A Textbook of Computer Based Numerical and Statistical Techniques

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