A Textbook of Computer Based Numerical and Statistical Techniques

Salient features of this work include: mathematical derivation of each method is given to build the students understanding of numerical analysis; a variety of solved examples are given; computer programs for almost all numerical methods discussed have been presented in 'C' language; error analysis for almost all methods are presented; each chapter begins with an introduction of concerned topic; and, exercise questions provide an opportunity to the students to test their understanding of the concepts. Cover......Page 1 Preface......Page 6 Contents ......Page 8 1.2 Accuracy of Numbers......Page 16 1.3 Errors ......Page 17 1.4 A General Formula for Error ......Page 18 1.4.3 Error in Product of Numbers......Page 19 1.4.5 Inverse Problem......Page 20 1.4.6 Error in Evaluating xk......Page 21 1.5 Error in Series Approximation ......Page 35 Problem Set 1.1 ......Page 38 1.6 Some Important Mathematical Preliminaries ......Page 39 1.7 Floating Point ......Page 40 1.8 Floating Point Arithmetic and their Computation ......Page 41 1.8.1 Arithmetic Operations on Floating Point Numbers......Page 42 Problem Set 1.2......Page 47 2.2.1 Direct Methods......Page 49 2.4 Bisection (or Bolzano) Method ......Page 50 2.4.2 Order of Convergence of Bisection Method......Page 51 2.5 False Position Method (or Regula Falsi Method) ......Page 61 2.5.2 Order (or Rate) of Convergence of False Position Method......Page 62 2.6.1 Procedure For Iteration Method To Find The Root of The Equation f(x) = 0......Page 76 2.6.2 Rate of Convergence of Iteration Method......Page 77 2.7 Newton–Raphson Method (or Newton’s Method) ......Page 82 2.7.2 Order (or Rate) of Convergence of Newton-Raphson Method......Page 83 Problem Set 2.4 ......Page 93 2.8.1 Procedure for Secant Method to Find the Root of f(x)=0......Page 94 2.8.2 Rate or Order of Convergence of Secant Method......Page 95 2.9 Methods for Complex Roots ......Page 101 2.10 Muller’s Method ......Page 102 2.11 Lin Bairstow Method ......Page 106 2.12 Quotient Difference Method ......Page 115 Problem Set 2.5 ......Page 118 3.4 Differences ......Page 119 3.4.1 Forward or Leading Differences......Page 120 3.4.2 Backward or Ascending Differences......Page 124 3.4.3 Central Differences......Page 127 3.4.5 Properties of Operators......Page 128 3.4.6 Relation between Different Operators......Page 129 Problem Set 3.1 ......Page 145 3.5 Fundamental Theorem on Differences of Polynomial ......Page 146 3.6 Estimation of Error By Difference Table......Page 147 3.7 Technique to Determine The Missing Term ......Page 151 Problem Set 3.2 ......Page 156 3.8 Separation of Symbols ......Page 157 3.9 Factorial Notations ......Page 162 3.10 Reciprocal Factorial Notation ......Page 164 3.11.1 Direct Method......Page 165 3.11.2 Method of Synthetic Division......Page 166 3.12 Errors in Polynomial Interpolation ......Page 171 3.13. Differences of Zeros ......Page 172 Problem Set 3.3 ......Page 174 4.1 Introduction ......Page 175 4.2 Newton’s Gregory Formula for Forward Interpolation ......Page 176 Problem Set 4.1 ......Page 186 4.3 Newton’s Gregory Formula for Backward Interpolation ......Page 187 Problem Set 4.2......Page 196 4.4.1 Gauss Forward Difference Formula......Page 197 4.4.2 Gauss Backward Difference Formula......Page 198 4.4.3 Stirling’s Formula......Page 199 4.4.4 Bessel’s Interpolation Formula......Page 200 4.4.5 Laplace-Everett’s Formula......Page 201 Problem Set 4.3 ......Page 206 Gauss