Computer Algebra Research Articles

Delivery of Integral Calculus at Maseno University: Is STACK Really Playing an Integral Part?

Abstract—Integral Calculus II is a second-year undergraduate course offered by the Department of Pure and Applied Mathematics at Maseno University. It is one of the 20 foundational courses with a high enrolment of about 800 students drawn from the school of Mathematics, Education, Science, and Business. Because of the large number of enrollments in these foundational courses, and the small number of teaching staff, instructors face challenges in providing immediate personalized feedback that can guide learning. Systems for Teaching and Assessment which use Computer Algebra Kernerl, STACK, a computer-aided assessment plug-in that provides a sophisticated assessment in mathematics-related disciplines, was used to deploy Continuous Assessment Tests (CAT) in the course for 370 students. This paper reports on the findings of the course based on a comparison between student test scores in STACK and in the final course exam.

Evaluating Engagement and Learning Based on a Student Categorization using STACK, Exam Data, Key Informant Interviews, and Focus Group Discussions

Abstract—Systems for Teaching and Assessment using Computer Algebra Kernel (STACK) is a computer-aided assessment plug-in for the Moodle learning management system that provides sophisticated tools for student assessment in mathematics and related disciplines, with emphasis on formative assessment. In the last four years, IDEMS international has supported the School of Mathematics at Maseno to integrate STACK and use it in the teaching, learning, and assessment of undergraduate students in nine courses. One of the courses was “Introduction to Complex Analysis'', a third-year course shared by students taking mathematics-related programs from different faculties within Maseno. This paper reports on an evaluation of learner behavior in the Complex Analysis course using data from the STACK weekly quizzes done in that course, the final exam, 20 key informant interviews, and 4 focus group discussions.

Prospective mathematics teachers learning complex numbers using technology

In this paper we studied the personal mathematical work of 14 students in initial teacher training, in their first year of studies at a public university in Chile, based on two tasks on complex numbers. These tasks were set in a Computer Aided Assessment System (CAA) based on Moodle and Wiris and it was proposed to solve them using different Computer Algebra Systems (CAS), such as GeoGebra, Symbolab, Photomath and Wolfram Alpha among others. Difficulties and potentialities were observed in the mathematical work of the students. In the difficulties, it was observed that the students had problems interpreting the information from the CAS and the CAA feedback because their previous mathematical knowledge was not solid enough to do so. In the potentialities, it was observed that different characteristics of the task, together with the articulation of two or more artifacts allowed students to give meaning to the mathematical objects involved.

Is Computer Algebra Ready for Conjecturing and Proving Geometric Inequalities in the Classroom?

Is Computer Algebra Ready for Conjecturing and Proving Geometric Inequalities in the Classroom?

Evolution equations of multi-planet systems with variable masses

In celestial mechanics and astrodynamics, the study  of the  dynamical evolution of exoplanetary systems is the   relevant topics.    For today more than 3,000 exoplanetary systems are known. In this paper, we study the dynamic evolution of extrasolar systems, when the leading factor of evolution  is the variability of the masses of gravitating bodies. The problem of spherically symmetric bodies with  variable masses is considered in a relative coordinate system, this bodies inter-gravitating according to Newton's law. The quasi-elliptical motions of planets whose orbits do not intersect during evolution are investigated. It is believed that the mass of bodies under consideration  varies isotropically by various known laws with different velocities. The mass of the parent star is considered to be the most massive than its planets and the origin of the relative coordinate system is in the center of the parent star.. Due to the variability of the masses, the differential equations of motion become non-autonomous and the task is difficult. The problem is investigated by methods of perturbation theory. The canonical perturbation theory based on a periodic motion over a quasi-canonical section is used.  Canonical equations of motion are obtained in analogues of the second Poincare system, which are effective in the case when the analogues of eccentricities and the analogues of the inclination of the orbital plane of planets are sufficiently small.  The secular perturbations of the planets, which determine the behavior of the orbital parameters over long time intervals, are studied. The evolutionary equations of many planetary systems with isotropically varying masses in analogues of the second system of Poincare variables are derived in an analytical form which are obtained using the Wolfram Mathematica computer algebra system. This takes into account the effects of the decreasing mass of the parent star and the growth of the masses of the planets due to the accretion of matter from the remnants of the protoplanetary disk. For the three-planetary problem of four bodies with variable masses, the evolutionary equations in dimensionless variables are obtained explicitly. In the future, these results will be used to study the dynamics of the three-planet system K2-3 in the non-stationary stage of its evolution.

