COMPUTING IN APPLIED SCIENCE

William J. Thompson

University of North Carolina, Chapel Hill

John Wiley & Sons

New York

Chichester

Brisbane

Toronto

Singapore

CONTENTS

MODULE A APPLICABLE MATHEMATICS |

Al

Introduction to Applicable Mathematics

What is applicable mathematics? Motivation for using computers;

Against the abuse of computer power

A2

Complex Numbers and Complex Exponentials

A2.1

Definitions: The algebra of complex numbers

4
Rules for complex numbers; Computers and complex numbers;
Complex numbers and geometry; Complex conjugation, modulus,
argument

A2.2

The complex plane: De Moivre 's theorem

10
Cartesian and plane-polar coordinates; De Moivre's theorem

A2.3

Complex exponentials: Euler's theorem

13
Deriving Euler's theorem; Applying Euler's theorem

A2.4

Hyperbolic functions

15

Definition of hyperbolic functions; Circular-hyperbolic analogs

A2.5

Phase angles and vibrations

18

A2.6

Diversion: The interpretation of complex numbers

Problems with complex numbers

19

20

Finding buried treasure; Proofs of De Moivre's theorem; Exponentials and the

FFT; Exponentialfun; Graphing hyperbolicfunctions; Analytic continuation

References on complex numbers

A3

Power Series Expansions

A3.1

Motivation for using power series

21

23
23

t Items in italics indicate extensive problems.

xv

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Contents

A3.2

Convergence of power series

24
The geometric series; Geometric series summed

A3.3

Taylor series and their interpretation

27
Proof of Taylor's theorem; Interpreting Taylor series

A3.4

Taylor expansions of useful functions

29
Expansion of exponentials; Series for circular functions;
Hyperbolic function expansions

A3.5

T h e binomial approximation
33
The binomial approximation derived; Interpreting the binomial
approximation; Applying the binomial approximation

A3.6

Diversion: Financial interest schemes

Problems with power series

35

36

Complex geometric series; Proof by induction; Numerical geometric series,

Series by analogy; Numerical exponentials; Logarithms in series; Euler and
Maclaurin; Numerical cosines and sines; Financing solar energy

References on power series

39

A4

Numerical Derivatives, Integrals,

and Curve Fitting
40

A4.1

The discreteness of data

40
Numerical mathematics; Discreteness

A4.2

Numerical noise
41
Round-off error; Truncation error; Unstable problems and
unstable methods; Subtractive cancellation; Numerical evaluation
of polynomials

A4.3

Approximation of derivatives
47
Forward-difference derivatives; Central-difference derivatives;
Error estimates for numerical derivatives; Numerical second
derivatives

A4.4

Numerical integration
51
Trapezoid formula; The Simpson formula; Comparing trapezoid
and Simpson formulas

A4.5

Diversion: Analytic evaluation by computer

54

Contents
A4.6

Curve fitting by splines

55
Introduction to splines; Derivation of spline formulas; Algorithm
for spline fitting; Spline properties; Development of splines and
computers

A4.7

The least-squares principle

61
Choice of least squares; Derivation of least-squares equations; Use
of orthogonal functions; Fitting averages and straight lines
Problems in numerical analysis

68

Round-off and truncation noise; Numerical convergence of series; Subtractive

cancellation in quadratics; Comparison ofpolynomial algorithms; Numerical
integration algorithms; Integral approximations to series; Natural splines have
minimal curvature; Program for cubic splines; Orthogonal polynomials for
least-squares fit; Stability of least-squares algorithms

References on numerical analysis

71

A5

Fourier Expansions

73

A5.1

Overview of Fourier expansions

73
Transformations; Nomenclature of Fourier expansions

A5.2

Discrete Fourier transforms

75
Derivation of the discrete transform; Properties of the discrete
transform

A5.3

Fourier series: Harmonic approximations

78'
From discrete transforms to series; Interpreting Fourier coefficients

A5.4

Examples of Fourier series

80 .
Square pulses; Fourier coefficients and symmetry conditions; The
wedge function; Window functions; Convergence of Fourier
series

A5.5

Diversion: The Gibbs phenomenon

87

A5.6

Fourier series for arbitrary intervals

Interval scaling; Magnitude scaling

88

A5.7

Fourier integral transforms

89
From Fourier series to Fourier integrals; Applying Fourier integral
transforms

xviii
A5.8

Contents
A fast Fourier transform algorithm
93
Derivation of FFT algorithm; Reordering the FFT coefficients;
Comparing FFT with conventional transforms
Problems on Fourier expansions

98

Relating Fourier coefficients through algebra; Relating Fourier coefficients

through geometry; Smoothing window functions; Gibbs phenomenon and
Lanczos damping factors; Dirac delta function; FFT in remote-sensing
applications

