# Introduction to numerical methods

**INTRODUCTION, APPROXIMATION & ERRORS**

**Chapter 01.01 Introduction to numerical methods 1**

Multiple-choice test 7

Problem set 9

**Chapter 01.02 Measuring errors 11**

True error 11

Relative true error 12

Approximate error 13

Relative approximate error 14

Significant digits 15

Multiple -choice test 17

Problem set 19

**Chapter 01.03 Sources of error 21**

What is round off error? 21

What problems can be created by round off errors? 21

What is truncation error? 22

Can you give me other examples of truncation error? 23

Multiple-choice test 27

Problem set 29

**Chapter 01.04 Binary representation of numbers 33**

Multiple-choice test 40

Problem set 42

**Chapter 01.05 Floating point representation 43**

Multiple-choice test 51

Problem set 53

**Chapter 01.06 Propagation of errors 54**

Multiple-choice test 57

**Chapter 01.07 Taylor theorem revisited 59
**Multiple-choice test 67

## DIFFERENTIATION

**Physical problems**

Chapter 02.00A Physical problem - general engineering 69

Chapter 02.00B Physical problem - chemical engineering 71

Chapter 02.00D Physical problem - computer engineering 73

Chapter 02.00E Physical problem - electrical engineering 77

Chapter 02.00F Physical problem - industrial engineering 81

Chapter 02.00G Physical problem - mechanical engineering 85

**Chapter 02.01 Primer on differential calculus (View it
on the web)**

Multiple-choice test 89

Problem set 91

**Chapter 02.02 Differentiation of continuous functions
93**

Forward difference approximation of the first derivative 93

Backward difference approximation of the first derivative 96

Forward difference approximation from the Taylor series 97

Finite difference approximation of higher derivatives 100

Multiple-choice test 105

Problem set 107

**Chapter 02.03 Differentiation of discrete functions 109**

Forward difference approximation of the first derivative 109

Direct fit polynomials 111

Lagrange polynomial 113

Multiple-choice test 115

Problem set 118

## NONLINEAR EQUATIONS

**Physical problems**

Chapter 03.00A Physical problem - general engineering 120

Chapter 03.00B Physical problem - chemical engineering 124

Chapter 03.00C Physical problem - civil engineering 127

Chapter 0.3.00D Physical problem - computer engineering 133

Chapter 03.00E Physical problem - electrical engineering 136

Chapter 03.00F Physical problem – industrial engineering 139

Chapter 03.00G Physical problem - mechanical engineering 145

**Chapter 03.01 Solution of quadratic equations 149
**Multiple-choice test 152

Problem set 154

**Chapter 03.03 Bisection method of solving a nonlinear
equation 156**

Bisection method 156

Algorithm for the bisection method 159

Advantages of bisection method 162

Drawbacks of bisection method 162

Multiple-choice test 165

Problem set 167

**Chapter 03.04 Newton-Raphson method of solving a
nonlinear equation 169**

Introduction 169

Derivation 169

Algorithm 170

Drawbacks of the Newton-Raphson method 173

What is an inflection point? 174

Derivation of Newton Raphson method from Taylor series 177

Multiple-choice test 178

Problem set 180

**Chapter 03.05 Secant method of solving nonlinear
equations 182**

What is the secant method and why would I want to use it instead of the

Newton-Raphson method? 182

Multiple-choice test 187

Problem set 189

## SIMULTANEOUS LINEAR EQUATIONS

**Physical problems**

Chapter 04.00A Physical problem - general engineering 191

Chapter 04.00B Physical problem - chemical engineering 194

Chapter 04.00C Physical problem - civil engineering 196

Chapter 04.00D Physical problem - computer engineering 201

Chapter 04.00E Physical problem - electrical engineering 206

Chapter 04.00F Physical problem – industrial engineering 212

Chapter 04.00G Physical problem - mechanical engineering 215

