Combining Programming and Mathematics Through Computer Simulation Problems

Abstract

ion, and algorithms). We provide the example of the use one of CAS (Mathematica) for the mathematical research on the D(s)-function associated with Riemann Zeta function. For solving this problem, we need to find the algorithm in order to get a mathematically correct results generated by Mathematica.

Keywords

• Computer algebra system (CAS)
• Mathematica
• Wolfram software
• computational thinking
• numerical computing
• Riemann zeta function
• zeta effect

Introduction

Today technically competent young people can easily use digital devices, know how to connect to GPRS, GPS and start streaming video. At the same time, educators say that traditional forms of educational cognitive activity have fallen. In the 20th century the core skills, that every person needed, were the abilities to read, to write and to count – so-called “3R’s” (Reading, wRiting, aRithmetic). In the 21st century another core skill – Computational Thinking (CT) – was added to these 3R’s. CT, which implies a new way to solve emerging problems with the methods of computer science and engineering, information technology, information systems. First the term “Computational thinking” was introduced in [1]. Seymour Papert discussed new pedagogical approaches in mathematical education in the paper [1]. This term denoted a way of thinking for the algorithmic solution of complex mathematical problems. Later Jeannette M. Wing in the paper [2] developed the computational thinking approach beyond mathematics.

In the paper [3] authors outlined that research team at MIT had developed computing environments designed to facilitate computational thinking (Logo, Scratch) and the use of computer as a computational object. Main components of computational thinking are Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). decomposition, pattern recognition, abstraction, and algorithms. Decomposition demonstrates how to divide complex problems into smaller problems. Pattern recognition shows how to find connections between similar problems and how to use previous experience. Abstraction helps to focus only on the important information without irrelevant details. Using algorithms, we can develop a step-by-step solution to the problem, or the road map to solve the problem.

Computational thinking and programming allow students to learn not only math and programming languages but help them to learn in order to become successful.

For some aspects of computer science students need to know mathematics that is a fundamental course in the educating process of CS professionals. Math helps programmers to solve a problem in an efficient way. Discrete math (set theory, logic, combinatorics), number theory (cryptography and security), geometry (geometric objects, transformations, rotations), linear algebra (matrices, series, differential equations), game theory etc. are math fields that are most important and commonly using in computer science. Math is not directly used in computer science. But computer science students have to think logically and analytically for being good programmers.

These are the same types of thinking in solving difficult mathematical problems.

Without math, students will face a longer learning curve in programming and vice versa.

In the papers [4], [5], [6], [7] authors provided an overview of educational aspects of math teaching and learning with integrated platforms and computer aided learning software.

Studies related to the effect of computer algebra systems (CAS) on learning efficiency of computer science students presented in papers [8], [9], [10]. The papers [11], [12], [13], [14], [15] presented how to use Mathematica and other CAS for solving math analytical and numerical problems. The ways of using Mathematica as a tool for visualization of the results of the different types of mathematical researches are described in [16], [17], [18], [19], [20], [21], [22], [23]. Some issues about organization of the workspace of a computational system are presented in [24]. There is a bibliography of publications about the Mathematica symbolic algebra language in the [25]. Special advanced math researches using Mathematica as a tool for computing are presented in papers [26], [27], [28], [29], [30]. Topics of the instability that is related to the well-known Gibbs phenomenon [31], [32] and is not in the specifics of the CAS, are presented in [26], [27], [28].

The paper is organized as follows. Section 2 details the advantages of using a computer algebraic software for solving math problems and presents a brief review and comparison of computer algebraic systems. Section 3 presents an advanced math problem that was solved using Mathematica as a main tool. In this section we illustrate the methods and results. 2

Programming, math and computer algebra systems

One way to implement the paradigm of the computational thinking is the use CAS for teaching mathematics and programming at CS departments.

Computer algebra system is a software that helps to manipulate mathematical expressions and mathematical objects, to provide symbolic or numeric computations, to plot different graphics and to visualize math objects. CAS may be divided into two classes: ─ specialized, that can be used for solving specific problems of mathematics or statistics; ─ general-purpose that can be used for a scientific domain that requires manipulation of mathematical expressions or objects.

The main features of general-purpose CAS are ─ a user interface, allowing to enter math formulas or data, and graphics capability; ─ a programming language and interpreter; ─ a memory manager and garbage collector; ─ a rewrite system for simplifying mathematics formulas; ─ a large library of mathematical algorithms, special functions, efficient data structures; ─ an arbitrary-precision (bignum) arithmetic, needed for calculations are performed on the huge size numbers; ─ a fast kernel.

You can see a comparison of most popular CAS in the table 1.