References on Fourier expansions

100

A6

Introduction to Differential Equations

From the particular to the general

A6.1

Differential equations and physical systems

101
Why differential equations? Notation and classification;
Homogeneous and linear equations

A6.2

Separable differential equations

104
Relating student scores and work; Generalizing separable
equations

A6.3

First-order linear equations: World-record sprints

107
Kinematics of world-record sprints; Limbering up; Physics and
physical activity; General first-order differential equation

A6.4

Diversion: T h e logarithmic century

111
Interpreting exponential behavior; Bell's decibels

A6.5

Nonlinear differential equations

1 14
The logistic growth curve; Exploring logistic growth

A6.6

Numerical methods for first-order equations

1 18
Predictor formulas; Initial values for solutions; Stability of
numerical methods
Problems on first-order differential equations

101

123

Acceleration and speed in sprints; World-class sprinters; Attenuation in solar

collectors; Predictions from exponential growth; Belles, bells and decibels;
Reaping Nature's bounty; The struggle for survival; Unstable methods for
differential equations

References on differential equations

128

Contents

A7

Second-Order Differential Equations

130

Why are second-order equations so common?

A7.1

Cables and hyperbolic functions

131
Getting the hang of it; Interpreting the cable parameter; Solving
the cable differential equation; Exercises with catenaries

A7.2

Diversion: History of the catenary

A7.3

Second-order linear differential equations

136
Mechanical and electrical analogs; Solving the equations for free
motion; Discussion of free-motion solutions

A7.4

Forced motion and resonances . 142

Differential equation with a source term; Alternative treatment by
Fourier transforms; Resonant oscillations; The Lorentzian function

A7.5

Electricity in nerve fibers

147
Modeling nerve fibers; Solution of the axon potential

A7.6

Numerical methods for second-order equations

150
Euler approximations; Numerov's method for linear equations;
Comparisons of Euler and Numerov algorithms

A7.7

Solution of stiff differential equations

155
What is a stiff differential equation? The Riccati transformation

136

Problems on second-order differential equations

158

Cables and arcs; Suspension bridges; Properties of Lorentzians; Electricity

in axons; Stability of Euler approximations; Madelung transformations for
stiff equations

References on differential equations

162

A8

Applied Vector Dynamics

163

A8.1

Kinematics in Cartesian and polar coordinates

163
Polar coordinate unit vectors; Velocity and acceleration in polar
coordinates

A8.2

Central forces and inverse-square forces 166

Angular momentum conservation: Kepler's first law; Areal
velocity: Kepler's second law

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Contents
i

A8.3

Satellite orbits
169
Inverse-square force differential equations; Analytic solution of
orbit equations; Some geometry of ellipses; Relation of period to
axes: Kepler's third law

A8.4

Diversion: Kepler's Harmony of the World

A8.5

Summary of Keplerian orbits

176
Kepler's three laws; The inverse-square laws from Kepler's laws
Problems on vector dynamics

175

178

Earth-bound projectiles; Electrostatic-force orbits; Solar properties;

Comets; Interstellar travel

References on vector dynamics

180

MODULE L LABORATORIES IN COMPUTING f

LI

Introduction to the Computing Laboratories

182
Analysis and simulation by computer; Coding is not programming
is not computing; The programming languages; Exploring with
the computer; References on languages

L2

Conversion Between Polar

and Cartesian Coordinates

L2.1

185

Cartesian coordinates from polar coordinates

185
Pascal program for conversion to Cartesian coordinates; Fortran
program for conversion to Cartesian coordinates; Exercises on
Cartesian from polar coordinates; Cartesian coordinates from polar
coordinates

L2.2

Polar coordinates from Cartesian coordinates

187
Pascal program for conversion to polar coordinates; Fortran
program for conversion to polar coordinates; Exercises on polar
from Cartesian coordinates; Polar coordinatesfrom Cartesian coordinates

L3

Numerical Approximation of Derivatives

L3.1

Forward and central difference methods

Extrapolation to the limit

| Items in italics indicate extensive programming exercises.

192
193

Contents

L3.2

Exercises in numerical differentiation

194

Numerical derivatives program structure; Exercises on numerical

derivatives; Numerical derivatives of simple functions; Extrapolating to
the limit for derivatives; Second derivatives numerically estimated; Other

differentiation techniques
References on numerical derivatives

197

L4

An Introduction to Computer Graphics

Why printer graphics?