Backward ......Page 207 Problem Set 4.4 ......Page 213 Stirling’s Formula......Page 214 Problem Set 4.5 ......Page 220 4.5 Bessel’s ......Page 221 Problem Set 4.6 ......Page 228 4.6 Laplace Everetts ......Page 229 Problem Set 4.7 ......Page 237 5.2 Lagrange’s Interpolation Formula ......Page 239 Problem Set 5.1 ......Page 250 5.3 Errors in Polynomial Interpolation ......Page 251 5.3.1 Error in Lagrange’s interpolation formula......Page 252 5.3.2 Inverse Interpolation......Page 254 5.3.3 Expression of Function as a Sum of Partial Fractions......Page 258 Problem Set 5.2 ......Page 259 5.4 Divided Difference ......Page 260 5.4.1 Properties of Divided Differences......Page 261 5.5 Newton’s Divided Difference Formula ......Page 262 Problem Set 5.3 ......Page 273 5.6. Hermite’s Interpolation Formula ......Page 274 Problem Set 5.4 ......Page 281 5.7.1 Some Remarkable Points about Chosen Different Interpolation Formulae......Page 284 5.7.2 Approximation of Function......Page 285 5.7.3 Spline Interpolation......Page 300 5.7.4 Cubic Spline Interpolation for Equally and Unequally Spaced Values......Page 301 Problem Set 5.5 ......Page 306 6.2 Numerical Differentiation ......Page 309 6.2.1 Derivation Using Newton’s Forward Interpolation Formula......Page 310 6.2.2 Derivatives Using Newton’s Backward Difference Formula......Page 311 6.2.4 Derivative Using Newton’s Divided Difference Formula......Page 313 Problem Set 6.1 ......Page 328 6.4 General Quadrature Formula ......Page 330 6.6 Simpson’s One-third Rule ......Page 331 6.8 Boole’s Rule ......Page 332 6.9 Weddle’s Rule ......Page 333 6.10 Euler-Maclaurin’s Formula ......Page 342 Problem Set 6.2 ......Page 345 7.2 Taylor’s Method ......Page 347 7.3 Picard’s Method of Successive Approximations......Page 351 7.4 Euler’s Method ......Page 357 7.5 Euler’s Modified Method ......Page 358 Problem Set 7.1 ......Page 366 7.6 Runge-Kutta Method ......Page 367 Milne’s Method......Page 376 7.8 Automatic Error Monitoring ......Page 384 7.9 Stability in the Solution of Ordinary Differential equation ......Page 385 Problem Set 7.2 ......Page 387 8.2 Gauss-Elimination Method ......Page 388 8.3 Gauss-Elimination with Pivoting Method ......Page 390 8.5 Iterative Refinement of The Solution by Gauss elimination Method......Page 391 8.6 Iterative Method for Solution of Simultaneouslinear Equation......Page 392 8.6.1 Jacobi’s Method or Gauss-Jacobi Method......Page 393 8.6.2 Guass-Seidel Method......Page 397 Problem Set 8.1......Page 400 9.2 Principle of Least Squares ......Page 402 9.2.1 Fitting of Straight Line......Page 403 9.2.3 Change of Scale......Page 404 9.2.5 Fitting of the Curve y=ax+bx2......Page 410 9.2.6 Fitting of the Curve y = ax+b/x ......Page 411 9.2.7 Fitting of the Curve......Page 412 Problem Set 9.1 ......Page 420 9.3.5 To obtain the Equation of Line of Regression of y on x......Page 422 9.3.6 To Obtain the Equation of Line of Regression of x on y......Page 423 9.3.7 Another Form of Equations of Lines of Regression......Page 426 9.3.8 Some Properties of Regression Coefficients......Page 430 9.4 Error of Prediction ......Page 436 9.5 Multiple Linear Regression ......Page 437 Problem Set 9.2 ......Page 439 10.2 Times Series Graph ......Page 440 10.3 Component of Time Series ......Page 441 10.4 Analysis of Time Series ......Page 442 10.4.1 Analysis of Trend or Secular Trend......Page 443 10.4.2 Analysis of Seasonal Variation......Page 452 10.5 Importance of Time Series......Page 460 Problem Set 10.1 ......Page 461 10.7.2 