Maple [33] was released by Maplesoft in 1982 as a symbolic and numeric computing environment. It is based on a kernel (written in C) and has libraries (written in Maple language) for performing technical and numeric computations. For storing symbolic expressions Maple uses such data structure as directed acyclic graphs. Maple’s interfaces are written in Java. Maple software allows to analyze, explore, visualize, and solve mathematical problems. It can be used in mathematics, smart document environment, application areas, application development, high performance computing, connectivity and education.

Mathcad [34] is computer software product of the Parametric Technology Corporation (PTC) first introduced in 1986. It is used for engineering calculations; results are stored as a notebook. Equations and expressions are created in a worksheet and manipulated in the same graphical format.

The Mathcad functionality contains: ─ numerous numeric functions covering such areas as statistics, data analysis and image processing; ─ systems of equations (including ordinary and partial differential equations); ─ roots of polynomials and functions finder; ─ symbolical calculation and manipulation of math expressions; ─ parametric, 2D and 3D plotting; ─ vector and matrix operations (including eigenvalues, eigenvectors); ─ statistical functions, regression analysis on experimental datasets.

Name of CAS / creator

Maple / Maplesoft

First /

Latest releases

1984 / 2020

Mathcad / Mathsoft, PTC 1985 / 2019

Mathematica / Wolfram Research

SageMath / William Arthur Stein

Symbolic Math Toolbox (MATLAB) / MathWorks

SymPy / Ondřej Čertík

Wolfram|Alpha /

Wolfram Research 1988 / 2020 2005 / 2020 2008 / 2020 2007 / 2020 2009 / 2020

Latest version $2,390 (Commercial), $2,265 (Government), $995

For symbolic and numeric (Academic), $239 Proprie

computing. (Personal Edition), tary

Written in С, Java, Maple $99 (Student), $79 (Student, 12

Month term) $1,600 (Commer- Includes some of the cacial), $105 (Stu- Proprie- pabilities of CAS. For nu

dent), Free tary merical computing of the (Express Edition) engineering problems $2,495 (Professional), $1095 (Edu- For solving problems in cation), $295 (Per- many technical, scientific, sonal), $140 engineering, mathema

Proprie(Student), $69.95 tical, and computing fi

tary (Student annual elds.

license), free on Written in Wolfram LanRaspberry Pi hard- guage, C/C++, Java ware

Wolfram Mathematica [35] is an application for mathematical symbolic calculations that consists of two parts – kernel (back end) and interface (front end). In general, Mathematica is a great tool for solving problems, it integrates all functionalities such as symbolic calculations, manipulations with equations, numeric and graphical outputs. Mathematica offers predefined functions for mathematics, physics, economy, biology and other areas. It is used for calculations in the scientific, engineering, mathematical and computer fields. The Mathematica is also called the CAS Mathematica uses the Wolfram Language. Wolfram Language is a multi-paradigm programming language developed by Wolfram Research for symbolic computing, functional and logical programming, which allows the implementation of arbitrary data structures.

SageMath [36] is free and an open source, python-based alternative to Mathematic, Mathcad, Maple. It uses many python packages, for example, Numpy, Matplotlib, Scipy, Pylab. SageMath has features covering many parts of mathematics – algebra, combinatorics, graph theory, numerical analysis, number theory, calculus and statistics.

MATLAB (MATrix LABoratory) [37] is a software package for high performance numerical computation. It provides an interactive environment with hundreds of built in functions for technical computation, graphics and animation and easy extensibility with its own high-level programming language. MATLAB contains a lot of tools for linear algebra computations, data analysis, signal processing, optimization, numerical solution of Ordinary Differential Equations (ODEs), quadrature and many other types of scientific computations. MATLAB also provides matrix manipulations, parametric, 2D and 3D plotting of functions and data, algorithms implementation, creation of GUI, and interfacing with programs written in other programming languages (C, C++, C#, Java, FORTRAN, Python).

SymPy [38] is an open-source library for symbolic computation that completely written in Python. It provides computer algebra (algebra, matrices, etc.) capabilities either as a standalone application, as a library to other applications, or live application on the web. The SymPy library is split into a core with many optional modules (polynomials, calculus, solving equations, discrete math, matrices, geometry, plotting, physics, statistics, combinatorics, printing).

Wolfram|Alpha [39] is a computational knowledge engine (answer engine) developed by WolframAlpha LLC. Wolfram|Alpha is an online service like a factbased engine. It answers factual queries directly by computing the answer and does not provide a list of documents or web pages like search engine. Wolfram|Alpha uses technologies that can be divided into four key general areas: a data curation pipeline, an algorithmic computation system, a linguistic processing system, an automated presentation system.

CAS can run under different operating systems natively without emulation. Like most modern apps, Mathematica (except mobile OS), SageMath (except mobile Android OS), MATHLAB (except mobile OS), SymPy run on almost all commonly used OS. Maple, Mathcad do not have versions that run under Android, iOS and SaaS. Moreover, Mathcad does not have versions running under masOS, Linux. We provide a list of OS supporting CAS discussed above (Table 2).

