Numerical Analysis. Introduction to. Rostam K. Saeed Karwan H.F. Jwamer Faraidun K. Hamasalh

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1 Iraq Kurdistan Region Ministry of Higher Education and Scientific Research University of Sulaimani Faculty of Science and Science Education School of Science Education-Mathematics Department Introduction to Numerical Analysis First Edition Rostam K. Saeed Karwan H.F. Jwamer Faraidun K. Hamasalh Sulaimani, Kurdistan Region- Iraq,

2 Introduction to Numerical Analysis First Edition Rostam K. Saeed Department of Mathematics, College of Science, Salahaddin University/Erbil, Kurdistan Region, Hawler, Iraq. Karwan H.F. Jwamer Department of Mathematics, School of Science, University of Sulaimani, Kurdistan Region, Sulaimani, Iraq. Faraidun K. Hamasalh Department of Mathematics, School of Science Education, University of Sulaimani,, Kurdistan Region, Sulaimani, Iraq. Sulaimani, Kurdistan Region- Iraq, 2015

3 Copyright 2015 by Rostam K. Saeed, Karwan H.F. Jwamer and Faraidun K. Hamasalh. First Edition published 2015 by University of Sulaimani, Sulaimani, Kurdistan Region- Iraq. Printed by Pasha print, 2015, Sulaimani, Kurdistan Region- Iraq. Bibliography: Includes index Numerical Analysis Rostam K. Saeed, Karwan H.F. Jwamer and Faraidun K. Hamasalh

4 Preface After several years as lecture in Numerical Analysis, we felt that the books that were available on the subject were written in such a way that the students found them difficult to understand. It s hard for students whose mother language is not English, so we decided that we should try to compose a book on Numerical Analysis that is suitable for students in the Mathematics and Physics departments in College of Science, College of Education, and College of Engineering. This book will primarily function as an abbreviation of the Numerical Analysis course that student in a number of different colleges study. It can be used as an introductory handbook for students from many different backgrounds and academic levels. It is a known fact that Numerical Analysis is not just a science, it requires picking through different methods for the sake of solving the problem at hand. Examples and solutions at the end of each chapter have been provided which would help the students. While writing the chapters, we tried to write in a suitable level that would be easy to understand, for English students and foreign ones alike. We have also provided several examples and solutions so that understanding would be even more efficient. In writing this book, we have tried to write at a level that would be easy to be understood by native and non-native English speaking students. And that helps them pay more attention to the lecturer, which in turn, increases their understanding. And to enrich the subject even more, a couple of new ways have been accomplished by the authors and they have been added into this book in I

5 several chapters. We hope that this book will be able to help students in this magnificent portion of the field of Mathematics and its vast array of applications, and provide a paved pathway to encourage them to learn more about the subject. Authors, 2015 II

6 ACKNOWLEDGMENTS We would like to express our appreciation to Faculty of Science and Science Education of University of Sulaimani and College of Science of University of Salahaddin/Erbil, for volunteering to use a preliminary version of this text. We also wish to acknowledge the valuable input from students who used this book. A deeply felt thank you goes to the following reviewers for their words of encouragement, criticisms, and thoughtful suggestions: We would like to thank Professor Dr. Fadhil Hameed Easif and assistance Professor Dr. Shazad Shawqi Ahmed Scientific evaluation and Dr.Bekhal Latif Muhedeen for language s evaluation and their feedback on this book. Finally, we thank the editorial and production staff, especially Professor Dr. Hamid Majid Ahmed head of the Central Committee for books approving and publishing at Sulaimani University for invaluable help, comments and proving in preparation of the book. Authors 2015 III

7 CONTENTS Preface ACKNOWLEDGMENTS I III Chapter 1: Basic Concepts and Computer Arithmetic Introduction Some Basic Concepts Taylor Series for Functions of a Single Variable Infinitely Differentiable Functions Mean Value Theorem Rolle s Theorem Intermediate Value Theorem Differentiable Functions Computer Arithmetic Floating-Point Number The Representation of Fractions Sources of Error Errors of Numerical Approximations Truncation Errors Rounding and Chopped Error Round-of Errors Inherent Error Measuring Errors True and Relative Errors Approximate Error and Relative Approximate Error Numerical Instability Big O and Small o Notation 16 EXERCISES 1 18 IV