A symbolic framework to obtain mid-fidelity models of flexible multibody systems with application to horizontal-axis wind turbines

Abstract. The article presents a symbolic framework (also called computer algebra program) that is used to obtain, in symbolic mathematical form, the linear and nonlinear equations of motion of a mid-fidelity multibody system including rigid and flexible bodies. Our approach is based on Kane's method and a nonlinear shape function representation for flexible bodies. The shape function approach does not represent the state of the art for flexible multibody dynamics but is an effective trade-off to obtain mid-fidelity models with few degrees of freedom, taking advantage of the separation of space and time. The method yields compact symbolic equations of motion with implicit account of the constraints. The general and automatic framework facilitates the creation and manipulation of models with various levels of complexity by adding or removing degrees of freedom. The symbolic treatment allows for analytical gradients and linearized equations of motion. The linear and nonlinear equations can be exported to Python code or dedicated software. There are multiple applications, such as time domain simulation, stability analyses, frequency domain analyses, advanced controller design, state observers, and digital twins. In this article, we describe the method we used to systematically generate the equations of motion of multibody systems and present the implementation of the framework using the Python package SymPy. We apply the framework to generate illustrative land-based and offshore wind turbine models. We compare our results with OpenFAST simulations and discuss the advantages and limitations of the method. The Python implementation is provided as an open-source project.

VCSR: An Efficient GPU Memory-Aware Sparse Format

The Sparse Matrix-Vector Multiplication (SpMV) kernel is used in a broad class of linear algebra computations. SpMV computations result in a performance bottleneck in many high performance applications, so optimizing SpMV performance is paramount. While implementing this kernel on a GPU can potentially boost performance significantly, current GPU libraries either provide modest performance gains or are burdened with high sparse format conversion overhead. In this paper we introduce the Vertical Compressed Sparse Row (VCSR) format, a novel memory-aware format that out-performs previous proposed formats on a GPU. We first motivate the design of our baseline VCSR format and then step through a series of enhancements that further improve VCSR's memory efficiency (VCSR-MEM) and performance (VCSR-INTRLV), while also considering conversion overhead. VCSR attempts to produce a high degree of thread-level parallelism and memory utilization by exploiting knowledge of GPU memory microarchitecture. VCSR can reduce the number of global memory transactions significantly, an issue not addressed by most other sparse formats. In addition, VCSR provides a novel reordering mechanism. It minimizes the size of the compressed matrix, handles both regular/irregular sparse matrices, and can be customized based on matrix size. VCSR also minimizes conversion overhead, as compared to full or partial row reordering. Our methodology is highly configurable and can be optimized for any sparse matrix. We have evaluated the VCSR format for the SpMV kernel when run on two different NVIDIA GPUs, the Kepler K40 and the Volta V100. We compare VCSR with NVIDIA's cuSPARSE library (the HYB format), a state-of-the-art sparse library. We also compare against other state-of-the-art CSR-based formats, including CSR5, merge-base SpMV and HOLA. We evaluate the benefits of VCSR over the entire University of Florida's SuiteSparse dataset collection. The VCSR-baseline format achieves an average speedup ranging from <inline-formula><tex-math notation="LaTeX">$1.10\times$</tex-math></inline-formula> to <inline-formula><tex-math notation="LaTeX">$1.39\times$</tex-math></inline-formula> when compared to the performance of the four state-of-the-art formats on an NVIDIA V100. While the VCSR-MEM format can save a significant amount of memory space, it is a bit slower than our VCSR-baseline. VCSR-INTRLV performs much better than the VCSR-baseline, and even when including the conversion overhead, achieves an average speedup of <inline-formula><tex-math notation="LaTeX">$1.08\times$</tex-math></inline-formula> as compared to HOLA (the best performing format among the prior schemes).

