Open Source Computer Algebra System Research Articles

Investigating difficulties and enhancing understanding in linear algebra: Leveraging SageMath and ChatGPT for (orthogonal) diagonalization and singular value decomposition.

We explored some common challenges faced by undergraduate students when studying linear algebra, particularly when dealing with algorithmic thinking skills required for topics such as matrix factorization, focusing on (orthogonal) diagonalization and singular value decomposition (SVD). To address these challenges, we introduced SageMath, a Python-based open-source computer algebra system, as a supportive tool for students performing computational tasks despite its static output nature. We further examined the potential of dynamic ChatGPT, an AI-based chatbot, by requesting examples or problem-solving assistance related to (orthogonal) diagonalization or the SVD of a specific matrix. By reinforcing essential concepts in linear algebra and enhancing computational skills through effective practice, mastering these topics can become more accessible while minimizing mistakes. Although static in nature, SageMath proved valuable for confirming calculations and handling tedious computations because of its easy-to-understand syntax and accurate solutions. However, although dynamic ChatGPT may not be fully reliable for solving linear algebra problems, the errors it produces can serve as a valuable resource for improving critical thinking skills.

Using a constraint-based expert model to provide step-level feedback for user-inputted mathematics equations

Studies across a variety of educational fields have shown the efficacy of feedback on student performance and learning. Web-based homework is a common feature of secondary and collegiate mathematics courses to provide such feedback. While web-based homework provides often instantaneous feedback to students as they complete assignments, the feedback is limited to the correctness of the final answer or to a limited number of pre-programmed common mistakes. Intelligent Tutoring Systems have the ability to provide step-by-step feedback and guidance tailored to individual students, however authoring content for such systems is generally considered a huge barrier to their wider use. This paper presents an alternative system, called SANYMS (Show and Name Your Math Steps), which combines the wide-scale usability of web-based homework with a Constraint-Based Intelligent Tutoring System Expert Model based on the open-source computer algebra system Maxima. With the integration of Maxima, SANYMS can provide step-by-step feedback to students constructing their own solutions to a wide variety of algebraic equations of their choice without the need for extensive content authoring inherent in many Intelligent Tutoring Systems.

ОПТИМІЗАЦІЯ РОЗТАШУВАННЯ ДЕМПФЕРІВ СУХОГО ТЕРТЯ ДЛЯ СЕЙСМІЧНОГО ЗАХИСТУ КАРКАСНИХ БУДІВЕЛЬ ЗА ДОПОМОГОЮ МЕТОДУ РОЮ ЧАСТИНОК

Abstract. A challenging problem of Civil Engineering is the protection of buildings against dynamic loads and earthquake impacts. The advanced solutions employ lightweight structures equipped with special damping devices. These devices can be active or passive and their application depends on the investments for the project. The active devices change their properties depending on the structural response and they are the most expensive ones. On the other hand, passive devices are essentially cheaper and, in many cases, require minimal costs of installation and maintenance. Last decades, passive friction dampers are widely used for the earthquake protection of multi-storey buildings. The friction dampers make use of the effect of solid friction to dissipate the mechanical energy and to reduce the amplitude of the vibration of the structure. The friction is developed between two solid bodies sliding in relation to one another. As usual, pairs of metal, polymer or concrete components can be utilized. Determination of the optimal location of the friction dampers inside the building presents a complicated task for the practical design. In this paper, a new approach is developed basing on the method of particle swarm optimization (PSO). The PSO method presents an artificial simulation of the phenomenon of collective intelligence, which is observed in many decentralized biological systems like ant colonies, bee swarms, flocks of birds and even social groups of human individuals. As an illustrative example, the 2D model of a ten-storey concrete frame building is considered. The purpose of the analysis is to minimize the objective function, which is the amplitude of the displacements of the top of the structure in a case of the resonance dynamic load with a frequency of the first normal mode. Non-stationary dynamic analysis is performed by the finite elements method using the program package «LIRA-SAPR» and its module «Dynamic-plus». The software implementation of the PSO procedure is developed using the open-source computer algebra system «Maxima». The optimal placement of friction dampers is determined providing the minimal displacements of the frame. The developed approach can be further extended to various problems of the optimal design of buildings and structures.

