Numerical and Statistical Techniques

Numerical and Statistical Techniques

Dr. Qazi Shoeb Ahmad Assistant Professor in Department of Mathematics Integral University, Lucknow Dr. Zubair Khan Lecturer in Department of Mathematics Integral University, Lucknow Shadab Ahmad Khan Lecturer in Department of Mathematics Integral University, Lucknow

Ane Books Pvt. Ltd. New Delhi i Chennai

Numerical and Statistical Techniques Qazi Shoeb Ahmad, Zubair Khan, Shadab Ahmad Khan © Authors First Edition: 2009 Reprint: 2014, 2017, 2018, 2019, 2020

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Preface

The authors feel great pleasure in presenting the first edition of the book “Numerical and Statistical Techniques”. This book is designed to meet the requirements of B.Tech, B.Tech (Biotech), B.C.A. students of Integral University & its study centres, Lucknow University, Jamia Hamdard and Various other Universities. The subject matter has been discussed in such a way that the students will find no difficulty to understand it. Each chapter of this book contains complete self-explanatory theory and a large number of solved examples, followed by a collection of good exercises. The language of the book is simple and easy to understand. The authors hope that the students, teachers and other readers will find the book interesting and to the point covering the whole course. We hope that the students will receive the book warmly. We have taken great care in eliminating the misprints, but if there are still any, we shall be highly obliged to those who will take trouble of pointing them out. Suggestions for the improvement of the book will be gratefully acknowledged. Authors

Acknowledgement

All praises and thanks to ‘ALLAH’, the Almighty, the Merciful, and the Omniscient whose blessings enabled us to complete this work in the present form. We feel immense pleasure in expressing our gratitude in the honour of Prof. S.W. Akhtar, Vice Chancellor, Prof. S.M. Iqbal, Pro-Vice Chancellor and Dr. I.A. Khan, Registrar, Integral University Lucknow for providing us all the necessary facilities to execute this manuscript. We are also thankful to Prof. Q.H. Ansari, Prof. Mursaleen, Prof. Zafar Ahsan, Prof. Afzal Beg, Prof. Mohd. Imdad, Dr. Rais Ahmad, Dr. Nabiullah Khan, Department of Mathematics and Prof. Abdul Bari, Department of Statistics & Operations Research, A.M.U., Aligarh for their fruitful suggestions and constant encouragement during this work. We are also greatly indebted to all our colleagues and friends especially Dr. Riyaz Ahmad Khan, HOD, Department of Mathematics, Integral University, Lucknow and Dr. Arshad Khan, Department of Mathematics, J.M.I., New Delhi for creating a healthy environment and sharing of ideas during the preparation of this manuscript. We are deeply indebted to our parents and family members for their tremendous patience, enthusiastic inspirations in pursuit of this project. We would like to accord special thanks to Mr. Shoeb Siddiqui, Department of Computer Science, Integral University Lucknow, for his valuable time during the preparation of this manuscript. In the last but not the least we are also thankful to Mr. Sunil Saxena, Mr. Jai Raj Kapoor, Mr. H. Rahman and their team of “Ane Books Pvt. Ltd” for their kind cooperation at every stage, without which it would not have been possible to bring out this book in such a fine format. Qazi Shoeb Ahmad Zubair Khan Shadab Ahmad Khan

Contents Preface ....................................................................................................................... v Acknowledgement .................................................................................................... vii 1. Error and Computer Arithmetic ........................................................................... 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Introduction ...................................................................................................... 1 Accuracy of Numbers ....................................................................................... 1 Errors ................................................................................................................ 2 Types of Error ................................................................................................... 2 General Formula for Error .................................................................................. 5 Errors in Numerical Computation ...................................................................... 5 Floating Point Representation of Numbers ....................................................... 9 Arithmetic Operations with Normalized Floating Point Numbers ................... 10

2. Solution of Algebraic and Transcendental Equations ........................................ 17 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18

Solution of Algebraic and Transcendental Equations .................................... 17 Bisection Method or Bolzano Method ........................................................... 17 Iteration Method or Successive Approxi-mation Method .............................. 26 Condition For Convergence of the Iterative Method ..................................... 27 Regula-Falsi Method or Method of False Position ......................................... 31 Order and Rate of Convergence of Regula-Falsi Method ............................... 33 Newton-Raphson Method or Newton's Method ............................................ 38 Order and Rate of Convergence of Newton-Raphson Method ...................... 39 Geometrical Interpretation of Newton-Raphson method ................................ 40 Solution of Simultaneous Linear Algebraic Equations .................................. 47 Gauss-Elimination Method ............................................................................. 48 Gauss-Elimination Method with Pivoting ....................................................... 52 Gauss-Jordan Method .................................................................................... 54 Matrix-Inversion Method ............................................................................... 59 Method of Triangularisation or Method of Factorization ............................... 62 Gauss-Jacobi Method or Jacobi Method of Iteration ..................................... 71 Gauss-Seidel Method ..................................................................................... 77 Lin-Bairstow’s method .................................................................................... 83

x Numerical and Statistical Techniques 3. Finite Differences ............................................................................................... 87 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14