Study of Bulk Properties Relation to SARA Composition Data of Various Vacuum Residues Employing Intercriteria Analysis

Twenty-two straight run vacuum residues extracted from extra light, light, medium, heavy, and extra heavy crude oils and nine different hydrocracked vacuum residues were characterized for their bulk properties and SARA composition using four and eight fractions (SAR-ADTM) methods. Intercriteria analysis was employed to determine the statistically meaningful relations between the SARA composition data and the bulk properties. The determined strong relations were modeled using the computer algebra system Maple and NLPSolve with the Modified Newton Iterative Method. It was found that the SAR-ADTM saturates, and the sum of the contents of saturates and ARO-1 can be predicted from vacuum residue density, while the SAR-ADTM asphaltene fraction content, and the sum of asphaltenes, and resins contents correlate with the softening point of the straight run vacuum residues. The softening point of hydrocracked vacuum residues was found to strongly negatively correlates with SAR-ADTM Aro-1 fraction, and strongly positively correlates with SAR-ADTM Aro-3 fraction. While in the straight run vacuum residues, the softening point is controlled by the content of SAR-ADTM asphaltene fraction in the H-Oil hydrocracked vacuum residues, the softening point is controlled by the content of SAR-ADTM Aro-3 fraction content. During high severity H-Oil operation, resulting in higher conversion, hydrocracked vacuum residue with higher SAR-ADTM Aro-3 fraction content is obtained, which makes it harder and more brittle.

Sympy2c: From symbolic expressions to fast C/C++ functions and ODE solvers in Python

Computer algebra systems play an important role in science as they facilitate the development of new theoretical models. The resulting symbolic equations are often implemented in a compiled programming language in order to provide fast and portable codes for practical applications. We describe sympy2c, a new Python package designed to bridge the gap between the symbolic development and the numerical implementation of a theoretical model. sympy2c translates symbolic equations implemented in the SymPy Python package to C/C++ code that is optimized using symbolic transformations. The resulting functions can be conveniently used as an extension module in Python. sympy2c is used within the PyCosmo Python package to solve the Einstein–Boltzmann equations, a large system of ODEs describing the evolution of linear perturbations in the Universe. After reviewing the functionalities and usage of sympy2c, we describe its implementation and optimization strategies. This includes, in particular, a novel approach to generate optimized ODE solvers making use of the sparsity of the symbolic Jacobian matrix. We demonstrate its performance using the Einstein–Boltzmann equations as a test case. sympy2c is general and potentially useful for various areas of computational physics. sympy2c is publicly available at https://cosmology.ethz.ch/research/software-lab/sympy2c.html.

Computing maximum likelihood estimates for Gaussian graphical models with Macaulay2

We introduce the package "GraphicalModelsMLE" for computing the maximum likelihood estimates (MLEs) of a Gaussian graphical model in the computer algebra system Macaulay2. This package allows the computation of MLEs for the class of loopless mixed graphs. Additional functionality allows the user to explore the underlying algebraic structure of the model, such as its maximum likelihood degree and the ideal of score equations.

The Weierstrass–Durand–Kerner root finder is not generally convergent

Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the root-finding method of Weierstrass, also known as the Durand–Kerner-method: this is a root finder that tries to approximate all roots of a given polynomial in parallel. This method has been introduced 130 years ago and has since then a good reputation for finding all roots in practice except in obvious cases of symmetry. Nonetheless, very little is known about its global dynamics and convergence properties. We show that the Weierstrass method, like the well-known Newton method, is not generally convergent: there are open sets of polynomials p p of every degree d ≥ 3 d \ge 3 such that the dynamics of the Weierstrass method applied to p p exhibits attracting periodic orbits. Specifically, all polynomials sufficiently close to Z 3 + Z + 175 Z^3 + Z + 175 have attracting cycles of period 4 4 . Here, period 4 4 is minimal: we show that for cubic polynomials, there are no periodic orbits of length 2 2 or 3 3 that attract open sets of starting points. We also establish another convergence problem for the Weierstrass method: for almost every polynomial of degree d ≥ 3 d\ge 3 there are orbits that are defined for all iterates but converge to ∞ \infty ; this is a problem that does not occur for Newton’s method. Our results are obtained by first interpreting the original problem coming from numerical mathematics in terms of higher-dimensional complex dynamics, then phrasing the question in algebraic terms in such a way that we could finally answer it by applying methods from computer algebra. The main innovation here is the translation into an algebraic question, which is amenable to (exact) computational methods close to the limits of current computer algebra systems.

