An Introduction to SAGE Programming E-Book

Razvan A. Mezei

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 198

Veröffentlichungsjahr: 2015

Beschreibung

Features a simplified presentation of numerical methods by introducing and implementing SAGE programs

An Introduction to SAGE Programming: With Applications to SAGE Interacts for Numerical Methods emphasizes how to implement numerical methods using SAGE Math and SAGE Interacts and also addresses the fundamentals of computer programming, including if statements, loops, functions, and interacts. The book also provides a unique introduction to SAGE and its computer algebra system capabilities; discusses second and higher order equations and estimate limits; and determines derivatives, integrals, and summations. Providing critical resources for developing successful interactive SAGE numerical computations, the book is accessible without delving into the mathematical rigor of numerical methods.

The author illustrates the benefits of utilizing the SAGE language for calculus and the numerical analysis of various methods such as bisection methods, numerical integration, Taylor’s expansions, and Newton’s iterations. Providing an introduction to the terminology and concepts involved, An Introduction to SAGE Programming: With Applications to SAGE Interacts for Numerical Methods also features:

• An introduction to computer programming using SAGE
• Many practical examples throughout to illustrate the application of SAGE Interacts for various numerical methods
• Discussions on how to use SAGE Interacts and SAGE Cloud in order to create mathematical demonstrations
• Numerous homework problems and exercises that allow readers to practice their programming skillset
• A companion website that includes related SAGE programming code and select solutions to the homework problems and exercises

An Introduction to SAGE Programming: With Applications to SAGE Interacts for Numerical Methods is an ideal reference for applied mathematicians who need to employ SAGE for the study of numerical methods and analysis. The book is also an appropriate supplemental textbook for upper-undergraduate and graduate-level courses in numerical methods.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Dedication

Preface

Chapter 1: Introduction

1.1 What is Sage Math?

1.2 Various Flavors of Sage Math

Chapter 2: Using Sage Math as a Calculator

2.1 First Sage Math Examples

2.2 Computations

2.3 Graphs

Chapter 3: Introduction to Programming in Sage

3.1 Variables

3.2 More on Operators

3.3 Making Decisions

3.4 Loops

3.5 Functions

3.6 Interacts

3.7 Application to Data Security: Caesar's Cipher. Interacts, Strings, and Encryption

3.8 Application to Business: Present Value of an Annuity. Amortization

3.9 Application to Elementary Statistics: Mean, Median, Histograms, and Bar Charts

Chapter 4: Sage Interacts for Numerical Methods

4.1 Equations of Lines

4.2 Tangent Lines and Plots

4.3 Taylor Polynomials

4.4 Riemann Sum and Definite Integrals

4.5 Trapezoidal Rule for Numerical Integration

4.6 Bisection Algorithm for Solving Equations

4.7 Newton–Raphson Algorithm for Solving Equations

4.8 Polynomial Interpolation

4.9 Linear Spline Interpolation

4.10 Cubic Spline Interpolation

4.11 Sage for Solving Differential Equations

4.12 Numerical Methods for Ordinary Differential Equations

4.13 Numerical Methods for Partial Differential Equations

4.14 Scatter Plots – Line of Best Fit and More

4.15 Matrices, Eigenvalues, and Eigenvectors

4.16 Solving Matrix Equations

References

Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Begin Reading

AN INTRODUCTION TO SAGE PROGRAMMING

With Applications to SAGE Interacts for Numerical Methods

RAZVAN A. MEZEI

Lenoir-Rhyne UniversityDonald & Helen Schort School of Mathematics and Computing SciencesHickory, NCUSA

Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

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Library of Congress Cataloging-in-Publication Data applied for.

ISBN:9781119122784

I would like to dedicate this work to all my instructors who passionately directed my interest into the great fields of Mathematics, Computer Science, and Statistics.