Particle swarm method for solving the problems of nonlinear optimization

Purpose. The optimal design of buildings and structures requires solving a number of important tasks related to determining the best topology and geometric shape of structures, physical properties of elements, connections of elements among themselves, etc. In this case, it is necessary to take into account the influence of many factors: the distribution of static and dynamic loads, corrosive effects, the nature of the operating conditions, the requirements for the reliability and durability of the unit. The complexity of solving such problems is connected with the fact that, as a rule, the objective functions of such problems are nonlinear.Methods. Various approaches are used to solve nonlinear optimization problems: stochastic search (Monte Carlo method); search methods in which the step is successively reduced, following to the given relation (halving, golden ratio, inverse Fibonacci numbers); gradient descent method; evolutionary algorithms; penalty function method and others. In recent years, a new class of methods of numerical optimization has been intensively developed, in various works it is called social-behavioral, population, or swarm. For practical verification of the particle swarm method, the finding of the extrema of the test functions of Rosenbrock and Rastrigin is considered. Results. A new software implementation of one of the methods of artificial collective intelligence, the particle swarm method, is proposed for solving nonlinear optimization problems in the Maxima open-source computer algebra system. The high computational efficiency of this method for finding global extrema of “ravine” and multimodal functions in those cases where the application of many classical algorithms can be difficult is shown. Scientific novelty. Compared to classical methods, swarm intelligence methods are especially effective for finding extrema of nonlinear multimodal functions, as well as for solving high-dimensional problems. Practical significance. The effect of the method parameters (particle number and weight coefficients) on the rate of practical convergence is investigated. The developed method can be used, inter alia, to solve the problems of optimal design of building structures, buildings and structures.

Symbolic and numerical analysis in general relativity with open source computer algebra systems

We study three computer algebra systems, namely SageMath (with SageManifolds package), Maxima (with ctensor package) and Python language (with GraviPy module), which allow tensor manipulation for general relativity calculations along with general algebraic calculations. We present a benchmark of these systems using simple examples. After the general analysis, we focus on the SageMath and SageManifolds system to derive, analyze and visualize the solutions of the massless Klein-Gordon equation and geodesic motion with Hamilton-Jacobi formalism. We compare our numerical result of the Klein-Gordon equation with the asymptotic form of the analytical solution to see that they agree.

Maximal subsemigroups of finite transformation and diagram monoids

We describe and count the maximal subsemigroups of many well-known transformation monoids, and diagram monoids, using a new unified framework that allows the treatment of several classes of monoids simultaneously. The problem of determining the maximal subsemigroups of a finite monoid of transformations has been extensively studied in the literature. To our knowledge, every existing result in the literature is a special case of the approach we present. In particular, our technique can be used to determine the maximal subsemigroups of the full spectrum of monoids of order- or orientation-preserving transformations and partial permutations considered by I. Dimitrova, V. H. Fernandes, and co-authors. We only present details for the transformation monoids whose maximal subsemigroups were not previously known; and for certain diagram monoids, such as the partition, Brauer, Jones, and Motzkin monoids.The technique we present is based on a specialised version of an algorithm for determining the maximal subsemigroups of any finite semigroup, developed by the third and fourth authors, and available in the Semigroups package for GAP, an open source computer algebra system. This allows us to concisely present the descriptions of the maximal subsemigroups, and to clearly see their common features.

SymPy: symbolic computing in Python

SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.

Do genome‐scale models need exact solvers or clearer standards?