8 Chapter2: Solutions of Equations in one Variable Introduction Locating the Oosition of Roots (Programming Method) Numerical Methods Bisection Method False position method(regula Falsi Method) Secant Method Newton-Raphson Method Chybeshev method Fixed Point Method Aitkin Method Müller's Method Horner Algorithm Bairstow's Method Rostam-Kawa Methods Rostam-Shno Methods 67 Exercises 2 71 Chapter 3: Solving linear System of Equations Introduction Direct Method Gauss Elimination Gauss Elimination with Partial Pivoting LU Factorization Method (Doolittle factorization) Cholesky Factorization Norms of Vectors and Matrices Indirect method (or Iterative methods) Jacobi Method Gauss-Seidel Method 107 V

9 Exercises Chapter 4: Solving System of Non-Linear Equations Introduction Numerical Methods Fixed-Point Iteration Newton-Raphson Method Modified Newton-Raphson Method 123 Exercise Chapter 5: Interpolation and Numerical Differentiation Introduction The Finite Difference Calculus Shifting Operator (E) Forward Difference Operator ( ) Backward Difference Operator ( ) Central Deference Operator (δ) Average Operator (µ) Divided Difference Operator ( ) Interpolation Interpolation Problem Lagrange Interpolation Polynomial Divided Difference Interpolation Formula Interpolation at Equally spaced nodes Newton Forward Differences Interpolation Formula Newton Backward Differences Interpolation Formula Bessel s Interpolation Formula Inverse Interpolation Numerical Differentiation Differentiation of Continuous Functions 155 VI

10 5.6.2 Forward Difference Approximation of the First Derivative Backward Difference Approximation of the First Derivative Forward Difference Approximation from Taylor Series Finite Difference Approximation of Higher Derivatives Differentiation of Discrete Functions 167 Exercises Chapter 6: Spline Approximations Introduction Interpolation by Spline Function Fist Degree Spline Spline of Degree two (Quadratic Spline) Natural Cubic Spline Lacunary Interpolation by Splines Function Quintic Spline Sixth Degree Spline Seventh Degree Spline Ninth Degree Spline Function 204 Exercises Chapter 7: Least square and Curve fitting Introduction linear Least Square Nonlinear Least Square Exponential Model Growth Model Polynomial Models Transforming the Data to use Linear Least Square Formulas Exponential Model 230 VII

11 7.4.2 Logarithmic Functions Power Functions Growth Model 238 Exercises Chapter 8: Numerical Integrations Introduction Trapezoidal Rule of Integration Derivation of the Trapezoidal Rule Multiple-Segment Trapezoidal Rule Error in Multiple-segment Trapezoidal Rule Simpson s 1/3 Rule : Error in Multiple-Segment Simpson s 1/3 rule Simpson 3/8 Rule for Integration Richardson s Extrapolation Formula for Trapezoidal Rule Romberg Integration Gauss Quadrature Rule of Integration Derivation of two-point Gauss Quadrature Rule Higher Point Gauss Quadrature Formulas Arguments and Weighing Factors for n-point Gauss Quadrature Rules Gauss Legendre Integration Methods Gauss-Chebyshev Integration Methods Gauss-Hermite Integration Methods 296 Exercises Chapter 9: Numerical Solutions of Ordinary Differential Equations Introduction Euler s Method for Ordinary Differential Equations Derivation of Euler s Method Modified Euler Method 309 VIII

12 9.4 Taylor s Series Method Runge-Kutta 2 nd Order Method Runge-Kutta 2 nd Order Method Runge-Kutta 4 th Order Method Finite Difference Method Shooting Method Predictor-Corrector Methods Adams-Moulton Predictor-Corrector Method 345 Exercises Chapter 10: Solving Higher Order Ordinary Differential Equations Euler s and Runge-Kutta methods for Higher Order Ordinary Differential Equations (ODEs) Lacunary Interpolation Methods for Higher order ODEs Ninth Degree Spline Method for Solving System of ODE s Fifth Degree Spline Method for Solving Initial Value Problems 365 Bibliography 370 Index 374 IX

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