Limiting Dynamics for Q-Learning with Memory One in Symmetric Two-Player, Two-Action Games

We develop a method based on computer algebra systems to represent the mutual pure strategy best-response dynamics of symmetric two-player, two-action repeated games played by players with a one-period memory. We apply this method to the iterated prisoner’s dilemma, stag hunt, and hawk-dove games and identify all possible equilibrium strategy pairs and the conditions for their existence. The only equilibrium strategy pair that is possible in all three games is the win-stay, lose-shift strategy. Lastly, we show that the mutual best-response dynamics are realized by a sample batch Q-learning algorithm in the infinite batch size limit.

Development of Student Worksheets Using Problem Based Learning Models on Algebraic Computing Operation Materials

The formulation of the problem in this study are: How is the LKPD Development using a problem-based learning learning model on the Algebraic Computing Operation material? How is the validity of the Student Worksheet with the problem-based learning model developed as a learning medium for students? How are students' responses to the Worksheet Students by using a problem-based learning model? This study aims to produce a Student Worksheet product using a Problem Based Learning model on valid and feasible algebraic arithmetic operations. This research is a type of development research and uses 4-D (four-D) research procedures which are modified into 3-D (three-D), namely the definition stage, the design stage, and the development stage. The results of this study: 1) Researchers have produced Student Worksheets using the Problem Based Learning model on algebraic arithmetic operations on the subject of addition, subtraction, multiplication, and division in algebraic form. 2) the results of the assessment of the Student Worksheets from material experts are 94.3% with a very valid validity level and the assessment of the Student Worksheets from media experts is 81.6% with a fairly valid validity level. 3) obtaining the results of the assessment through a student questionnaire with an average percentage of 88.25% with a very positive category. From the results of the assessment of material experts, media experts, and student responses, the developed Student Worksheets are valid and suitable to be used as learning media by students.

The Laplace Transform of Composed Functions and Bivariate Bell Polynomials

The problem of computing the Laplace transform of composed functions has not found its way into the literature because it was customarily believed that there were no suitable formula to solve it. Actually, it has been shown in previous work that by making use of Bell polynomials, efficient approximations can be found. Moreover, using an extension of Bell’s polynomials to bivariate functions, it is also possible to approximate the Laplace transform of composed functions of two variables. This topic is solved in this paper and some numerical verifications, due to the first author using the computer algebra system Mathematica©, are given proving the effectiveness of the proposed method.

Simultaneous Rational Function Reconstruction with errors: Handling multiplicities and poles

In this paper, we focus on extensions of methods for interpolating rational functions from their evaluations, in the context of erroneous evaluations. This problem can be seen both from a computer algebra and a coding theory point of view. In computer algebra, this is a generalization of Simultaneous Rational Function Reconstruction with errors and multiprecision evaluations. From an error correcting codes point of view, this problem is related to the decoding of some algebraic codes such as Reed-Solomon, Derivatives or Multiplicity codes. We give conditions on the inputs of the problem which guarantee the uniqueness of the interpolant.Since we deal with rational functions, some evaluation points may be poles: a first contribution of this work is to correct any error in a scenario with poles and multiplicities that extends (Kaltofen et al., 2020). Our second contribution is to adapt rational function reconstruction for random errors, and provide better conditions for uniqueness using interleaving techniques as in (Guerrini et al., 2021).

Noncommutative Bispectral Algebras and Their Presentations

We prove a general result on presentations of finitely generated algebras and apply it to obtain nice presentations for some noncommutative algebras arising in the matrix bispectral problem. By “nice presentation”, we mean a presentation that has as few as possible defining relations. This, in turn, has potential applications in computer algebra implementations and examples. Our results can be divided into three parts. In the first two, we consider bispectral algebras with the eigenvalue in the physical equation to be scalar-valued for 2×2 and 3×3 matrix-valued eigenfunctions. In the third part, we assume the eigenvalue in the physical equation to be matrix-valued and draw an important connection with Spin Calogero–Moser systems. In all cases, we show that these algebras are finitely presented. As a byproduct, we answer positively a conjecture of F. A. Grünbaum about these algebras.

