Numerical Methods in Engineering and Science: (C, C++, and MATLAB) by B. S. Grewal, ISBN-13: 978-1683921288

Description

Numerical Methods in Engineering and Science: (C, C++, and MATLAB) by B. S. Grewal, ISBN-13: 978-1683921288

[PDF eBook eTextbook]

• Publisher: ‎ Mercury Learning and Information (September 4, 2018)
• Language: ‎ English
• 950 pages
• ISBN-10: ‎ 1683921283
• ISBN-13: ‎ 978-1683921288

This book is intended as an introduction to numerical methods for scientists and engineers. Providing an excellent balance of theoretical and applied topics, it shows the numerical methods used with C, C++, and MATLAB.

* Provides a balance of theoretical and applied topics

* Shows the numerical methods used with C, C++, and MATLAB

Table of Contents:

Chapter 1 Approximations and Errors

in Computation 1

1.1 Introduction 1

1.2 Accuracy of Numbers 2

1.3 Errors 3

1.4 Useful Rules for Estimating Errors 4

Exercises 1.1 7

1.5 Error Propagation 8

1.6 Error in the Approximation of a Function 11

1.7 Error in a Series Approximation 13

1.8 Order of Approximation 14

1.9 Growth of Error 15

Exercises 1.2 16

1.10 Objective Type of Questions 17

Exercises 1.3 17

Chapter 2 Solution of Algebraic and Transcendental Equations 19

2.1 Introduction 20

2.2 Basic Properties of Equations 20

Exercises 2.1 25

2.3 Transformation of Equations 25

2.4 Synthetic Division of A Polynomial By A Linear Expression 29

Exercises 2.2 32

2.5 Iterative Methods 32

2.6 Graphical Solution of Equations 33

Exercises 2.3 37

2.7 Rate of Convergence 38

2.8 Bisection Method 38

2.9 Method of False Position or Regula-Falsi Method or

Interpolation Method 43

2.10 Secant Method 47

2.11 Iteration Method 50

Exercises 2.4 56

2.12 Newton-Raphson Method 57

2.13 Some Deductions From Newton-Raphson Formula 63

Exercises 2.5 66

2.14 Muller’s Method 68

Exercises 2.6 70

vi • CONTENTS

2.15 Roots of Polynomials Equations 70

Exercises 2.7 75

2.16 Multiple Roots 75

2.17 Complex Roots 77

2.18 Lin-Bairstow’s Method 78

2.19 Graeffe’s Root Squaring Method 84

Exercises 2.8 88

2.20 Comparison of Iterative Methods 89

2.21 Objective Type of Questions 90

Exercises 2.9 90

Chapter 3 Solution of Simultaneous Algebraic Equations 93

3.1 Introduction t to Determinants 93

Exercises 3.1 99

3.2 Introduction To Matrices 100

Exercises 3.2 113

3.3 Solution of Linear Simultaneous Equations 114

3.4 Direct Methods of Solution 115

Exercises 3.3 130

3.5 Iterative Methods of Solution 131

Exercises 3.4 145

3.6 Ill-Conditioned Equations 146

Exercises 3.5 148

3.7 Comparison of Various Methods 149

3.8 Solution of Non-Linear Simultaneous Equations 149

Exercises 3.6 152

3.9 Objective Type of Questions 153

Exercises 3.7 153

Chapter 4 Matrix Inversion and Eigenvalue Problem 155

4.1 Introduction 155

4.2 Matrix Inversion 156

4.3 Gauss Elimination Method 156

4.4 Gauss-Jordan Method 157

4.5 Factorization Method 159

4.6 Partition Method 162

4.7 Iterative Method 165

Exercises 4.1 166

4.8 Eigenvalues and Eigenvectors 168

4.9 Properties of Eigenvalues 171

4.10 Bounds for Eigenvalues 172

CONTENTS • vii

4.11 Power Method 174

Exercises 4.2 178

4.12 Jacobi’s Method 179

4.13 Given’s Method 183

4.14 House-Holder’s Method 186

Exercises 4.3 189

4.15 Objective Type of Questions 190

Chapter 5 Empirical Laws and Curve-Fitting 193

5.1 Introduction 193

5.2 Graphical Method 194

5.3 Laws Reducible to the Linear Law 195

Exercises 5.1 199

5.4 Principle of Least Squares 200

5.5 Method of Least Squares 202

Exercises 5.2 208

5.6 Fitting A Curve of the Type 209

5.7 Fitting of Other Curves 212

5.8 Most Plausible Values of Unknowns 216

Exercises 5.3 218

5.9 Method of Group Averages 219