198

L4.1

Plotting using printers

199
Formulas for scales and origins; Printer plotting procedure
structure

L4.2

Sample programs for printer plots

201
Printer plots from Pascal; Printer plots from Fortran; Plotting
exercises; Graphic examples, Improving your image; A pot-pourri of plots

L4.3

Other graphics techniques

209
Video-screen and interactive graphics; Static graphics
References on computer graphics

210

L5

Electrostatic Potentials by Integration

212

L5.1

Analytic derivation of line-charge potential

212
Electrostatic potential of a line charge; On-axis potential of a line
charge; Symmetries and reflections; Scaling the line-charge
potential formulas

L5.2

Line-charge potential using trapezoid formula

216
Structure of the electrostatic potential program; Analytic evolution of
the potential; Trapezoid program for line-charge potential

L5.3

Potential integral from Simpson's formula

219

Simpson 's formula exercises; Simpson program for line-charge potential

L5.4

Displaying equipotential distributions

220
Equipotentials of a line charge; Programming equipotentials; Using
logarithmic scales for displays

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Contents

L6

Monte-Carlo Simulations

L6.1

Generating and testing pseudo-random numbers

223
Power-residue random-number generator; Random numbers in a

223

given range; Program for array ofpseudo-random numbers; The random

walk
L6.2

Stimulating simulations in mathematics

227
Simulation of round-off errors; Random numbers and arithmetic
error; Estimating integrals and areas; Estimating n by MonteCarlo simulation

L6.3

The approach to thermodynamic equilibrium

229
Analytic method; Entropy and the approach to equilibrium;
Monte-Carlo method; Program for approach to equilibrium

L6.4

Simulation of nuclear radioactivity

235
Analytic formula for radioactivity; Monte-Carlo simulation;
Simulating radioactivity; Other Monte-Carlo simulations
Random references on Monte-Carlo methods

238

L7

Spline Fitting and Interpolation

239

L7.1

Sample programs for spline fitting

239
Structure of the spline procedures; Spline fitting from Pascal;
Spline fitting from Fortran

L7.2

Examples using splines

246
Boundary condition and interpolation exercises; Natural and other
spline boundary conditions; Interpolation using cubic splines; Exercises on
spline derivatives; Derivatives from cubic splines; Spline integration
exercises; Integration by cubic splines

References on spline fitting

249

L8

Least-Squares Analysis of Data

250

L8.1

Straight-line fits with errors in both variables

Straight-line least squares; Least-squares formulas

L8.2

Sample programs for straight-line fitting

253
Straight-line fitting in Pascal; Fitting to straight lines from Fortran

250

Contents

L8.3

Quarks, radiocarbon dating, solar cells, and warfare

257

Evidence for fractional charges; Least-squares analysis of fractional

charge data; Radiocarbon dating and Egyptian antiquities; Regression
analysis of radiocarbon and historical dates; Solar cell efficiency in
space; Degrading performance of solar cells; Is war on the increase?
Least-squares analysis of battle deaths

A minimal reading list on least squares

263

L9

Fourier Analysis of an EEG

265

L9.1

Introduction to encephalograplry
266
What is an EEG? Salmon EEG and socioeconomics;
The clinical record; Fourier analysis program structure

L9.2

Frequency spectrum analysis of the EEG 269

Coding the Fourier amplitude calculation; Fourier transform spectrum;
FFT programs in Pascal and Fortran; Predicting voltages from
Fourier amplitudes; Recomposing EEGs; Testing Wiener-Khinchin

L9.3

The Nyquist criterion and noise

278
The Nyquist criterion; Testing the Nyquist criterion; Effects of noise;
Analyzing noise effects

L9.4

Autocorrelation analysis of the EEG 280

Properties of autocorrelations; Noise and autocorrelations;
Autocorrelation analysis program

References on Fourier analysis and the EEG

L10

Analysis of Resonance Line Widths

283

285

L10.1 A brief introduction to atomic clocks

285
Atomic motions as clocks; Resonances and clocks
L10.2 T h e inversion resonance in ammonia
287
Microwave absorption experiments; Data for the ammonia
inversion resonance
L10.3 Fourier-transform analysis of a resonance
289
Derivation of the transform relation; Interpreting the Fourier
transform; Resonance analysis program structure; Exercises on
resonance analysis; Resonance line width program

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Contents

LI0.4 Error analysis for the Fourier transform

294
Finite-range-of-data errors; Finite-step-size errors; Error analysis
for integral transforms

References on resonances and atomic clocks

Lll

Space-Vehicle Orbits and Trajectories

296

298

LI 1.1 Space vehicles, satellites and computers

298
Uses of space vehicles; Communications satellites; Data on Earth
satellites
LI 1.2

Numerical methods for orbits and trajectories

301
For observables to orbital parameters; Program for orbits; Numerical
integration for trajectories; Structure of the space-vehicle program;
Program for trajectories; Files for space-vehicle data

LI 1.3 Display of satellite orbits

309
Geometry of the orbit display; Illusory ellipses
References on space vehicles and satellites

INDEX

313

310