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4.1 Overview 281 4.2 Port Orgonizotion 283

CHAPTER I3

Lineor Regression 284 13.I Stotistics Review 286

13.2 Lineor Leost-Squores Regression 292,l3.3 Lineorizotion of Nonlineor Relotionships 300 13.4 Computer Applicotions 304

13.5 Cose Siudy: Enzyme Kinetics 307rroDtems 5 | z

CHAPTER I4

Generol lineqr Leosf-Squores qnd Nonlineqr Regression 316

14. I Polynomiol Regression 31 6 14.2 Multiple Lineor Regression 32O

14.3 Generol Lineor Leost Souores 322 14.4 QR Foctorizotion ond the Bockslosh Ooerotor 325 14.5 Nonlineor Regression 326

14.6 Cose Sludy: Fitting Sinusoids 328rrontcms .1.1 /

'I 5.'l lntroduction to Interpotofion 336I5.2 Newron Interpoloring polynomiol 33gj: .j tosron9e tnrerpoloring polynomiol 347rJ.4 tnverse tnterpolotion 350 I J.J txkopolotion ond Oscillotions 351Problems 355 cHAPTER t6

Splines ond piecewise Inferpofofion 359

I6. I lnkoduction to Splines 359 lo.z Linear Splines 361

16.3 Quodroiic Splines 365 16.4 Cubic Splines 368 l9: liTewise Inrerpotorion in MATLAB 374i 6.6 Multidimensionol Interpolotion 37gl6 Z Cose Study: Heot Tronsfer 3g2rrobtems 386

Pnnr Fvr Infegrotion ond Differentiotion 3g9

5.1 Overview 3g9 5.2 Port Orgonizotion 39O

!H,\PTER | 7

Numericof fnfegrofion Formutos Sg2'l Z.J lnhoduction ond Bocrground 393I7.2 Newton-Cotes Formutos 39617.3 The Tropezoidol Rule 39g 17.4 Simpson,s Rules 405

17.5 Higher-Order Newfon_Cotes Formulos 4j jl7 6 lntegration with Unequol Segments 41217.7 OpenMerhods 416 I7 8 Muh,ple Integrols 4j 6

;lJ;:r";udy: Compuring Work wirh Numericol lnregrorion 4j9 i{.'l 8.I Introducfion 426 18.2 Romberq Inieqrotion 427

Numericof Integrotion of Functions

18.3 Gouss Quodroture 432 I8.4 Adoptive Quodroture 439

I8.5 Cose Study: Root-Meon-Squore Current 440 Problems 4

CHAPTER I9

Numericql Differentiqtion 448

19. I Inhoduction ond Bockground 449 19.2 High-Accurocy Differentiotion Formulos 452

I9.3 Richordson Extropolotion 455,l9.4 Derivotives of Unequolly Spoced Doto 457,l9.5 Derivotives ond lntegrols for Dofo with Errors 458 19.6 Portiol Derivotives 459

I9.2 Numericol Differentiotion with MATLAB 460 I9.8 Cose Study: Visuolizing Fields 465 Problems 467

Pnnr 5x Ordinory Differentiol Equotions 473

6.1 Overview 473 6.2 Porl Orgonizofion 477

CHAPTER 20

Initiol-Volue Problems 479

20.I Overview 481 20.2 Euleis Method 481

20.3 lmprovemenls of Euler's Method 487 20.4 Runge-Kutfo Methods 493 20.5 Systems of Equotions 498

20.6 Cose Study: Predotory-Prey Models ond Choos 50A Problems 509

CHAPTER 2I

Adopfive Merhods ond Stiff Systems 514

21 .'l Adoptive Runge-Kutto Methods 514 2l .2 Multistep Methods 521 2l .3 Stiffness 525

2l .4 MATLAB Applicotion: Bungee Jumper with Cord 531 2l .5 Cose Study: Pliny's lntermittent Fountoin 532 rroDlems 3J/

CONTENTS xl

CHAPTER 2

Boundory-Volue Problems 540

2.1 lntrodvction ond Bockground 541 2.2 lhe Shooting Method 545 2.3 Finite-Difference Methods 552 Problems 559

APPENDIX A: EIGENVALUES 565 APPENDIX B: MATLAB BUILT-IN FUNCTIONS 576 APPENDIX €: MATIAB M-FltE FUNCTIONS 578 BIBLIOGRAPHY 579 rNDEX 580

Modqling, CoTpute''i ,qnd Erior Anolysis t.t MoTtvATtoN

What are numerical methods and why should you stridy them?

Numerical methods are techniques by which mathematical problems are formulated so that they can be solved with arithmetic and logical operations. Because digital computers excel at perform.ing such operations. numerical methods are sometimes referred to as computer mathematics.

In the pre-computer era, the time and drudgery of implementing such calculations se.-- riously limited their practical use. However, with the advent of fast, inexpensive digttul computers, the role of numerical methods in engineering and scientific problem solving has exploded. Because they figure so prominently in,:' much of our work, I believe that numerical methods should be a part of every engineer's and scientist's basic education. Just as we a.l must have solid foundations in the other areas of mathematics and science, we should also have a fundamental understanding of numerical methods. In particular, we should have a solid appreciation of both their capabilities and their limitations. Beyond contributing to your overall education.

.thog T9 several additibnat reasons why you shoutO study numerical methods: .",,,,r,,,r,,,r,,,,,,

1. Numerical methods greatly expqld the types of , problems you can address. They are capable of handling large systems of equations. nonlineari-

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