**Chapter 4.1 Introduction to matrix algebra 221**

What is a matrix? 221

What are the special types of matrices? 222

Square matrix 223

Upper triangular matrix 223

Lower triangular matrix 223

Diagonal matrix 224

Identity matrix 224

Zero matrix 224

Tridiagonal matrices 225

When are two matrices considered to be equal? 225

How do you add two matrices? 226

How do you subtract two matrices? 227

How do I multiply two matrices? 228

What is a scalar product of a constant and a matrix? 230

what is a linear combination of matrices ? 231

What are some of the rules of binary matrix operations ? 231

Transpose of a matrix 234

Symmetric matrix 234

Matrix algebra is used for solving system of equations. Can you illustrate

this concept? 235

Can you divide two matrices? 237

Can I use the concept of the inverse of a matrix to find the solution of a set

of equations [A] [X] = [C]? 238

How do I find the inverse of a matrix? 238

If the inverse of a square matrix [A] exists, is it unique? 241

Multiple-choice test 242

Problem set 245

**Chapter 04.06 Gaussian elimination 249**

How are a set of equations solved numerically? 249

Forward elimination of unknowns 250

Back substitution 251

Are there any pitfalls of Naïve Gauss elimination method? 252

Round-off error 256

What are the techniques for improving Naïve Gauss elimination method?

258

How does Gaussian elimination with partial pivoting differ from Naïve

Gauss elimination? 258

Can we use Naïve Gauss elimination methods to find the determinant of a

square matrix? 261

What if I cannot find the determinant of the matrix using Naive Gauss

elimination method, for example, if I get division by zero problems during

Naïve Gauss elimination method? 262

Multiple-choice test 264

Problem set 267

**Chapter 04.07 LU decomposition 26**9

I hear about LU decomposition used as a method to solve a set of

simultaneous linear equations? What is it and why do we need to learn

different methods of solving a set of simultaneous linear equations ? 269

How do I decompose a non-singular matrix [A], that is, how do I find

[A] = [L] [U]? 271

How do I find the inverse of a square matrix using LU decomposition?

275

Multiple-choice test 279

Problem set 283

**Chapter 04.08 Gauss-Seidel method 285**

Why do we need another method to solve a set of simultaneous linear

equations? 285

The above system of equations does not seem to converge. Why? 290

Multiple-choice test 295

Problem set 299

## INTERPOLATION

**Physical problems**

Chapter 05.00A Physical problem - general engineering 300

Chapter 05.00B Physical problem - chemical engineering 302

Chapter 05.00C Physical problem - civil engineering 306

Chapter 05.00D Physical problem - computer engineering 309

Chapter 05.00E Physical problem - electrical engineering 312

Chapter 05.00F Physical problem - industrial engineering 315

Chapter 05.00G Physical problem - mechanical engineering 317

**Chapter 05.01 Background of interpolation**

Multiple-choice test 321

**Chapter 05.02 Direct method of interpolation 323**

What is interpolation? 323

Direct method 324

Multiple-choice test 331

Problem set 333

**Chapter 05.03 Newton’s divided difference interpolation
335**

What is interpolation? 335

Newton’s divided difference polynomial method 335

Linear interpolation 336

Quadratic interpolation 338

General form of Newton’s divided difference polynomial 341

Multiple-choice test 346

Problem set 348

**Chapter 05.05 Spline method of interpolation 350
**What is interpolation? 350

Linear spline interpolation 353

Quadratic splines 355

Multiple-choice test 360

Problem set 363

**Chapter 05.06 Extrapolation is a bad idea 365
Chapter 05.07 Higher order interpolation is a bad idea 369
Chapter 05.08 Why do we need splines? 372
Chapter 05.10 Shortest path of a robot 375**

