Thompson - Computing for Scientists and Engineers, страница 2

Introduction to Differential Equations7.1 Differential equations and physical systems 222Why are there differential equations? 222Notation and classification 223Homogeneous and linear equations 224Nonlinear differential equations 2257.2 First-order linear equations: World-record sprints 225Kinematics of world-record sprints 226Warming up to the problem 227Program for analyzing sprint data 229Women sprinters are getting faster 2347.3 Nonlinear differential equations: Logistic growth 235The logistic-growth curve 235Exploring logistic-growth curves 238Generalized logistic growth 239221CONTENTS xi7.4 Numerical methods for first-order equations 241Presenting error values 241Euler predictor formulas 242Testing the Euler predictors 244Adams predictor formulas 2457.5 Project 7: Program for solving first-order equations 247Programming the differential equation solver 247Exploring numerical first-order equations 252References on first-order equations 2558.

Second-Order Differential Equations8.1 Forces, second-order equations, resonances 258257Forces and second-order equations 258Mechanical and electrical analogs 259Solving and interpreting free-motion equations 261Forced motion and resonances 2658.2 Catenaries, cathedrals, and nuptial arches 269The equation of the catenary 270Catenaries of various shapes and strengths 273Demonstrating arches 278Practical arches and catenaries 2798.3 Numerical methods for second-order differential equations 279Euler-type algorithms for second-order equations 280Removing first derivatives from second-order linear equations 284Deriving the Noumerov algorithm for second-order equations 2 8 58.4 Project 8A: Progamming second-order Euler methods 287Programming the Euler algorithms 287Euler algorithms and the exponential function 291Euler algorithms and the cosine function 2938.5 Project 8B: Noumerov method for linear second-order equations 294Programming the Noumerov method 294Testing Noumerov for exponentials and cosines 2 9 7The quantum harmonic oscillator 299Noumerov solution of the quantum oscillator 3018.6 Introduction to stiff differential equations 304What is a stiff differential equation? 305The Riccati transformation 306Programming the Riccati algorithm 308Madelung’s transformation for stiff equations 311References on second-order equations 3129.

Discrete Fourier Transforms and Fourier Series9.1 Overview of Fourier expansions 316The uses of Fourier expansions 316Types and nomenclature of Fourier expansions 316315CONTENTSxii9.2Discrete Fourier transforms 3 18Derivation of the discrete transform 318Properties of the discrete transform 320Exponential decay and harmonic oscillation 3229.3The fast Fourier transform algorithm 329Deriving the FFT algorithm 329Bit reversal to reorder the FFT coefficients 332Efficiency of FFT and conventional transforms 3339.4 Fourier series: Harmonic approximations 334From discrete transforms to series 334Interpreting Fourier coefficients 336Fourier series for arbitrary intervals 3369.5 Some practical Fourier series 337The square-pulse function 338Program for Fourier series 340The wedge function 343The window function 345The sawtooth function 3479.6 Diversion: The Wilbraham-Gibbs overshoot 349Fourier series for the generalized sawtooth 350The Wilbraham-Gibbs phenomenon 353Overshoot for the square pulse and sawtooth 356Numerical methods for summing trigonometric series 3599.7Project 9A: Program for the fast Fourier transform 360Building and testing the FFT function 360Speed testing the FFT algorithm 3649.8Project 9B: Fourier analysis of an electroencephalogram 365Overview of EEGs and the clinical record 365Program for the EEG analysis 368Frequency spectrum analysis of the EEG 372Filtering the EEG data: The Lanczos filter 373References on Fourier expansions 37510.

Fourier Integral Transforms10.1 From Fourier series to Fourier integrals 377The transition from series to integrals 378Waves and Fourier transforms 379Dirac delta distributions 37910.2 Examples of Fourier transforms 380Exponential decay and harmonic oscillation 380The square-pulse function 383Fourier transform of the wedge function 384Gaussian functions and Fourier transforms 386Lorentzian functions and their properties 389Fourier integral transform of a Lorentzian 39110.3 Convolutions and Fourier transforms 393Convolutions: Definition and interpretation 393Convoluting a boxcar with a Lorentzian 394Program for convoluting discretized functions 398377CONTENTSxiiiFourier integral transforms and convolutions 401Convolutions of Gaussians and of Lorentzians 402Convoluting Gaussians with Lorentzians: Voigt profile 40610.4 Project 10: Computing and applying the Voigt profile 411The numerics of Dawson’s integral 412Program for series expansion of profile 413Program for direct integration of profile 418Application to stellar spectra 419References on Fourier integral transforms 419EPILOGUE421APPENDIX: TRANSLATING BETWEEN C, FORTRAN,AND PASCAL LANGUAGES423INDEX TO COMPUTER PROGRAMS429INDEX433COMPUTING FORSCIENTISTS ANDENGINEERSPrevious Home NextChapter 1INTRODUCTION TO APPLICABLEMATHEMATICS AND COMPUTINGThe major goal for you in using this book should be to integrate your understanding,at both conceptual and technical levels, of the interplay among mathematical analysis, numerical methods, and computer programming and its applications.

