Computer Algebra and Its Applications
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Plotting Direction Fields in Matlab and Maxima – a Short Tutorial
Plotting direction fields in Matlab and Maxima – a short tutorial Luis Carvalho Introduction A first order differential equation can be expressed as dx x0(t) = = f(t; x) (1) dt where t is the independent variable and x is the dependent variable (on t). Note that the equation above is in normal form. The difficulty in solving equation (1) depends clearly on f(t; x). One way to gain geometric insight on how the solution might look like is to observe that any solution to (1) should be such that the slope at any point P (t0; x0) is f(t0; x0), since this is the value of the derivative at P . With this motivation in mind, if you select enough points and plot the slopes in each of these points, you will obtain a direction field for ODE (1). However, it is not easy to plot such a direction field for any function f(t; x), and so we must resort to some computational help. There are many computational packages to help us in this task. This is a short tutorial on how to plot direction fields for first order ODE’s in Matlab and Maxima. Matlab Matlab is a powerful “computing environment that combines numeric computation, advanced graphics and visualization” 1. A nice package for plotting direction field in Matlab (although resourceful, Matlab does not provide such facility natively) can be found at http://math.rice.edu/»dfield, contributed by John C. Polking from the Dept. of Mathematics at Rice University. To install this add-on, simply browse to the files for your version of Matlab and download them. -
A Note on Integral Points on Elliptic Curves 3
Journal de Th´eorie des Nombres de Bordeaux 00 (XXXX), 000–000 A note on integral points on elliptic curves par Mark WATKINS Abstract. We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writ- ing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have non- degenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of Macsyma, with there being a unique rational nondegenerate solution. For the second case we found that resultants and/or Gr¨obner bases were not very efficacious. Instead, at the suggestion of Elkies, we used multidimensional p-adic Newton iteration, and were able to find a nondegenerate solution, albeit over a quartic number field. Due to our methodol- ogy, we do not have much hope of proving that there are no other solutions. For the third case we found a solution in a nonic number field, but we were unable to make much progress with the fourth case. We make a few concluding comments and include an ap- pendix from Elkies regarding his calculations and correspondence with Zagier. Resum´ e.´ A` la suite de Zagier et Elkies, nous recherchons de grands points entiers sur des courbes elliptiques. En ´ecrivant une solution polynomiale g´en´erique et en ´egalisant des coefficients, nous obtenons quatre cas extr´emaux susceptibles d’avoir des so- lutions non d´eg´en´er´ees. Chacun de ces cas conduit `aun syst`eme d’´equations polynomiales, le premier ´etant r´esolu par Elkies en 1988 en utilisant les r´esultants de Macsyma; il admet une unique solution rationnelle non d´eg´en´er´ee. -
Utilizing MATHEMATICA Software to Improve Students' Problem Solving
International Journal of Education and Research Vol. 7 No. 11 November 2019 Utilizing MATHEMATICA Software to Improve Students’ Problem Solving Skills of Derivative and its Applications Hiyam, Bataineh , Jordan University of Science and Technology, Jordan Ali Zoubi, Yarmouk University, Jordan Abdalla Khataybeh, Yarmouk University, Jordan Abstract Traditional methods of teaching calculus (1) at most Jordanian universities are usually used. This paper attempts to introduce an innovative approach for teaching Calculus (1) using Computerized Algebra Systems (CAS). This paper examined utilizing Mathematica as a supporting tool for the teaching learning process of Calculus (1), especially for derivative and its applications. The research created computerized educational materials using Mathematica, which was used as an approach for teaching a (25) students as experimental group and another (25) students were taught traditionally as control group. The understandings of both samples were tested using problem solving test of 6 questions. The results revealed the experimental group outscored the control group significantly on the problem solving test. The use of Mathematica not just improved students’ abilities to interpret graphs and make connection between the graph of a function and its derivative, but also motivate students’ thinking in different ways to come up with innovative solutions for unusual and non routine problems involving the derivative and its applications. This research suggests that CAS tools should be integrated to teaching calculus (1) courses. Keywords: Mathematica, Problem solving, Derivative, Derivative’s applivations. INTRODUCTION The major technological advancements assisted decision makers and educators to pay greater attention on making the learner as the focus of the learning-teaching process. The use of computer algebra systems (CAS) in teaching mathematics has proved to be more effective compared to traditional teaching methods (Irving & Bell, 2004; Dhimar & Petal, 2013; Abdul Majid, huneiti, Al- Nafa & Balachander, 2012). -