Mathematical modeling of typical electrical circuits, oscillators and pendulums

The work is devoted to mathematical and simulation modeling of electromechanical systems, in which oscillations in electrical circuits, oscillators and pendulums are studied under the action of forces of various physical nature. To compile the equations of electromechanical systems, the apparatus of analytical mechanics was used, in which the electromagnetic and mechanical quantities characterizing the system appear as formally equal. The equations of motion are based on the Lagrange-Maxwell equations. The simulation was performed using computer algebra programs.

Using Semicontinuity for Standard Bases Computations

We present new results on standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring over a computable field K. We prove the semicontinuity of the “highest corner” in a family of ideals, parametrized by the spectrum of a Noetherian domain A. This semicontinuity is used to design a new modular algorithm for computing a standard basis of I if K is the quotient field of A. It uses the computation over the residue field of a “good” prime ideal of A to truncate high order terms in the subsequent computation over K. We prove that almost all prime ideals are good, so a random choice is very likely to be good, and whether it is good is detected a posteriori by the algorithm. The algorithm yields a significant speed advantage over the non-modular version and works for arbitrary Noetherian domains. The most important special cases are perhaps A={mathbb {Z}} and A=k[t], k any field and t a set of parameters. Besides its generality, the method differs substantially from previously known modular algorithms for A={mathbb {Z}}, since it does not manipulate the coefficients. It is also usually faster and can be combined with other modular methods for computations in local rings. The algorithm is implemented in the computer algebra system Singular and we present several examples illustrating its power.

Measurement of multilayered conductive discs using eddy current method

This paper presents an analytical solution that was derived for eddy current testing of multilayer discs. The final formulas, obtained by using the truncated region eigenfunction expansion (TREE) method, have been presented in a closed form, therefore they can be implemented in any programming language or computer algebra system. The developed solution makes it possible to model discs of any thickness and diameter, and its form is independent of the parameters of the employed sensor. The calculation results obtained for three-layer discs made of brass, copper and bronze were verified by experiments. The measurements of the impedance components were carried out for air-core, I-core and pot-core type E sensors. A very good agreement was achieved within the whole analysed frequency range from 1 kHz to 50 kHz. The difference between the results of the calculations and measurements for both the resistance and reactance of each of the sensors was less than 3 %.

Challenging Examples of the Wise Use of Computer Tools for the Sustainability of Knowledge and Developing Active and Innovative Methods in STEAM and Mathematics Education

The rapid changes in information and communication technology (ICT), the increasing availability of processing power, and the complexity of mathematical software demand a radical re-thinking of science, technology, engineering, arts, and mathematics (STEAM), as well as mathematics education. In the transition to technology-based classrooms, the constant use of educational software is a requirement for sustainable STEAM and mathematics education. This software supports a collaborative and actionable learning environment, develops 21st-century skills, and promotes the adoption of active and innovative methodologies. This paper focuses on learning and teaching mathematics and analyzes the role and utility of ICT tools in education as computer algebra systems (CAS) and dynamic geometry systems (DGS) in implementing active and innovative teaching methodologies related to sustainable STEAM education. Likewise, it highlights the necessity for learners to have extensive knowledge of mathematical theory, an essential asset to ensure the reliable and effective use of mathematical software. Through a practical experiment, this study aims to highlight that a mixed teaching method can significantly improve the sustainability of math knowledge. It provides various solid examples of CAS and DGS applications to emphasize its usage rooted in a mathematical background to enable learners to identify when the computer solution is unreliable. The study highlights that the proper use of CAS and DGS is an efficient method of deepening our understanding of mathematical notions and solving tasks in STEAM subjects and real-life applications. This paper’s goal is to direct our attention to the proper and intelligent use of computer tools, especially symbolic calculators, such as CAS and DGS, without providing an in-depth analysis of the challenges of these technologies. The outcomes of the paper should offer educators and learners new elements of active strategies and innovative learning models that can be immediately applied in education.