To demonstrate the development of CT skills in teaching CS students advanced topics of mathematics, as well as the development of a heuristic, logical and algorithmic thinking, we used Mathematica 12.0 to solve math problems, which may result in meaningful mathematical problems that lie outside the capabilities of the Wolfram Mathematica, or programming problems will appear that also lie outside the scope of the Wolfram Mathematica, which the CS student have to learn how to solve. We used CAS (Mathematica 12.0 [35]) for investigating the D(s)-function associated with Riemann Zeta function. Results of the project are presented in paper [26].

Let us consider the function D(s) of the complex argument s=+it, formed with the use of a certain procedure of a transition to the limit: ( 1 ) ( 2 ) ( 3 )  1  N 1  .

D(s)  1  s Nlim  N 1s  1 n1 ns 

D(s)  s  1 (s), Re s  1 .

It is obvious that for σ>1 the limit of the sum in ( 1 ) turns into the Riemann zeta function, and, accordingly:

 1 Where  (s)   s is Riemann zeta function.

n1 n In the paper [26] was showed that for Re s<1:

D(s)  1 (Re s<1).

Therefore, the function D(s), defined by formula ( 1 ) in the entire complex plane (with the exception of the straight line =1), in the right-hand half-plane (>1) in accordance with formula ( 2 ) differs from the Riemann zeta function only by the factor (1–s), and in the left half-plane, in accordance with formula ( 3 ), it is equal to one. Formula ( 1 ), in  a certain sense, extends the Riemann series  ns to the entire complex plane s n1 (except for the straight line =1).

It is convenient to divide the real and imaginary parts of the function D: where

D(s)  R , t   iI  , t  R , t   t   , t   (  1) , t ,



I  , t   t   , t   (  1) , t ,   and  are the real Riemann series:

N cost ln n   , t   Nlim n1 n ,

N sint ln n   , t   Nlim n1 n .    ,t   1,   .

   ,t   0,

R , t    1,

 I  , t   t. 

N cost ln n   N  , t   n1 n ,

N sint ln n  N  , t   n1 n .  ( 4 ) ( 5 ) ( 6 ) ( 7 ) When computing the sums AN and BN we usually fix N in the range between N = 105 and N = 106. A calculation with a smaller value of N introduced noticeable distortions It follows from ( 6 ) that the series  , t  and  , t  have very simple asymptotics for large values of : Consequently, in accordance with formulas ( 5 ), the asymptotics of D(s) for   . and fixed t has the following form: When calculating the function D(s) in the right half-plane of the complex argument s (for >1) we used finite-dimensional approximations of the oscillating real Riemann series ( 6 ). Instead of formulas ( 6 ) containing the limiting transition N   , we used finite sums: in the results. Computations with large values N required an unacceptably high timeconsuming result. For N in the range 105  106 one calculation, – for example, plotting the dependence of R(t) for fixed  and 0  t  50 – requires, depending on  and N from several tens of minutes to several work hours for Mathematica. You can see codes of plotting the R-function at listing 1, and symbolic results at Fig. 1.

Listing 1. Codes for plotting the R-function as the function of the argument t for >1. Where β is the damping parameter:   1. (In the calculations, we used the value β=5). Exponential β-damping ( 8 ) does not significantly affect the contribution of the majority of “low-frequency” harmonics with n  N and substantially reduces the contribution of terms with large numbers n, approaching to n=N. This method smooths out the effect of a sharp break in the Riemann series ( 5 ) for n=N.

Fig. 3 demonstrates the effect of exponential β-damping on the numerical results of calculating the function D(s). This figure shows the same graph shown earlier in Fig. 2: dependence of the function R on the argument t for =1.05 and N  6 105 . In Fig. 3 this dependence is calculated by the formulas ( 14 ), taking into account β-damping (β=5). The “smoothed” function R(t) in this figure completely repeats the function R(t) of Fig. 2 in all that concerns real large-scale slow oscillations, but is practically free of parasitic oscillations generated by the zeta effect. The trace of these parasitic oscillations remained only for small t (t < 1). The suppression of the zeta-effect in the region of small t requires an increase in the damping parameter β. An increase in β can cause distortion in real large-scale oscillations of the function R(t). Here, the researcher must compromise, determining what is more important in a particular task – total suppression of the zeta-effect at small t or preservation of correct results for large t. Of the many problems that the author had to solve (on her own or in collaboration with colleagues and students) using various CAS packages, the author chose one problem here for purely illustrative purposes. In this task, the mathematics package used by itself works flawlessly, but you need to reflect on the algorithm in order to get a mathematically correct result using this package that suppresses some kind of pure instability. This instability is inherent in the task itself, and not in the specifics of the Mathematics package. In some sense, this instability is related to the well-known Gibbs phenomenon, which consists in a certain instability of the process of numerical summation of Fourier series in some situations.

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