5.10 Laws Containing Three Constants 222

Exercises 5.4 226

5.11 Method of Moments 228

Exercises 5.5 230

5.12 Objective Type of Questions 231

Exercises 5.6 231

Chapter 6 Finite Differences 233

6.1 Introduction 233

6.2 Finite Differences 234

6.3 Differences of A Polynomial 240

Exercises 6.1 241

6.3 Factorial Notation 242

6.5 Reciprocal Factorial Function 246

6.6 Inverse Operator of D 247

6.7 Effect of an Error on a Difference Table 248

Exercises 6.2 251

6.8 Other Difference Operators 252

6.9 Relations Between the Operators 252

6.10 To Find One or More Missing Terms 257

Exercises 6.3 263

6.11 Application to Summation of Series 265

Exercises 6.4 268

6.12 Objective Type of Questions 269

Exercises 6.5 269

Chapter 7 Interpolation 273

7.1 Introduction 274

7.2 Newton’s Forward Interpolation Formula 274

7.3 Newton’s Backward Interpolation Formula 276

Exercises 7.1 283

7.4 Central Difference Interpolation Formulae 286

7.5 Gauss’s Forward Interpolation Formula 287

7.6 Gauss’s Backward Interpolation Formula 288

7.7 Stirling’s Formula 289

7.8 Bessel’s Formula 290

7.9 Laplace-Everett’s Formula 291

7.10 Choice of an Interpolation Formula 292

Exercises 7.2 304

7.11 Interpolation with Unequal Intervals 306

7.12 Lagrange’s Interpolation Formula 306

Exercises 7.3 311

7.13 Divided Differences 313

7.14 Newton’s Divided Difference Formula 314

7.15 Relation Between Divided and Forward Differences 315

Exercises 7.4 319

7.16 Hermite’s Interpolation Formula 320

Exercises 7.5 325

7.17 Spline Interpolation 326

Exercises 7.6 331

7.18 Double Interpolation 331

7.19 Inverse Interpolation 332

7.20 Lagrange’s Method 332

7.21 Iterative Method 334

Exercises 7.7 336

7.22 Objective Type of Questions 337

Exercises 7.8 337

Chapter 8 Numerical Differentiation and Integration 339

8.1 Numerical Differentiation 339

8.2 Formulae for Derivatives 340

8.3 Maxima and Minima of a Tabulated Function 352

Exercises 8.1 355

CONTENTS • ix

8.4 Numerical Integration 358

8.5 Newton-Cotes Quadrature Formula 359

Exercises 8.2 369

8.6 Errors in Quadrature Formulae 372

8.7 Romberg’s Method 375

8.8 Euler-Maclaurin Formula 380

8.9 Method of Undetermined Coefficients 383

8.10 Gaussian Integration 385

Exercises 8.3 390

8.11 Numerical Double Integration 392

Exercises 8.4 393

8.12 Objective Type of Questions 394

Exercises 8.5 394

Chapter 9 Difference Equations 397

9.1 Introduction 397

9.2 Definition 398

9.3 Formation of Difference Equations 399

Exercises 9.1 401

9.4 Linear Difference Equations 401

9.5 Rules for Finding the Complementary Function 402

Exercises 9.2 406

9.6 Rules for Finding the Particular Integral 407

Exercises 9.3 411

9.7 Difference Equations Reducible to Linear Form 412

Exercises 9.4 414

9.8 Simultaneous Difference Equations with Constant Coefficients 414

Exercises 9.5 415

9.9 Application to Deflection of a Loaded String 415

Exercises 9.6 417

9.10 Objective Type of Questions 417

Exercises 9.7 417

Chapter 10 Numerical Solution of Ordinary

Differential Equations 419

10.1 Introduction 420

10.2 Picard’s Method 421

10.3 Taylor’s Series Method 424

Exercises 10.1 429

10.4 Euler’s Method 429

10.5 Modified Euler’s Method 432

Exercises 10.2 437

x • CONTENTS

10.6 Runge’s Method 438

10.7 Runge-Kutta Method 440

Exercises 10.3 447

10.8 Predictor-Corrector Methods 448

10.9 Milne’s Method 448

Exercises 10.4 455

10.10 Adams-Bashforth Method 456

Exercises 10.5 462

10.11 Simultaneous First Order Differential Equations 463

10.12 Second Order Differential Equations 468

Exercises 10.6 472

10.13 Error Analysis 473

10.14 Convergence of a Method 476

10.15 Stability Analysis 476