Multiplicative Model......Page 463 10.9 Smoothing of Curve ......Page 464 11.1.1 Difference between Diagrams and Graphs......Page 466 11.2 Line Diagram ......Page 467 11.3 Bar Diagram ......Page 468 11.4 One Dimensional Diagram ......Page 471 11.5 Two Dimensional Diagrams ......Page 472 11.7 Pictograms ......Page 475 11.9.1 Graphs of Frequency Distribution......Page 476 11.10 Statistical Quality Control......Page 479 11.10.2 Types of Quality Control......Page 481 11.11 Control Charts......Page 482 11.13 Types of Control Chart ......Page 483 11.13.1 Control Chart for Variable......Page 484 11.13.2 Control Chart for Attributes......Page 488 Problem Set 11.1 ......Page 503 12.1 Introduction ......Page 507 12.2 Some Important Definitions ......Page 508 12.3 Understanding the Type of Test ......Page 509 12.5 Standard Error ......Page 511 12.6 Test of Significance for Large Samples ......Page 512 Problem Set 12.1 ......Page 536 12.7.1 Chi-Square (χ2) Test......Page 538 12.7.2 Student’s t-distribution......Page 553 12.7.3 Snedecor’s Variance Ratio Test or F-test......Page 560 12.7.4 Fisher’s Z-test......Page 561 Problem Set 12.2 ......Page 563 Elements of Real Programming Languages......Page 568 Compiler Terminology......Page 569 Basic Data Types and Operators......Page 570 Function Calls......Page 571 Statements And Control Flow......Page 572 Boolean Expressions......Page 573 Functions and Program Structure......Page 575 Assignment Operators......Page 576 Pointers and Arrays......Page 577 13.3 Programming for Bisection Method ......Page 578 13.5 Programming for False Position Method ......Page 581 13.7 Programming for Iteration Method ......Page 583 13.8 Algorithm for Newton’s Raphson Method ......Page 586 13.9 Programming for Newton Raphson Method ......Page 587 13.10 Programming for Muller’s Method ......Page 588 13.11Algorithm For Newton’s Forward Interpolation method......Page 590 13.12 Program for Constructing Difference Table ......Page 591 13.13 Programming For Newton’s Forward Interpolation method......Page 592 13.15 Programming for Newton’s Backward Interpolation method......Page 594 13.16 Algorithm for Gauss Forward Interpolation Method ......Page 596 13.17 Programming for Gauss Forward Interpolation method ......Page 597 13.19 Programming for Gauss Backward Interpolation Method......Page 599 13.21 Programming for Stirling’s Method ......Page 602 13.23 Programming for Bessel’s Method ......Page 605 13.24 Algorithm for Laplace Everett Method ......Page 607 13.25 Programming for Laplace Everett Method ......Page 608 13.27 Programming for Lagrange’s Interpolation Method ......Page 611 13.29 Programming for Trapezoidal Rule ......Page 613 13.30 Algorithm for Simpson’s 1/3 Rule ......Page 614 13.31 Programming for Simpson’s 1/3 Rule......Page 615 13.33 Programming For Simpson’s 3/8 Rule ......Page 616 13.34 Algorithm for Fitting A Straight Line of the Form Y = a + bX ......Page 618 About the Book: Application of Numerical Analysis has become an integral part of the life of all the modern engineers and scientists. The contents of this book covers both the introductory topics and the more advanced topics such as partial differential equations. This book is different from many other books in a number of ways. Salient Features: Mathematical derivation of each method is given to build the students understanding of numerical analysis. A variety of solved examples are given. Computer programs for almost all numerical methods discussed have been presented in C̀ ̀langu.