We present a version of the REDUCE computer algebra system as it was in the early 1970s. We show how this historical version of REDUCE may be built and run in very modest present-day environments and outline some of its capabilities.
In this work, we introduce a semi-algebraic model for automatic parallelization of perfectly nested polynomial loops, which generalizes the classical polyhedral model. This model supports the basic tasks for automatic loop parallelization, such as the representation of the nested loop, the dependence analysis, the computation of valid schedules, as well as the transformation of the loop program with a valid schedule.
This paper presents a generalised symbolic algorithm for solving systems of linear algebraic equations with multi-diagonal coefficient matrices. The algorithm is given in a pseudocode. A theorem which gives the condition for correctness of the algorithm is formulated and proven. Formula for the complexity of the multi-diagonal numerical algorithm is obtained.
David Montes de Oca Zapiain, Mitchell A. Wood, Nicholas Lubbers
et al.
AbstractAdvances in machine learning (ML) have enabled the development of interatomic potentials that promise the accuracy of first principles methods and the low-cost, parallel efficiency of empirical potentials. However, ML-based potentials struggle to achieve transferability, i.e., provide consistent accuracy across configurations that differ from those used during training. In order to realize the promise of ML-based potentials, systematic and scalable approaches to generate diverse training sets need to be developed. This work creates a diverse training set for tungsten in an automated manner using an entropy optimization approach. Subsequently, multiple polynomial and neural network potentials are trained on the entropy-optimized dataset. A corresponding set of potentials are trained on an expert-curated dataset for tungsten for comparison. The models trained to the entropy-optimized data exhibited superior transferability compared to the expert-curated models. Furthermore, the models trained to the expert-curated set exhibited a significant decrease in performance when evaluated on out-of-sample configurations.
Nicolas Lazzari, Andrea Poltronieri, Valentina Presutti
This paper describes a pattern to formalise context-free grammars in OWL and its use for sequence classification. The proposed approach is compared to existing methods in terms of computational complexity as well as pragmatic applicability, with examples in the music domain.
The report is devoted to the current state of the MathPartner computer algebra web project. We discuss the main directions of development of the project and give several examples of using it to solve selected problems.
The Naive Angle Method, used by Geometry Expressions for solving problems which involve only angle constraints, represents a geometrical configuration as a sparse linear system. Linear systems with the same underlying matrix structure underpin a number of different geometrical theorems. We use a graph theoretical approach to define a generalization of the matrix structure.
In this short article I introduce the stokes package which provides functionality for working with tensors, alternating forms, wedge products, and related concepts from the exterior calculus. Notation and spirit follow Spivak. Stokes's generalized integral theorem, viz $\int_{\partial X}φ=\int_Xdφ$, is demonstrated here using the package; it is available on CRAN athttps://CRAN.R-project.org/package=stokes.
A lot of information concerning solutions of linear differential equations can be computed directly from the equation. It is therefore natural to consider these equations as a data-structure, from which mathematical properties can be computed. A variety of algorithms has thus been designed in recent years that do not aim at "solving", but at computing with this representation. Many of these results are surveyed here.
We describe how the extension of a solver for linear differential equations by Kovacic's algorithm helps to improve a method to compute the inverse Mellin transform of holonomic sequences. The method is implemented in the computer algebra package HarmonicSums.
Sergei A. Abramov, Marko Petkovšek, Helena Zakrajšek
While Liouvillian sequences are closed under many operations, simple examples show that they are not closed under convolution, and the same goes for d'Alembertian sequences. Nevertheless, we show that d'Alembertian sequences are closed under convolution with rationally d'Alembertian sequences, and that Liouvillian sequences are closed under convolution with rationally Liouvillian sequences.
In this paper we take a look at Automatic Differentiation through the eyes of Tensor and Operational Calculus. This work is best consumed as supplementary material for learning tensor and operational calculus by those already familiar with automatic differentiation. To that purpose, we provide a simple implementation of automatic differentiation, where the steps taken are explained in the language tensor and operational calculus.
In this paper, we suggest a new efficient algorithm in order to compute S-polynomial reduction rapidly in the known algorithm for computing Grobner bases, and compare the complexity with others.
We provide bounds on the size of operators obtained by algorithms for executing D-finite closure properties. For operators of small order, we give bounds on the degree and on the height (bit-size). For higher order operators, we give degree bounds that are parameterized with respect to the order and reflect the phenomenon that higher order operators may have lower degrees (order-degree curves).
We present an algorithm which computes a cylindrical algebraic decomposition of a semialgebraic set using projection sets computed for each cell separately. Such local projection sets can be significantly smaller than the global projection set used by the Cylindrical Algebraic Decomposition (CAD) algorithm. This leads to reduction in the number of cells the algorithm needs to construct. We give an empirical comparison of our algorithm and the classical CAD algorithm.
In this paper, the author present a reliable symbolic computational algorithm for inverting a general comrade matrix by using parallel computing along with recursion. The computational cost of our algorithm is O(n^2). The algorithm is implementable to the Computer Algebra System (CAS) such as MAPLE, MATLAB and MATHEMATICA. Three examples are presented for the sake of illustration.
We introduce a majorization order on monomials. With the help of this order, we derive a necessary condition on the positive termination of a general successive difference substitution algorithm (KSDS) for an input form $f$.
This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast high-precision numerical evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our descriptions contain improvements over existing algorithms. We also provide references to relevant ideas not currently used in NumGfun.