Exercises 10.7 479

10.16 Boundary Value Problems 479

10.17 Finite-Difference Method 480

10.18 Shooting Method 485

Exercises 10.8 487

10.19 Objective Type of Questions 488

Exercises 10.9 488

Chapter 11 Numerical Solution of

Partial Differential Equations 491

11.1 Introduction 491

11.2 Classification of Second Order Equations 492

Exercises 11.1 493

11.3 Finite Difference Approximations to Partial Derivatives 494

11.4 Elliptic Equations 495

11.5 Solution of Laplace’s Equation 496

11.6 Solution of Poisson’s Equation 508

Exercises 11.2 511

11.7 Solution of Elliptic Equations by Relaxation Method 513

Exercises 11.3 520

11.8 Parabolic Equations 521

11.9 Solution of One Dimensional Heat Equation 522

11.10 Solution of Two Dimensional Heat Equation 530

Exercises 11.4 534

11.11 Hyperbolic Equations 535

11.12 Solution of Wave Equation 535

Chapter 12 Linear Programming 547

12.1 Introduction 548

12.2 Formulation of the Problem 548

Exercises 12.11 552

12.3 Graphical Method 555

12.4 Some Exceptional Cases 562

Exercises 12.2 566

12.5 General Linear Programming Problem 569

12.6 Canonical and Standard Forms of L.P.P. 570

12.7 Simplex Method 573

Exercises 12.3 577

12.8 Working Procedure of the Simplex Method 578

Exercises 12.4 589

12.9 Artificial Variable Techniques 591

12.10 Exceptional Cases 599

Exercises 12.5 602

12.11 Duality Concept 603

Exercises 12.6 606

12.12 Duality Principle 607

Exercises 12.7 612

12.13 Dual Simplex Method 613

Exercises 12.8 618

12.14 Transportation Problem 619

12.15 Working Procedure for Transportation Problems 621

12.16 Degeneracy in Transportation Problems’ 627

Exercises 12.9 629

12.17 Assignment Problem 632

Exercises 12.10 639

12.18 Objective Type of Questions 642

Exercises 12.11 642

Chapter 13 A Brief Review of Computers 645

13.1 Introduction 645

13.2 Structure of a Computer 646

13.3 Computer Representation of Numbers 647

13.4 Floating Point Representation of Numbers 649

13.5 Computer Calculations 652

Exercises 13.1 655

13.6 Program Writing 655

xii • CONTENTS

Chapter 14 Numerical Methods Using C Language 657

14.1 Introduction 657

14.2 An Overview of “C” Features 658

14.3 Bisection Method (Section 2.7) 674

14.4 Regula-Falsi Method (Section 2.8) 676

14.5 Newton Raphson Method (Section 2.11) 679

14.6 Muller’s Method (Section 2.13) 681

14.7 Multiplication of Matrices [Section 3.2 (3)4] 684

14.8 Gauss Elimination Method [Section 3.4(3)] 687

14.9 Gauss-Jordan Method [Section 3.4(4)] 689

14.10 Factorization Method [Section 3.4(5)] 691

14.11 Gauss-Seidal Iteration Method [Section 3.5(2)] 695

14.12 Power Method (Section 4.11) 699

14.13 Method of Least Squares (Section 5.5) 703

14.14 Method of Group Averages (Section 5.9) 706

14.15 Method of Moments (Section 5.11) 708

14.16 Newton’s Forward Interpolation Formula (Section 7.2) 711

14.17 Lagrange’s Interpolation Formula (Section 7.12) 714

14.18 Newton’s Divided Difference Formula (Section 7.14) 716

14.19 Derivatives Using Forward Difference Formulae [Section 8.2 (1)] 718

14.20 Trapezoidal Rule (Section 8.5—I) 724

14.21 Simpson’s Rule (Section 8.5—II) 726

14.22 Euler’s Method (Section 10.4) 727

14.23 Modified Euler’s Method (Section 10.5) 729

14.24 Runge-Kutta Method (Section 10.7) 732

14.25 Milne’s Method (Section 10.9) 734

14.26 Adams-Bashforth Method (Section 10.10) 736

14.27 Solution of Laplace Equation (Section 11.5) 740

14.28 Solution of Heat Equation (Section 11.9) 745

14.29 Solution of Wave Equation (Section 11.12) 748

14.30 Linear Programming—Simplex Method (Section 12.8) 750

Exercises 14.1 754

Chapter 15 Numerical Methods Using C++ Language 757

15.1 Introduction 757

15.2 An Overview of C++ Features 758