The purpose of this chapter is to introduce you to the major viewpoints that I have abouthow you can best accomplish this integration.Beginning in Section 1.1, is my summary of what I mean by applicable mathematics and my opinions on its relations to programming. The nested hierarchy (arare bird indeed) of computing, programming, and coding is described in Section 1.2. There I also describe why I am using C as the programming language,then how to translate from to Fortran or Pascal from C, if you insist. The text hastwelve projects on computing, so I also summarize their purpose in this section.Section 1.3 has remarks about the usefulness of graphics, of which there are manyin this book.

Various ways in which the book may be used are suggested in Section 1.4, where I also point out the guideposts that are provided for you to navigatethrough it. Finally, there is a list of general references and many references onlearning the C language, especially for those already familiar with Fortran or Pascal.1.1WHAT IS APPLICABLE MATHEMATICS?Applicable mathematics covers a wide range of topics from diverse fields of mathematics and computing. In this section I summarize the main themes of this book.First I emphasize my distinctions between programming and applications of programs, then I summarize the purpose of the diversion sections, some new paths to12INTRODUCTIONfamiliar destinations are then pointed out, and I conclude with remarks about common topics that I have omitted.Analysis, numerics, and applicationsIn computing, the fields of analysis, numerics, and applications interact in complicated ways.

I envisage the connections between the areas of mathematical analysis,numerical methods, and computer programming, and their scientific applications asshown schematically in Figure 1.1.FIGURE 1.1 Mathematical analysis is the foundation upon which numerical methods andcomputer programming for scientific and engineering applications are built.You should note that the lines connecting these areas are not arrows flying upward. The demands of scientific and engineering applications have a large impact onnumerical methods and computing, and all of these have an impact on topics andprogress in mathematics.

Therefore, there is also a downward flow of ideas andmethods. For example, numerical weather forecasting accelerates the developmentof supercomputers, while topics such as chaos, string theory in physics, and neuralnetworks have a large influence on diverse areas of mathematics.Several of the applications topics that I cover in some detail are often found inbooks on mathematical modeling, such as the interesting books by Dym and Ivey,by Meyer, and by Mesterton-Gibbons.

However, such books usually do not emphasize much computation beyond the pencil-and-paper level. This is not enoughfor scientists, since there are usually many experimental or observational data to behandled and many model parameters to be estimated. Thus, interfacing the mathematical models to the realities of computer use and the experience of program writingis mandatory for the training of scientists and engi neers. I hope that by working thematerials provided in this book you will become adept at connecting formalism topractice.1.1WHAT IS APPLICABLE MATHEMATICS?3By comparison with the computational physics books by Koonin (see also Koonin and Meredith), and by Gould and Tobochnik, I place less emphasis on the physics and more emphasis on the mathematics and general algorithms than these authorsprovide. There are several textbooks on computational methods in engineering andscience, such as that by Nakamura.

These books place more emphasis on specificproblem-solving techniques and less emphasis on computer use than I provide here.Data analysis methods, as well as mathematical or numerical techniques that maybe useful in data analysis, are given significant attention in this book. Examples arespline fitting (Chapter 5), least-squares analyses (Chapter 6), the fast Fourier transform (Chapter 9), and convolutions (Chapter 10). My observation is that, as partof their training, many scientists and engineers learn to apply data-analysis methodswithout understanding their assumptions, formulation, and limitations.

I hope toprovide you the opportunity to avoid these defects of training. That is one reasonwhy I develop the algorithms fairly completely and show their relation to other analysis methods. The statistics background to several of the data-analysis methods isalso provided.Cooking school, then recipesThose who wish to become good cooks of nutritious and enjoyable food usually goto cooking school to learn and practice the culinary arts.

After such training they areable to use and adapt a wide variety of recipes from various culinary traditions fortheir needs, employment, and pleasure. I believe that computing — including analysis, numerics, and their applications — should be approached in the same way. Oneshould first develop an understanding of analytical techniques and algorithms. Aftersuch training, one can usually make profitable and enlightened use of numericalrecipes from a variety of sources and in a variety of computer languages.The approach used in this book is therefore to illustrate the processes throughwhich algorithms and programming are derived from mathematical analysis of scientific problems. The topics that I have chosen to develop in detail are those that I believe contain elements common to many such problems.