Forward Differences ....................................................................................... 87 Backward Differences ..................................................................................... 89 Central Differences ......................................................................................... 89 Different Types of Operators .......................................................................... 90 Relation Between Operators ........................................................................... 91 Differences of a Polynomial ............................................................................ 93 Factorial Notation ......................................................................................... 108 Reciprocal or Negative Factorial Notation .................................................... 108 Differences of a Factorial Function ............................................................... 109 Polynomial in Factorial Notation ................................................................... 110 Computation of Missing Terms .................................................................... 110 Finite Integration (or Inverse Operator ∆–1) .................................................. 118 Summation of Series ..................................................................................... 119 Montmort’s Theorem .................................................................................... 120

4. Interpolation ..................................................................................................... 128 4.1 Interpolation ................................................................................................. 128 4.2 Newton’s-Gregory Forward Interpolation Formula ....................................... 129 4.3 Error in polynomial Interpolation .................................................................. 131 4 .4 Error in Newton′s Gregory Forward Interpolation Formula ..................... 131 4.5 Newton′s Gregory Backward Interpolation Formula ..................................... 142 4.6 Error in Newton’s Gregory Backward Interpolation formula ......................... 145 4.7 Central Difference Interpolation Formula ...................................................... 155 4.8 Gauss’s Forward Interpolation Formula ........................................................ 156 4.9 Gauss’s Backward Interpolation Formula ..................................................... 158 4.10 Stirling’s formula ........................................................................................... 165 4.11 Bessel’s formula ............................................................................................ 166 4.12 Laplace-Everett formula ................................................................................ 167 4.13 Relation Between Bessel’s and Everett’s Formula ........................................ 168 4.14 Advantages of Central Difference Interpolation Formula ............................. 169 4.15 Interpolation with Unequal Intervals ............................................................ 177 4.16 Divided Differences ...................................................................................... 178 4.17 Properties of Divided Differences ................................................................. 179 4.18 Relation Between Divided Differences and Forward Differences ................. 181 4.19 Newton’s General Interpolation Formula (for Unequal Intervals) or Newton’s .. Divided Difference Interpolation Formula .................................................... 181 4.20 Lagrange’s Interpolation Formula (for Unequal Intervals) ............................ 188 4.21 Inverse Interpolation .................................................................................... 195 4.22 Cubic Spline .................................................................................................. 197

Contents xi

5. Numerical Differentiation and Integration ...................................................... 203 5.1 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14

Numerical Differentiation .............................................................................. 203 Newton’s Forward Difference Formula to get the Derivative ........................ 203 Newton’s Forward Difference Formula to get the Derivative ........................ 203 Newton’s Backward Difference Formula to get the Derivative ..................... 205 Stirling’s Interpolation Formula to get the Derivative ................................... 207 Bessel’s Formula to get the Derivative ......................................................... 209 Numerical Integration ................................................................................... 219 Newton-Cote's Quadrature Formula ............................................................. 220 Trapezoidal Rule (for n = 1) ........................................................................... 220 Simpson's 1/3rd Rule (for n = 2) ..................................................................... 221 Simpson's 3/8th Rule (for n = 3) ...................................................................... 222 Boole's Rule (for n = 4) .................................................................................. 223 Weddle's Rule (for n = 6) ............................................................................... 224 Romberg's Method ....................................................................................... 233 Euler-Maclaurin's Formula ............................................................................ 235 Gaussian Quadrature Formula ...................................................................... 241

6. Numerical Solution of Ordinary Differential Equations ................................. 247 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15

Introduction .................................................................................................. 247 Inttial and Boundary Value Problems ............................................................ 248 Numerical Methods of Solving Ordinary Differential Equations .................. 248 Picard’s Method of Successive Approximations .......................................... 248 Picard’s Method for Simultaneous First Order Differential Equations .......... 254 Taylor’s Series Method ................................................................................. 256 Taylor’s Method for Simultateous First Order Differential Equations .......... 256 Euler’s Method ............................................................................................. 263 Modified Euler’s Method .............................................................................. 267 Runge-kutta Method .................................................................................... 272 First Order Runge-kutta Method .................................................................. 272 Second Order Runge-kutta Method ............................................................. 272 Third Order Runge-kutta Method ................................................................. 272 Fourth Order Runge-kutta Method ............................................................... 273 Runge-kutta Method for Simultaneous First Order Differential Equations 284

7. Curve Fitting .................................................................................................... 288 7.1 7.2 7.3 7.4 7.5

Introduction .................................................................................................. 288 Method of Least Squares ............................................................................. 288 Fitting of a Straight Line by Method of Least Squares ................................. 289 Change of Origin and Scale .......................................................................... 290 Normal Equations for Different forms of Curve ............................................ 290

xii Numerical and Statistical Techniques 8. Regression Analysis ........................................................................................ 303 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