Performance Analysis of Effective Symbolic Methods for Solving Band Matrix Slaes
Performance Analysis of Effective Symbolic Methods for Solving Band Matrix SLAEs Milena Veneva ∗1 and Alexander Ayriyan y1 1Joint Institute for Nuclear Research, Laboratory of Information Technologies, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia Abstract This paper presents an experimental performance study of implementations of three symbolic algorithms for solving band matrix systems of linear algebraic equations with heptadiagonal, pen- tadiagonal, and tridiagonal coefficient matrices. The only assumption on the coefficient matrix in order for the algorithms to be stable is nonsingularity. These algorithms are implemented using the GiNaC library of C++ and the SymPy library of Python, considering five different data storing classes. Performance analysis of the implementations is done using the high-performance computing (HPC) platforms “HybriLIT” and “Avitohol”. The experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed. An analysis of the three algorithms is performed. 1 Introduction Systems of linear algebraic equations (SLAEs) with heptadiagonal (HD), pentadiagonal (PD) and tridiagonal (TD) coefficient matrices may arise after many different scientific and engineering prob- lems, as well as problems of the computational linear algebra where finding the solution of a SLAE is considered to be one of the most important problems. On the other hand, special matrix’s char- acteristics like diagonal dominance, positive definiteness, etc. are not always feasible. The latter two points explain why there is a need of methods for solving of SLAEs which take into account the band structure of the matrices and do not have any other special requirements to them. One possible approach to this problem is the symbolic algorithms. -
The Yacas Book of Algorithms
The Yacas Book of Algorithms by the Yacas team 1 Yacas version: 1.3.6 generated on November 25, 2014 This book is a detailed description of the algorithms used in the Yacas system for exact symbolic and arbitrary-precision numerical computations. Very few of these algorithms are new, and most are well-known. The goal of this book is to become a compendium of all relevant issues of design and implementation of these algorithms. 1This text is part of the Yacas software package. Copyright 2000{2002. Principal documentation authors: Ayal Zwi Pinkus, Serge Winitzki, Jitse Niesen. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts and no Back-Cover Texts. A copy of the license is included in the section entitled \GNU Free Documentation License". Contents 1 Symbolic algebra algorithms 3 1.1 Sparse representations . 3 1.2 Implementation of multivariate polynomials . 4 1.3 Integration . 5 1.4 Transforms . 6 1.5 Finding real roots of polynomials . 7 2 Number theory algorithms 10 2.1 Euclidean GCD algorithms . 10 2.2 Prime numbers: the Miller-Rabin test and its improvements . 10 2.3 Factorization of integers . 11 2.4 The Jacobi symbol . 12 2.5 Integer partitions . 12 2.6 Miscellaneous functions . 13 2.7 Gaussian integers . 13 3 A simple factorization algorithm for univariate polynomials 15 3.1 Modular arithmetic . 15 3.2 Factoring using modular arithmetic . -
Rapid Research with Computer Algebra Systems
doi: 10.21495/71-0-109 25th International Conference ENGINEERING MECHANICS 2019 Svratka, Czech Republic, 13 – 16 May 2019 RAPID RESEARCH WITH COMPUTER ALGEBRA SYSTEMS C. Fischer* Abstract: Computer algebra systems (CAS) are gaining popularity not only among young students and schol- ars but also as a tool for serious work. These highly complicated software systems, which used to just be regarded as toys for computer enthusiasts, have reached maturity. Nowadays such systems are available on a variety of computer platforms, starting from freely-available on-line services up to complex and expensive software packages. The aim of this review paper is to show some selected capabilities of CAS and point out some problems with their usage from the point of view of 25 years of experience. Keywords: Computer algebra system, Methodology, Wolfram Mathematica 1. Introduction The Wikipedia page (Wikipedia contributors, 2019a) defines CAS as a package comprising a set of algo- rithms for performing symbolic manipulations on algebraic objects, a language to implement them, and an environment in which to use the language. There are 35 different systems listed on the page, four of them discontinued. The oldest one, Reduce, was publicly released in 1968 (Hearn, 2005) and is still available as an open-source project. Maple (2019a) is among the most popular CAS. It was first publicly released in 1984 (Maple, 2019b) and is still popular, also among users in the Czech Republic. PTC Mathcad (2019) was published in 1986 in DOS as an engineering calculation solution, and gained popularity for its ability to work with typeset mathematical notation in combination with automatic computations. -