15.3 Bisection Method (Section 2.7) 776

15.4 Regula-Falsi Method (Section 2.8) 778

15.5 Newton Raphson Method (Section 2.11) 780

15.6 Muller’s Method (Section 2.13) 781

15.7 Multiplication of Matrices [Section 3.2 (3)4] 784

15.8 Gauss Elimination Method [Section 3.4 (3)] 786

CONTENTS • xiii

15.9 Gauss-Jordan Method [Section 3.4 (4)] 788

15.10 Factorization Method [Section 3.4 (5)] 790

15.11 Gauss-Seidal Iteration Method [Section 3.5 (2)] 793

15.12 Power Method (Section 4.11) 796

15.13 Method of Least Squares (Section 5.5) 799

15.14 Method of Group Averages (Section 5.9) 801

15.15 Method of Moments (Section 5.11) 803

15.16 Newton’s Forward Interpolation Formula (Section 7.2) 805

15.17 Lagrange’s Interpolation Formula (Section 7.12) 807

15.18 Newton’s Divided Difference Formula (Section 7.14) 808

15.19 Derivatives Using Forward Difference Formulae

(Section 8.2) 809

15.20 Trapezoidal Rule (Section 8.5—I) 813

15.21 Simpson’s Rule (Section 8.5—II) 814

15.22 Euler’s Method (Section 10.4) 815

15.23 Modified Euler’s Method (Section 10.5) 816

15.24 Runge-Kutta Method (Section 10.7) 818

15.25 Milne’s Method (Section 10.9) 819

15.26 Adams-Bashforth Method 821

15.27 Solution of Laplace’s Equation (Section 11.5) 823

15.28 Solution of Heat Equation (Section 11.9) 827

15.29 Solution of Wave Equation (Section 11.12) 829

15.30 Linear Programming—Simplex Method (Section 12.8) 830

Exercises 15.1 834

Chapter 16 Numerical Methods Using MATLAB 837

16.1 Introduction 837

16.2 An Overview of MATLAB Features 838

16.3 Bisection Method (Section 2.7) 844

16.4 Regula-Falsi Method (Section 2.8) 846

16.5 Newton Raphson Method (Section 2.11) 848

16.6 Muller’s Method (Section 2.13) 849

16.7 Multiplication of Matrices [Section 3.2 (3)4] 850

16.8 Gauss Elimination Method [Section 3.4 (3)] 851

16.9 Gauss-Jordan Method [Section 3.4 (4)] 852

16.10 Factorization Method [Section 3.4 (5)] 853

16.11 Gauss Siedel Iteration Method [Section 3.5 (2)] 856

16.12 Power Method (Section 4.11) 858

16.13 Method of Least Squares (Section 5.5) 860

16.14 Method of Group Averages (Section 5.9) 861

16.15 Method of Moments (Section 5.11) 863

16.16 Newton’s Forward Interpolation Formula (Section 7.2) 864

16.17 Lagrange’s Interpolation Formula (Section 7.12) 866

16.18 Newton’s Divided Difference Formula (Section 7.14) 867

16.19 Derivatives Using Forward Difference Formula

[Section 8.2] 868

16.20 Trapezoidal Rule (Section 8.5-1) 870

16.21 Simpson’s Rule (Section 8.5-II) 871

16.22 Euler’s Method (Section 10.4) 871

16.23 Modified Euler’s Method (Section 10.5) 872

16.24 Runge-Kutta Method (Section 10.7) 873

16.25 Milne’s Method (Section 10.9) 874

16.26 Adams-Bashforth Method (Section 10.10) 876

16.27 Solution of Laplace’s Equation (Section 11.5) 877

16.28 Solution of Heat Equation (Section 11.9) 881

16.29 Solution of Wave Equation (Section 11.12) 882

16.30 Linear Programming-Simplex Method (Section 12.8) 884

Exercises 16.1 887

Appendix A Useful Information 891

I Basic Information and Errors 891

II Solution of Algebraic and Trancendental Equations 892

III Solution of Simultaneous Algebraic Equations 893

IV Finite Differences and Interpolation 895

V Interpolation 895

VI Numerical Differentiation 897

VII Numerical Integration 898

VIII Number Solution of ordinary Differential Equations 900

IX Number Solution of Partial Differential Equations 901

Appendix B Answers to Exercises 903

Appendix C Bibliography 929

Index 931

B. S. Grewal (late), had written numerous articles and books about mathematics and had received several awards and grants during his career.

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