Constraint‐based analysis of genome‐scale models (GEMs) arose shortly after the first genome sequences became available. As numerous reviews of the field show, this approach and methodology has proven to be successful in studying a wide range of biological phenomena (McCloskey et al, 2013; Bordbar et al, 2014). However, efforts to expand the user base are impeded by hurdles in correctly formulating these problems to obtain numerical solutions. In particular, in a study entitled “An exact arithmetic toolbox for a consistent and reproducible structural analysis of metabolic network models” (Chindelevitch et al, 2014), the authors apply an exact solver to 88 genome‐scale constraint‐based models of metabolism. The authors claim that COBRA calculations (Orth et al, 2010) are inconsistent with their results and that many published and actively used (Lee et al, 2007; McCloskey et al, 2013) genome‐scale models do support cellular growth in existing studies only because of numerical errors. They base these broad claims on two observations: (i) three reconstructions (iAF1260, iIT341, and iNJ661) compute feasibly in COBRA, but are infeasible when exact numerical algorithms are used by their software (entitled MONGOOSE); (ii) linear programs generated by MONGOOSE for iIT341 were submitted to the NEOS Server (a Web site that runs linear programs through various solvers) and gave inconsistent results. They further claim that a large percentage of these COBRA models are actually unable to produce biomass flux. Here, we demonstrate that the claims made by Chindelevitch et al (2014) stem from an incorrect parsing of models from files rather than actual problems with numerical error or COBRA computations.

More Power through Symbolic Computation: Extending Stata by using the Maxima Computer algebra system

Maxima is a free and open-source computer algebra system that can perform symbolic computations such as solving equations, determining derivatives of functions, obtaining Taylor series, and manipulating algebraic expressions. In this article, I present the Maxima Bridge System, which is a collection of software programs that allows Stata to interface with Maxima so that Maxima can be used for symbolic computation to transfer data from Stata to Maxima and to retrieve results from Maxima. The cooperation between Stata and Maxima provides an environment for statistical analysis in which symbolic computation can be easily used together with all the facilities supplied by Stata. In this environment, symbolic computation algorithms can be used to manage the complexity of algebra and calculus, whereas numerical computation can be used when speed matters. I also discuss the software architecture of the Maxima Bridge System, and I present several examples to illustrate how to develop new Stata commands that exploit the capabilities of Maxima.

Code optimization in FORM

We describe the implementation of output code optimization in the open source computer algebra system Form. This implementation is based on recently discovered techniques of Monte Carlo tree search to find efficient multivariate Horner schemes, in combination with other optimization algorithms, such as common subexpression elimination. For systems for which no specific knowledge is provided it performs significantly better than other methods we could compare with. Because the method has a number of free parameters, we also show some methods by which to tune them to different types of problems.

Recent developments on FORM

We report on recent developments on the open source computer algebra system Form. Especially we focus on the code optimization implemented after the release of Form version 4.0 in March 2012.

Using an Open-Source Computer Algebra System to Teach Economics

Computer Algebra Systems can be combined with conventional teaching methods to provide students a more hands-on understanding of economic concepts. Maxima and its interface wxMaxima are open source (and therefore zero-price) alternatives to expensive programs like Maple and Mathematica. This paper argues for the benefits of using wxMaxima, and presents several examples for instructors to use in classes. Many of these examples are from Hodgin (1999), but we provide additional examples, and enhance some of Hodgin’s examples by adding animation. We also provide several simple exercises for students to try, and explain why these exercises can deepen students' understanding.

Automatic deduction in (dynamic) geometry: Loci computation

A symbolic tool based on open source software that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction is presented. The prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction, namely with the open source dynamic geometry system GeoGebra or using the common file format for dynamic geometry developed by the Intergeo project. Locus computation algorithms based on Automatic Deduction techniques are recalled and presented as basic for an efficient treatment of advanced methods in dynamic geometry. Moreover, an algorithm to eliminate extraneous parts in symbolically computed loci is discussed. The algorithm, based on a recent work on the Gröbner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Several examples are shown in detail.

