the presence of non-linear constraints, this task may simply become an undecidable problem [12, 15]. However, studying the integer solutions of linear systems of equations and inequalities is of practical importance in various areas of scientific computing. Two such areas are combinatorial optimization (in particular, integer linear programming) and compiler optimization (in particular, the analysis, transformation, and scheduling of nested loops in computer programs), where a variety of algorithms solve questions related to the points with integer coordinates in a given polyhedron. Another area is at the crossroads of computer algebra and polyhedral geometry, with topics such as toric ideals and Hilbert bases, see [16], as well as the manipulation of Laurent series, see [1].
An in-progress pipeline based on B-series is proposed to solve reachability analysis of ordinary differential equations. First, a symbolic step generates a B-series and its associated truncation error. This later is symbolically optimized in order to reduce the over-estimation introduced by computation with sets. Finally, the B-series is evaluated with affine arithmetic and intervals to produce a guaranteed reachable tube.
Transcendental Liouvillian extensions are differential fields in which one can model polylogarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension, we construct a complement, over the subfield of constants, to the subspace of derivatives and develop an algorithm that decomposes any element of the field into the sum of a derivative and a component in this complement. Consequently, an element is a derivative if and only if its complementary component vanishes. Moreover, the algorithm enables us to determine elementary integrability over the extension by computing parametric logarithmic parts. It also yields a reduction-based approach to constructing telescopers for elements of the extension, provided that an a priori order bound is available.
S -packing coloring is a generalization of proper coloring of graphs, introduced more than a decade ago. In this paper, we present algebraic and combinatorial algorithms for the problem of S -packing coloring of finite undirected and unweighted graphs. We assess the upper bounds for the complexity of our algorithms and of an existing algebraic algorithm by Maarouf [Electron. J. Combin. 2017]. We also provide a comparative evaluation of these algorithms.