Regression .................................................................................................... 303 Linear Regression ......................................................................................... 303 Lines of Regression ...................................................................................... 303 Properties of Regression Coefficients .......................................................... 304 Angle Between two Lines of Regression ...................................................... 306 Angle Between two Lines of Regression ...................................................... 307 Linearization ................................................................................................. 307 Multiple Regression ..................................................................................... 307

9. Time Series and Forecasting ........................................................................... 321 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11

Introduction .................................................................................................. 321 Analysis of Time Series ................................................................................ 321 Applications of Time series .......................................................................... 322 Components of Time Series .......................................................................... 322 Forecasting ................................................................................................... 323 Forecasting Models ...................................................................................... 323 Forecasting Methods ................................................................................... 324 Measurement of Trend ................................................................................. 325 Measurement of Seasonal Variations ............................................................ 332 Measurement of Cyclical Variations ............................................................. 339 Measurement of Random or Irregular Variations .......................................... 340

10. Test of Significance and Analysis of Variance ............................................... 345 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15

Introduction .................................................................................................. 345 Parameter and Statistic .................................................................................. 345 Sampling Distribution ................................................................................... 345 Standard Error ............................................................................................... 346 Uses of standard error .................................................................................. 346 Hypothesis Testing ...................................................................................... 347 Null Hypothesis ............................................................................................ 347 Alternative Hypothesis ................................................................................ 347 Level of Significance ..................................................................................... 348 Critical Region .............................................................................................. 348 Critical Value ................................................................................................. 348 One tailed and Two Tailed Tests ................................................................... 348 Type-I Error and Type-II Error ...................................................................... 349 Power of the Test .......................................................................................... 350 Procedure for Testing of Hypothesis ............................................................ 350

Contents

10.16 10.17 10.18 10.19 10.20 10.21 10.22 10.23 10.24 10.25 10.26 10.27 10.28 10.29 10.30 10.31 10.32 10.33 10.34 10.35 10.36 10.37 10.38 10.39 10.40 10.41 10.42 10.43 10.44

xiii

Student's t-Test ............................................................................................. 350 Assumptions for Student's t-Test ................................................................. 350 t-Test for Single Mean .................................................................................. 351 t-Test for Difference of Means ...................................................................... 353 Paired t-Test for Difference of Means ........................................................... 347 Z-Test ............................................................................................................ 361 Test of Significance for Attributes ................................................................ 361 Test for Number of Successes ...................................................................... 361 Test for Single Proportion ............................................................................. 362 Test for Difference of Proportions ................................................................ 364 Test of Significance for Variables .................................................................. 370 Test of Significance for Single Mean ............................................................ 370 Test of Significance for Difference of Means ................................................. 373 Test of Significance for Difference of Standard Deviations .......................... 376 F-Test ........................................................................................................... 380 Procedure of F-Test ...................................................................................... 381 Assumptions for F-Test ............................................................................... 381 Critical Values of F-Distribution ................................................................... 382 Chi-Square Test ............................................................................................ 386 Chi-Square Test to Test the Goodness of Fit ................................................ 386 Chi-Square Test to Test the Independence of Attributes ............................. 392 Conditions for χ2 Test ................................................................................... 394 Uses of χ2 Test .............................................................................................. 394 Analysis of Variance ..................................................................................... 401 Assumptions in the Analysis of Variance ..................................................... 401 Technique of Analysis of Variance ............................................................... 401 The basic Principle of Analysis of Variance .................................................. 402 Analysis of Variance in one way Classification ............................................ 402 Analysis of Variance in two way Classification .............................................. 409 Appendix ............................................................................................... 419-428 Index ..................................................................................................... 429-433

Chapter 1

Error and Computer Arithmetic 1.1 INTRODUCTION In this chapter, we examine the sources of various types of errors. A number of different types of errors arise during the process of numerical computation, some are avoidable, and some are not. These errors contribute to the total error in the final result. Errors, in numerical computation, can be made as small as we please, by taking the number to as many figures as we desired. Therefore, we can assume that the calculations are always carried out in such a manner as to make the errors of calculation negligible. 1.2

ACCURACY OF NUMBERS (i) Exact numbers: The numbers in which, there is no uncertainty and no approximation, are said to be exact numbers. 5 6 1 , , , 1.45, 8.30,... are exact numbers. 2 3 4 (ii) Approximate numbers: The numbers which are not exact are approximate numbers. These numbers contain infinitely many digits.

e.g. 3, 4, 6,

1 e.g. 3 = 1.73205 ..., = 0.333333 ...; π = 3.141592 ... are 3 approximate numbers. (iii) Significant figures: The digits used to express a number are called significant digits. Here, we have that all the digits 1, 2, 3, ..., 9 are significant figures and 0 is a significant figure except when it is used to fix the decimal point or to fill the places of unknown digits i.e. 0 may or may not be a significant figure.

Remark: The zeroes used between two non-zero digits are always significant figures.