## REGRESSION

**Physical problems**

Chapter 06.00A Physical problem - general engineering 380

Chapter 06.00B Physical problem - chemical engineering 384

Chapter 06.00C Physical problem - civil engineering 387

Chapter 06.00D Physical problem - computer engineering 390

Chapter 06.00E Physical problem - electrical engineering 393

Chapter 06.00F Physical problem - industrial engineering 397

Chapter 06.00G Physical problem - mechanical engineering 399

**Chapter 06.01 Statistics background of regression
analysis 404**

Review of statistical terminologies 404

Elementary statistics 404

A brief history of regression 408

**Chapter 06.02 Introduction of regression analysis 410
**What is regression analysis? 410

Comparison of regression and correlation 411

Uses of regression analysis 411

Abuses of regression analysis 411

Extrapolation 411

Least squares methods 414

Why minimize the sum of the square of the residuals? 414

Multiple-choice test 416

Problem set 418

**Chapter 06.03 Linear regression 419**

Why minimize the sum of the square of the residuals? 419

Multiple-choice test 432

Problem set 434

**Chapter 06.04 Nonlinear models for regression 436**

Nonlinear models using least squares 436

Exponential model 436

Growth model 440

Polynomial models 442

Linearization of data 446

Exponential model 446

Logarithmic functions 449

Power functions 452

Multiple-choice test 457

Problem set 459

**Chapter 06.05 Adequacy of models for regression 463**

Quality of fitted model 463

Caution in the use of r2 467

What else should I check for the adequacy of the model in example 1?

467

Adequacy of coefficient of regression 469

Problem set 470

## INTEGRATION

**Physical problems**

Chapter 07.00A Physical problem - general engineering 473

Chapter 07.00B Physical problem - chemical engineering 476

Chapter 07.00C Physical problem - civil engineering 479

Chapter 07.00D Physical problem - computer engineering 485

Chapter 07.00E Physical problem - electrical engineering 496

Chapter 07.00F Physical problem – industrial engineering 501

Chapter 07.00G Physical problem - mechanical engineering 505

**Chapter 07.01 Primer on integration (View it on the
web)**

Multiple-choice test 509

Problem set 511

**Chapter 07.02 Trapezoidal rule of integration 514**

What is integration? 514

What is the trapezoidal rule? 514

Derivation of the trapezoidal rule 515

Multiple-segment trapezoidal rule 521

Error in multiple-segment trapezoidal rule 527

Multiple-choice test 530

Problem set 532

**Chapter 07.03 Simpson’s 1/3 rule of integration 535**

What is integration? 535

Simpson’s 1/3 rule 535

Multiple-segment Simpson’s 1/3 rule 542

Error in multiple-segment Simpson’s 1/3 rule 545

Multiple-choice test 547

Problem set 549

**Chapter 07.05 Gauss quadrature 551**

What is integration? 551

Gauss quadrature rule 552

Derivation of two-point Gaussian quadrature rule 553

Higher point Gaussian quadrature formulas 555

Arguments and weighing factors for n-point Gauss quadrature rules 556

Multiple-choice test 565

Problem set 568

**Chapter 07.06 Integrating discrete functions 570**

What is integration? 570

Integrating discrete functions 571

Trapezoidal rule for discrete functions with unequal segments 575

Problem set 578

**Chapter 07.07 Integrating improper functions 581**

What is integration? 581

What is an improper integral? 582

Problem set 592

## ORDINARY DIFFERENTIAL EQUATIONS

**Physical problems**

Chapter 08.00A Physical problem - general engineering 593

Chapter 08.00B Physical problem - chemical engineering 597

Chapter 08.00C Physical problem - civil engineering 599

Chapter 08.00D Physical problem - computer engineering 601

Chapter 08.00E Physical problem - electrical engineering 605

Chapter 08.00F Physical problem – industrial engineering 610

Chapter 08.00G Physical problem - mechanical engineering 616

**Chapter 08.01 Primer for ordinary differential
equations (View it on web)**

Multiple-choice test 622

Problem set 624

**Chapter 08.02 Euler’s method for ordinary differential
equations 626**

What is Euler’s method? 626

Derivation of Euler’s method 627

Multiple-choice test 635

Problem set 638

**Chapter 08.03 Runge-Kutta 2nd order method 642**

What is the Runge-Kutta 2nd order method? 643

Heun’s method 645

Midpoint method 645

Ralston’s method 646

How do these three methods compare with results obtained if we found

f'(x,y) directly? 649

How do we get the 2nd order Runge-Kutta method equations? 650

Multiple-choice test 653

Problem set 656

**Chapter 08.04 Runge-Kutta 4th order method 660**

What is the Runge-Kutta 4th order method? 660

How does one write a first order differential equation in the above form?

660

Multiple-choice test 667

Problem set 671

**Chapter 08.05 On Solving higher order equations 675**

Problem set 684

**Chapter 08.07 Finite difference method 686**

What is the finite difference method? 686

Multiple-choice test 694

Problem set 699

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