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Numerical Analysis, Modelling and Simulation Numerical Analysis, Modelling and Simulation Griffin Cook Numerical Analysis, Modelling and Simulation Numerical Analysis, Modelling and Simulation Edited by Griffin Cook Numerical Analysis, Modelling and Simulation Edited by Griffin Cook ISBN: 978-1-9789-1530-5 © 2018 Library Press Published by Library Press, 5 Penn Plaza, 19th Floor, New York, NY 10001, USA Cataloging-in-Publication Data Numerical analysis, modelling and simulation / edited by Griffin Cook. p. cm. Includes bibliographical references and index. ISBN 978-1-9789-1530-5 1. Numerical analysis. 2. Mathematical models. 3. Simulation methods. I. Cook, Griffin. QA297 .N86 2018 518--dc23 This book contains information obtained from authentic and highly regarded sources. All chapters are published with permission under the Creative Commons Attribution Share Alike License or equivalent. A wide variety of references are listed. Permissions and sources are indicated; for detailed attributions, please refer to the permissions page. Reasonable efforts have been made to publish reliable data and information, but the authors, editors and publisher cannot assume any responsibility for the validity of all materials or the consequences of their use. Copyright of this ebook is with Library Press, rights acquired from the original print publisher, Larsen and Keller Education. Trademark Notice: All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. The publisher’s policy is to use permanent paper from mills that operate a sustainable forestry policy. Furthermore, the publisher ensures that the text paper and cover boards used have met acceptable environmental accreditation standards. Table of Contents Preface VII Chapter 1 Introduction to Numerical Analysis 1 Chapter 2 Understanding Simulation 11 • Simulation 11 • Computer Simulation 13 • Dynamic Simulation 43 • Discrete Event Simulation 44 • List of Computer Simulation Software 47 Chapter 3 Modeling: An Overview 62 • Mathematical Model 62 • Conceptual Model 85 • Conceptual Model (Computer Science) 94 • Multiscale Modeling 94 • Ontology (Information Science) 97 • Statistical Model 104 Chapter 4 Theorems in Approximation Theory 108 • Approximation Theory 108 • Stone–Weierstrass Theorem 112 • Fejér’s Theorem 116 • Bernstein’s Theorem (Approximation Theory) 117 • Favard’s Theorem 118 • Müntz–Szász Theorem 118 Chapter 5 Methods and Techniques of Numerical Analysis 120 • Numerical Methods for Ordinary Differential Equations 120 • Series Acceleration 128 • Minimum Polynomial Extrapolation 130 • Richardson Extrapolation 131 • Shanks Transformation 136 • Interpolation 139 • Van Wijngaarden Transformation 145 • Matrix Splitting 146 • Gaussian Elimination 151 • Convex Optimization 158 Chapter 6 Essential Aspects of Numerical Analysis 164 • Numerical Integration 164 • Monte Carlo Method 172 • Monte Carlo Integration 183 • Mathematical Optimization 190 • Optimization Problem 203 ________________________ WORLD TECHNOLOGIES VI Contents • Multi-objective Optimization 205 • Eigendecomposition of A Matrix 217 • Singular Value Decomposition 226 • System of Linear Equations 243 Chapter 7 Various Numerical Analysis Softwares 256 • List of Numerical Analysis Software 256 • TK Solver 262 • LAPACK 264 • DataMelt 266 • Analytica (Software) 272 • GNU Octave 275 • Julia (Programming Language) 281 Chapter 8 Applications of Simulation 285 • Flight Simulator 285 • Robotics Suite 291 • Reservoir Simulation 291 • UrbanSim 296 • Traffic Simulation 297 • Stochastic Simulation 302 Permissions Index ________________________ WORLD TECHNOLOGIES Preface Numerical analysis, modeling and simulation are very complex and intricate subjects, which play a crucial role in the visual (2D and 3D) representation of concepts and objects that are otherwise not visible at a phenomenal level. They are used in the fields like systems theory, science education, knowledge visualization and even in philosophy of science. The text presents these difficult subjects in the most comprehensible and easy to understand language. It includes topics, which are important for the holistic understanding of the subject matter. It studies, analyses and upholds the pillars of the subject and its utmost significance in modern times. Coherent flow of topics, student-friendly language and extensive use of examples make this textbook an invaluable source of knowledge. Given below is the chapter wise description of the book: Chapter 1- Numerical analysis is a topic in mathematics that concerns itself with the study of algorithms. Numerical analysis has applications in all fields of engineering; recently it has been adopted by life sciences also. This chapter will provide an integrated understanding of numerical analysis. Chapter 2- The process of re-creating the operations of the processes that occur in the real world over time is referred to as simulation. Simulation can only take place when the model it is imitating has already been fully developed. This section on simulation offers an insightful focus, keeping in mind the subject matter. Chapter 3- Mathematical modeling is the description of systems that uses mathematical concepts. The process of developing this model is known as mathematical modeling. The major elements of mathematical modeling are governing equations, constitutive equations, constraints, kinematics equations etc. The major components are discussed in this section. Chapter 4- Approximation theory concerns itself with how functions can be valued with simpler functions. The other theorems explained in this section are Stone-Weierstrass theorem, Fejér’s theorem, Bernstein’s theorem and Favard’s theorem. The chapter strategically encompasses and incorporated the theorems used in approximation theory, providing a complete understanding. Chapter 5- The methods and techniques of numerical analysis are series acceleration, minimum polynomial extrapolation, Richardson extrapolation, Shanks transformation and interpolation. Series acceleration improves the rate of convergence of a series. It is also used to obtain a variety of identities on special functions. The aspects elucidated in this chapter are of vital importance, and provides a better understanding of numerical analysis. Chapter 6- Numerical integration is a broad family that involves algorithms for calculation. The calculation is done for calculating the numerical value of a definite integral. The aspects of numerical analysis explained in this section are Monte Carlo method, Monte Carlo integration, mathematical optimization, optimization problem, singular value decomposition etc. The topics discussed in the section are of great importance to broaden the existing knowledge on numerical analysis. ________________________ WORLD TECHNOLOGIES VIII Preface Chapter 7- This chapter lists the numerical analysis softwares; some of these softwares are TK Solver, LAPACK, DataMelt, Analytica and GNU Octave. TK Solver is a mathematical modeling software that is based on declarative and rule-based language. Analytica is a software that is developed by Lumia decision systems for creating and analyzing quantitative decision models. The section serves as a source to understand all the numerical analysis softwares. Chapter 8- Simulation has numerous applications; some of these applications are flight simulators, robotics suites, reservoir simulations, UrbanSim and traffic simulation. Flight simulators are devices that artificially re-creates aircraft flight and along with this also re-creates the environment in which it flies. This is used for pilot training. This chapter helps the readers in understanding the applications of simulation in today’s time. At the end, I would like to thank all those who dedicated their time and efforts for the successful completion of this book. I also wish to convey my gratitude towards my friends and family who supported me at every step. Editor ________________________ WORLD TECHNOLOGIES 1 Introduction to Numerical Analysis Numerical analysis is a topic in mathematics that concerns itself with the study of algorithms. Numerical analysis has applications in all fields of engineering; recently it has been adopted by life sciences also. This chapter will provide an integrated understanding of numerical analysis. Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Col- lection (YBC 7289), which gives a sexagesimal numerical approximation of 2 , the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in astronomy, carpentry and construc- tiotn. Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of 2, modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in
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