CAS (Computer Algebra System) Mathematica
CAS (Computer Algebra System) Mathematica- UML students can download a copy for free as part of the UML site license; see the course website for details From: Wikipedia 2/9/2014 A computer algebra system (CAS) is a software program that allows [one] to compute with mathematical expressions in a way which is similar to the traditional handwritten computations of the mathematicians and other scientists. The main ones are Axiom, Magma, Maple, Mathematica and Sage (the latter includes several computer algebras systems, such as Macsyma and SymPy). Computer algebra systems began to appear in the 1960s, and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence. A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics Martin Veltman, who designed a program for symbolic mathematics, especially High Energy Physics, called Schoonschip (Dutch for "clean ship") in 1963. Using LISP as the programming basis, Carl Engelman created MATHLAB in 1964 at MITRE within an artificial intelligence research environment. Later MATHLAB was made available to users on PDP-6 and PDP-10 Systems running TOPS-10 or TENEX in universities. Today it can still be used on SIMH-Emulations of the PDP-10. MATHLAB ("mathematical laboratory") should not be confused with MATLAB ("matrix laboratory") which is a system for numerical computation built 15 years later at the University of New Mexico, accidentally named rather similarly. The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma; a popular copyleft version of Macsyma called Maxima is actively being maintained. -
SNC: a Cloud Service Platform for Symbolic-Numeric Computation Using Just-In-Time Compilation
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCC.2017.2656088, IEEE Transactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING 1 SNC: A Cloud Service Platform for Symbolic-Numeric Computation using Just-In-Time Compilation Peng Zhang1, Member, IEEE, Yueming Liu1, and Meikang Qiu, Senior Member, IEEE and SQL. Other types of Cloud services include: Software as a Abstract— Cloud services have been widely employed in IT Service (SaaS) in which the software is licensed and hosted in industry and scientific research. By using Cloud services users can Clouds; and Database as a Service (DBaaS) in which managed move computing tasks and data away from local computers to database services are hosted in Clouds. While enjoying these remote datacenters. By accessing Internet-based services over lightweight and mobile devices, users deploy diversified Cloud great services, we must face the challenges raised up by these applications on powerful machines. The key drivers towards this unexploited opportunities within a Cloud environment. paradigm for the scientific computing field include the substantial Complex scientific computing applications are being widely computing capacity, on-demand provisioning and cross-platform interoperability. To fully harness the Cloud services for scientific studied within the emerging Cloud-based services environment computing, however, we need to design an application-specific [4-12]. Traditionally, the focus is given to the parallel scientific platform to help the users efficiently migrate their applications. In HPC (high performance computing) applications [6, 12, 13], this, we propose a Cloud service platform for symbolic-numeric where substantial effort has been given to the integration of computation– SNC. -
Nonlinear Evolution Equations and Solving Algebraic Systems: the Importance of Computer Algebra
- Y'M?S </#i/ ИНСТИТУТ ядерных исследований дубна Е5-89-62Ц V.P.Gerdt, N.A.Kostov, A.Yu.Zharkov* NONLINEAR EVOLUTION EQUATIONS AND SOLVING ALGEBRAIC SYSTEMS: THE IMPORTANCE OF COMPUTER ALGEBRA Submitted to International Conference "Solitons and its Applications", Dubna, August 25-27, 1989 * Saratov State University, USSR 1989 1.INTRODUCTION A wide-spread problem in the theory of nonlinear evolution equations (NEE) is to find the exact solutions of complicated algebraic systems. For example, let us consider the following evolution systems U - AU + F(U,U , ..U ), U=U(x,t)-(U1, . ,UH) , U=D1(U), D=d/dx IN 1 N-1 I /ii F«(F\ . -FH) , A==diag(A , . .A) , A *0, A *A , (i*j). The general symmetry approach to checking up the integrability and classification of integrable NEE (see the reviews [1,2] and references therein) allows to obtain the integrability conditions which are related to existence of higher order symmetries and conservation laws in a fully algorithmic way. The implementation of these algorithms in a form of FORMAC program FORMINT have been given for scalar equations (M=l in (1)) in [3 j and for the general case (M>1) in [4]. Using this program one can automatically obtain the equations in right hand side of (l) which follow from the necessary integrability conditions. The integrabiiity conditions for the right hand side with arbitrary functions have the form of systems of differential equations, the procedure of solving these equations is not generally algorithmic. But in the important particular cases when the right hand side F are polynomials and the integrability conditions are reduced to a system of nonlinear algebraic equations in coefficients of the polynomials. -
