=Paper= {{Paper |id=Vol-2732/20200869 |storemode=property |title=Combining Programming and Mathematics through Computer Simulation Problems |pdfUrl=https://ceur-ws.org/Vol-2732/20200869.pdf |volume=Vol-2732 |authors=Zarema Seidametova |dblpUrl=https://dblp.org/rec/conf/icteri/Seidametova20 }} ==Combining Programming and Mathematics through Computer Simulation Problems== https://ceur-ws.org/Vol-2732/20200869.pdf

Combining Programming and Mathematics
Through Computer Simulation Problems

Zarema Seidametova[0000-0001-7643-6386]

Crimean Engineering and Pedagogical University named after Fevzi Yakubov,
8 Uchbovyi Ln., Simferopol, Crimea
z.seydametova@gmail.com

Abstract. Nowadays, educators often use different computer algebra systems for
teaching advanced math topics for CS students, and as a tool for solving math
problems and providing research. Computer algebra systems allow the students
to practice skills both programming and mathematics, that help to develop main
components of computational thinking (decomposition, pattern recognition,
abstraction, 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.

1 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.
Table 1. Popular computer algebra systems
First /
Name of CAS / Latest
Latest Price License Notes
creator version
releases
$2,390 (Commer-
cial), $2,265 (Go-
vernment), $995
2020.0 For symbolic and numeric
Maple / 1984 / (Academic), $239 Proprie-
(March computing.
Maplesoft 2020 (Personal Edition), tary
2020) Written in С, Java, Maple
$99 (Student), $79
(Student, 12-
Month term)
$1,600 (Commer- Includes some of the ca-
6.0.0.0
Mathcad / 1985 / cial), $105 (Stu- Proprie- pabilities of CAS. For nu-
(October
Mathsoft, PTC 2019 dent), Free tary merical computing of the
2019)
(Express Edition) engineering problems
$2,495 (Professio-
nal), $1095 (Edu- For solving problems in
cation), $295 (Per- many technical, scientific,
Mathematica / 12.1.0 sonal), $140 engineering, mathema-
1988 / Proprie-
Wolfram (March (Student), $69.95 tical, and computing fi-
2020 tary
Research 2020) (Student annual elds.
license), free on Written in Wolfram Lan-
Raspberry Pi hard- guage, C/C++, Java
ware
Open-source system with
features covering many
aspects of mathematics,
including algebra,
SageMath / 9.1
2005 / GNU combinatorics, graph
William Arthur (May Free
2020 GPL theory, numerical analy-
Stein 2020)
sis, number theory, calcu-
lus and statistics.
Written in Python, Cy-
thon
$3,150 (Commer- For solving and manipula-
Symbolic Math
R2020a cial), $99 (Student ting symbolic math ex-
Toolbox 2008 / Proprie-
(March Suite), $700 (Aca- pressions and performing
(MATLAB) / 2020 tary
2020) demic), $194 (Ho- variable-precision arith-
MathWorks
me) metic.
mo-
1.6 Open-source Python lib-
SymPy / Ondřej 2007 / dified
(May Free rary for symbolic compu-
Čertík 2020 BSD li-
2020) tation.
cense
Online computational
Pro version: $4.99
platform or toolkit that en-
Wolfram|Alpha / per month, Pro
2009 / Proprie- compasses computer al-
Wolfram 2020 version for stu-
2020 tary gebra, symbolic, nume-
Research dents: $2.99 per
rical computation, visua-
month
lization.
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 fact-
based 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.

Table 2. Operating systems supporting CAS

OS
CAS
DOS Windows macOS Linux Android iOS SaaS
Maple – + + + – – +
Mathcad + + – – – – –
Mathematica – + + + – – +
SageMath – + + + – + +
Symbolic Math Toolbox (MATLAB) – + + + + – +
SymPy – + + + + + +

3 Math project “On the function D(s) associated with
Riemann Zeta function”

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:

 N
1 1 
D(s)  1  s  lim  1 s   . (1)

N  N  1 n 1 n s 


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

D(s)  s  1 ( s ), Re s  1 . (2)

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

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

D(s)  1 (Re s<1). (3)

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

s
a certain sense, extends the Riemann series n
n 1
to the entire complex plane s

(except for the straight line =1).
It is convenient to divide the real and imaginary parts of the function D:

D(s)  R , t   iI  , t  (4)

where

R , t   t   , t   (  1)  , t ,
 (5)
I  , t   t   , t   (  1) , t , 

 and  are the real Riemann series:
N
cost ln n  
 , t   lim  ,
N 
n 1 n 
N  (6)
sin t ln n  
 , t   lim  .
N  n 
n 1 

It follows from (6) that the series  , t  and  , t  have very simple asymptotics for
large values of :

  , t   1,    .

  , t   0, 

Consequently, in accordance with formulas (5), the asymptotics of D(s) for   .
and fixed t has the following form:

R , t     1,

I  , t   t. 

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:
N
cost ln n  
 N  , t    ,
n 1 n 
N  (7)
sin t ln n  
 N  , t    .
n 
n 1 

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
in the results. Computations with large values N required an unacceptably high time-
consuming 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.

Fig. 1. Symbolic results of the R-function of the argument t for =1.05 (N=6105).

Fig. 2 demonstrates, in addition to real “slow” t-oscillations of sufficiently large
amplitude, also the presence of an interesting effect of short-period “parasitic”
oscillations of small amplitude. This effect (we called it the “zeta effect”) is generated
by a sharp break in the Riemann series (6) for a finite (albeit sufficiently large) value
of n, equal to the “cut-off parameter” N N  105  106 . In Fig. 2, these zeta
oscillations significantly deform the dependence R  R (t ) up to t  15 , but are also
noticeable at t>15.
A qualitative explanation of the nature of the “zeta effect” is given in the paper [26].
In some extent, this “zeta effect” is one of the version of the Gibbs phenomenon related
to the Fourier series theory [32].
How can we suppress these parasitic zeta-oscillations generated by the termination
of infinite Riemann series (5)?
To suppress the zeta effect, we used “exponential β-damping”, replacing each term
in the Riemann sums (7) with its “damped” expression:
n
cost ln n    N 
N
 N  , t    N , d  , t ,     e ,
n 1 n 
N n 
(8)
sin t ln n    N 
 N  , t    N , d  , t ,     e .
n 1 n 

Where β is the damping parameter:   1 . (In the calculations, we used the value β=5).

Fig. 2. The R-function as the function of the argument t for =1.05 (N=6105).

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.
Fig. 3. The dependence of the function R of argument t for =1.05 (N=6105). The dependence
is calculated using the exponential β-damping procedure described in the article for β=5.

4 Conclusions

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.
In other problems, there may be situations where Wolfram Mathematica cannot
perform certain actions at all (for example, plotting a curve containing cusp [27]) or
perform calculations with the necessary accuracy in a reasonable time (for example,
constructing fractal curves defined by Riemann-Weierstrass series).

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