A computer algebra system for analysis and control of nonlinear time-delay systems

An open-source computer algebra system toolbox devoted to the analysis and synthesis for a wide class of nonlinear time-delay systems is introduced. This contribution provides a practical way to carry out all the computations used to characterize several properties of the systems under consideration, which involve elements in a non-commutative ring of polynomials, and an extended version of the Lie brackets. The package usage will be illustrated with some examples.

Adapted hyperbolic polygons and symplectic representations for group actions on Riemann surfaces

We prove that given a finite group G together with a set of fixed geometric generators, there is a family of special hyperbolic polygons that uniformize the Riemann surfaces admitting the action of G with the given geometric generators. From these special polygons, we obtain geometric information for the action: a basis for the homology group of surfaces, its intersection matrix, and the action of the given generators of G on this basis. We then use the Frobenius algorithm to obtain a symplectic representation G of G corresponding to this action. The fixed point set of G in the Siegel upper half-space corresponds to a component of the singular locus of the moduli space of principally polarized abelian varieties.We also describe an implementation of the algorithm using the open source computer algebra system SAGE.

Using a free open source software to teach mathematics

ABSTRACTWe present the experience of the authors teaching mathematics to freshmen engineering students with the help of the open source computer algebra system Sage. We describe some teaching resources and present an ad hoc distribution of Sage used by the authors.© 2012 Wiley Periodicals, Inc. Comput Appl Eng Educ 22:728–735, 2014; View this article online at wileyonlinelibrary.com/journal/cae; DOI 10.1002/cae.21565

Automatic Deduction in Dynamic Geometry using Sage

We present a symbolic tool that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction. The main prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction. In one worksheet, diagrams constructed with the open source dynamic geometry system GeoGebra are accepted. In this worksheet, Groebner bases are used to either compute the equation of a geometric locus in the case of a locus construction or to determine the truth of a general geometric statement included in the GeoGebra construction as a boolean variable. In the second worksheet, locus constructions coded using the common file format for dynamic geometry developed by the Intergeo project are accepted for computation. The prototype and several examples are provided for testing. Moreover, a third Sage worksheet is presented in which a novel algorithm to eliminate extraneous parts in symbolically computed loci has been implemented. The algorithm, based on a recent work on the Groebner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Detailed examples are discussed.

Open source computer algebra systems

This survey will look at SymPy, a free and open source computer algebra system started in 2005 by the second author (O.Č.). It is written entirely in Python, available from http://sympy.org. SymPy is licensed under the "modified BSD" license, as is its beautiful logo designed by Fredrik Johansson.

A flexible, extensible online testing system for mathematics

<span>An online testing system developed for entry-skills testing of first-year university students in algebra and calculus is described. The system combines the open-source computer algebra system </span><em>Maxima</em><span> with computer scripts to parse student answers, which are entered using standard mathematical notation and conventions. The answers can involve structures such as: lists, variable-precision-floating-point numbers, integers and algebra. This flexibility allows more sophisticated testing designs than the multiple choice, or exact match, paradigms common in other systems, and the implementation is entirely vendor neutral. Experience using the system and ideas for further development are discussed.</span>

An implementation of the method of brackets for symbolic integration

In spite of being a classical problem, the current techniques available for Symbolic Integration are not sufficient to evaluate a variety of integrals coming from Mathematical Physics, such as Bessel functions. The Method of Brackets [2, 3], a heuristic process appearing in the evaluation of Feynman diagrams, can be used to evaluate symbolically a large class of single or multiple integrals. It represents an extension of the so-called Ramanujan Master Theorem [1]. The first implementation of the Method of Brackets has been written by the author in the open-source computer algebra system Sage. This implementation allows experimentation with representations of the integrand, which can affect output and efficiency. An algorithm that chooses the best representation of the integrand is being developed.