Analysis and Verification of Arithmetic Circuits Using Computer Algebra Approach
University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses March 2020 ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH TIANKAI SU University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_2 Part of the VLSI and Circuits, Embedded and Hardware Systems Commons Recommended Citation SU, TIANKAI, "ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH" (2020). Doctoral Dissertations. 1868. https://doi.org/10.7275/15875490 https://scholarworks.umass.edu/dissertations_2/1868 This Open Access Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH A Dissertation Presented by TIANKAI SU Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY February 2020 Electrical and Computer Engineering c Copyright by Tiankai Su 2020 All Rights Reserved ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH A Dissertation Presented by TIANKAI SU Approved as to style and content by: Maciej Ciesielski, Chair George S. Avrunin, Member Daniel Holcomb, Member Weibo Gong, Member Christopher V. Hollot, Department Head Electrical and Computer Engineering ABSTRACT ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH FEBRUARY 2020 TIANKAI SU B.Sc., NORTHEAST DIANLI UNIVERSITY Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Maciej Ciesielski Despite a considerable progress in verification of random and control logic, ad- vances in formal verification of arithmetic designs have been lagging. -
The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems
The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them? Antonio J. Durán, Mario Pérez, and Juan L. Varona Introduction by a typical research mathematician, but let us Nowadays, mathematicians often use a computer explain it briefly. It is not necessary to completely algebra system as an aid in their mathematical understand the mathematics, just to realize that it research; they do the thinking and leave the tedious is typical mathematical research using computer calculations to the computer. Everybody “knows” algebra as a tool. that computers perform this work better than Our starting point is a discrete positive measure people. But, of course, we must trust in the results on the real line, µ n 0 Mnδan (where δa denotes = ≥ derived via these powerful computer algebra the Dirac delta at a, and an < an 1) having P + systems. First of all, let us clarify that this paper is a sequence of orthogonal polynomials Pn n 0 f g ≥ not, in any way, a comparison between different (where Pn has degree n and positive leading computer algebra systems, but a sample of the coefficient). Karlin and Szeg˝oconsidered in 1961 (see [4]) the l l Casorati determinants current state of the art of what mathematicians × can expect when they use this kind of software. Pn(ak)Pn(ak 1):::Pn(ak l 1) Although our example deals with a concrete system, + + − Pn 1(ak)Pn 1(ak 1):::Pn 1(ak l 1) 0 + + + + + − 1 we are sure that similar situations may occur with (1) det . -
Addition and Subtraction
A Addition and Subtraction SUMMARY (475– 221 BCE), when arithmetic operations were per- formed by manipulating rods on a flat surface that was Addition and subtraction can be thought of as a pro- partitioned by vertical and horizontal lines. The num- cess of accumulation. For example, if a flock of 3 sheep bers were represented by a positional base- 10 system. is joined with a flock of 4 sheep, the combined flock Some scholars believe that this system— after moving will have 7 sheep. Thus, 7 is the sum that results from westward through India and the Islamic Empire— the addition of the numbers 3 and 4. This can be writ- became the modern system of representing numbers. ten as 3 + 4 = 7 where the sign “+” is read “plus” and The Greeks in the fifth century BCE, in addition the sign “=” is read “equals.” Both 3 and 4 are called to using a complex ciphered system for representing addends. Addition is commutative; that is, the order numbers, used a system that is very similar to Roman of the addends is irrelevant to how the sum is formed. numerals. It is possible that the Greeks performed Subtraction finds the remainder after a quantity is arithmetic operations by manipulating small stones diminished by a certain amount. If from a flock con- on a large, flat surface partitioned by lines. A simi- taining 5 sheep, 3 sheep are removed, then 2 sheep lar stone tablet was found on the island of Salamis remain. In this example, 5 is the minuend, 3 is the in the 1800s and is believed to date from the fourth subtrahend, and 2 is the remainder or difference.