To name just a few of them, in chronological order: Aurel Netea, Petru Dragos, Barnabas Bede, Alina Alb Lupas, Dan Noje, Mircea Balaj, Sorin Gal, Mircea Dragan, Alexandru Bica, Ioan Fechete, Ioan Dzitac, George Anastassiou, Maria Botelho, James Campbell, E. Olusegun George, Nikos Frantzikinakis, and not the least Seok Wong. There are many more, but the named ones modeled my thinking and gave me a direction to where I am today and influenced me the most.

Thank you from all my heart for your dedication, support, and friendship.

Razvan A. Mezei

Preface

This work is intended to be a gentle introduction to programming in Sage Math and Sage Interacts. It assumes no programming background from the reader, and it is specifically tailored for Mathematics, Mathematics Education, and Engineering students and instructors.

The book starts with a description on how one can use Sage Math as a calculator. It also explains how one can use it for computations and for plotting. Then, it covers a brief and gentle introduction to programming in Sage Math. You will learn how to create your own methods in Sage Math and how to create Sage Interacts. The book ends with a chapter that gives several examples on how one can use Sage Interacts for various Numerical Methods.

If you have no programming background yet, your programming skills need some improvements, you want to learn how to use Sage Math to program some numerical methods, or you want to create neat interactive representations of some mathematical concepts, then this book is for you.

The book, however, does not address in detail the mathematical topics covered in the given Sage Interacts examples. In particular, no mathematical proofs are given in here. If you want to study the Mathematical side of Numerical Analysis, we recommend pairing this book with such a textbook. See, for example, [2; 7; 8].

If you are still wondering whether to use a programming language such as Python™, C, C++, Java™, or some computational software such as Maple™, Mathematica®, Octave, or R, look no further. We strongly recommend you Sage Math. It is FREE, open-source, and it uses a Python-like syntax. This last phrase contains some of the strongest arguments why you may want to choose Sage Math.

To elaborate on the last statement, free open-source software allows you to use the software for free and also allows you to access the source code, which can be a great source of inspiration and information. It also allows you to obtain the entire Sage Math's source code and change it to better address your own needs.

The Sage Math syntax is similar to that of Python. Python has become very popular for being one of the easiest introductory programming languages. It became so trendy that many (if not most, by the time you read this book) universities in the United States are using Python to teach Introductory to Programming courses. It is that easy! In fact, most of the Sage Math source code was written in Python. An interesting article one may want to read in this direction is “Python bumps off Java as top learning language,” by Joab Jackson (http://www.javaworld.com/article/2452940/learn-java/python-bumps-off-java-as-top-learning-language.html). It says that “Python has surpassed Java as the top language used to introduce U.S. students to programming and computer science.”

We hope we got your interest in learning Sage Math. You can use it as a Computer Algebra System, as a Programming Tool, or to create nice interactive mathematical demonstrations (using Sage Interacts). You can download and install Sage Math on your own machine, or you can use it over the Internet: for this, you may either choose Sage Cloud, or you may prefer Sage Cell. Sage is accessible from your desktop computer or from your smartphone.

We recommend this book to all undergraduate Mathematics, Mathematics Education, Computer Science and Information Technology, pre-Engineering, and Science students and instructors. Sage Math can be used in most Mathematics courses, in Introduction to Programming courses, as well as other computational courses.

Razvan A. MezeiUSA

Chapter 2Using Sage Math as a Calculator

Sage Math can easily be used as a calculator, one that has lots of “features.” My favorite one is the fact that you can program your own algorithms in a programming language that is very easy to learn, but we'll learn this in the next chapter. In this chapter, we get introduced on how to use basic arithmetic expressions, as well as some Sage Math library functions that are already available to the user. We focus mostly on functions that are useful in a Numerical Analysis course such as solving equations, taking derivative and antiderivatives of a function, and finding the Taylor polynomial of degree of a given function. Then we describe different functions and options that can be used to easily and efficiently obtain 2D and 3D plots.