COMPUTING FOR SCIENTISTS AND ENGINEERS
CONTENTS 1 INTRODUCTION TO APPLICABLE MATHEMATICS AND COMPUTING 1 1 WHAT IS APPLICABLE MATHEMATICS? ANALYSIS, NUMERICS, AND APPLICATIONS COOKING SCHOOL, THEN RECIPES DIVERSIONS AND NEW ROUTES ROADS NOT TAKEN 1 2 COMPUTING, PROGRAMMING, CODING THE C LANGUAGE FOR THE PROGRAMS LEARNING TO PROGRAM IN C TRANSLATING TO FORTRAN OR PASCAL FROM C THE COMPUTING PROJECTS AND THE PROGRAMS WHY AND WHEN YOU SHOULD USE GRAPHICS IMPRESSIVE GRAPHICS, OR PRACTICAL GRAPHICS 1 4 SUGGESTIONS FOR USING THIS BOOK LINKS BETWEEN THE CHAPTERS THE EXERCISES AND PROJECTS REFERENCES FOR THE INTRODUCTION GENERAL REFERENCES REFERENCES ON LEARNING AND USING C ALGEBRA AND COMPUTING WITH COMPLEX NUMBERS THE ALGEBRA OF COMPLEX NUMBERS PROGRAMMING WITH COMPLEX NUMBERS COMPLEX CONJUGATION, MODULUS, ARGUMENT A PROGRAM FOR COMPLEX CONJUGATE AND MODULUS THE COMPLEX PLANE AND PLANE GEOMETRY CARTESIAN AND PLANE-POLAR COORDINATES DE MOIVRE’S THEOREM AND ITS USES 2 3 FUNCTIONS OF COMPLEX VARIABLES COMPLEX EXPONENTIALS: EULER’S THEOREM 3 APPLICATIONS OF EULER’S THEOREM HYPERBOLIC FUNCTIONS AND THEIR CIRCULAR ANALOGS TRAJECTORIES IN THE COMPLEX PLANE 2 4 PHASE ANGLES, VIBRATIONS, AND WAVES PHASE ANGLES AND PHASORS 4 VIBRATIONS AND WAVES 2 5 DIVERSION: INTERPRETING COMPLEX NUMBERS ARE COMPLEX NUMBERS REAL? ANALYTIC CONTINUATION PROJECT 2: PROGRAM TO CONVERT BETWEEN COORDINATES STEPPING INTO THE CORRECT QUADRANT CODING, TESTING, AND USING THE PROGRAM REFERENCES ON COMPLEX NUMBERS MOTIVATION FOR USING SERIES: TAYLOR’S THEOREM THE GEOMETRIC SERIES PROGRAMMING GEOMETRIC SERIES ALTERNATING SERIES TAYLOR’S THEOREM AND ITS PROOF INTERPRETING TAYLOR SERIES TAYLOR EXPANSIONS OF USEFUL FUNCTIONS EXPANSION OF EXPONENTIALS COMPUTING THE EXPONENTIAL SERIES SERIES FOR CIRCULAR FUNCTIONS INVERSE CIRCULAR FUNCTIONS HYPERBOLIC FUNCTION EXPANSIONS LOGARITHMS IN SERIES EXPANSIONS SERIES EXPANSION OF X IN(X) 3 3 THE BINOMIAL APPROXIMATION DERIVING THE BINOMIAL APPROXIMATION APPLICATIONS OF THE BINOMIAL APPROXIMATION LINEARIZED SQUARE-ROOT APPROXIMATIONS FINANCIAL INTEREST SCHEMES DIVERSION: REPETITION IN MATHEMATICS AND COMPUTING ITERATION RECURRENCE RECURSION PROJECT 3: TESTING THE CONVERGENCE OF SERIES CODING AND CHECKING EACH SERIES EXPANSION INCLUDING THE HYPERBOLIC FUNCTIONS FILE OUTPUT AND GRAPHICS OPTIONS THE COMPOSITE PROGRAM FOR THE FUNCTIONS USING THE PROGRAM TO TEST SERIES CONVERGENCE REFERENCES ON POWER SERIES THE WORKING FUNCTION AND ITS PROPERTIES PROPERTIES OF THE WORKING FUNCTION A C FUNCTION FOR HOMER’S ALGORITHM PROGRAMMING THE WORKING FUNCTION DISCRETE DATA AND NUMERICAL MATHEMATICS THE DISCRETENESS OF DATA NUMERICAL MATHEMATICS 4 3 NUMERICAL NOISE IN COMPUTING ROUNDOFF AND TRUNCATION ERRORS UNSTABLE PROBLEMS AND UNSTABLE METHODS ERRORS FROM SUBTRACTIVE CANCELLATION PROGRAM FOR ROOTS OF QUADRATIC EQUATIONS 4 4 HOW TO APPROXIMATE DERIVATIVES FORWARD-DIFFERENCE DERIVATIVES DERIVATIVES BY CENTRAL DIFFERENCES NUMERICAL SECOND DERIVATIVES BETTER ALGORITHMS FOR SECOND DERIVATIVES PROJECT 4A: COMPUTING DERIVATIVES NUMERICALLY DERIVATIVES OF THE EXPONENTIAL FUNCTION DIFFERENTIATING THE COSINE FUNCTION 4 6 NUMERICAL INTEGRATION METHODS TRAPEZOID FORMULA AND PROGRAM FOR INTEGRATION SIMPSON FORMULA AND PROGRAM FOR INTEGRALS INTEGRALS WITH COSINES HIGHER-ORDER POLYNOMIAL INTEGRATION PROJECT 4B: ELECTROSTATIC POTENTIAL FROM A CHARGED WIRE POTENTIALS BY ANALYTICAL INTEGRATION POTENTIALS BY NUMERICAL-INTEGRATION METHODS REFERENCES ON NUMERICAL DERIVATIVES AND INTEGRALS HOW TO FIT CURVES USING SPLINES WHAT IS A SPLINE? PROPERTIES FOR SPLINE FITS DERIVING THE SPLINE EQUATIONS THE SPLINE ALGORITHM 5 2 BOUNDARY CONDITIONS FOR SPLINE FITTING NATURAL SPLINES PROJECT 5: PROGRAM FOR SPLINE FITTING THE MAIN PROGRAM, CUBIC SPLINES THE FUNCTION SPLINEFIT 5 4 INTERPOLATING BY SPLINES INTERPOLATING VALUES AND DERIVATIVES THE C FUNCTION SPLINELNTERP INTERPOLATING WORKING-FUNCTION VALUES AND DERIVATIVES INTERPOLATING COSINE VALUES AND DERIVATIVES 5 5 INTEGRATION METHODS USING SPLINES DERIVING THE INTEGRATION ALGORITHM THE C FUNCTION FOR SPLINE INTEGRATION INTEGRATING THE WORKING FUNCTION AND COSINE 5 6 DIVERSION: COMPUTERS, SPLINES, AND GRAPHICS REFERENCES ON SPLINE FITTING INTRODUCTION TO THE LEAST-SQUARES CRITERION MAXIMUM LIKELIHOOD AND LEAST SQUARES LEAST SQUARES AND THE OBJECTIVE FUNCTION ORTHOGONAL FUNCTIONS AND LINEAR LEAST SQUARES WHAT ARE ORTHOGONAL FUNCTIONS? ORTHOGONALITY AND LEAST SQUARES ERRORS IN BOTH VARIABLES: STRAIGHT-LINE LEAST SQUARES WEIGHTING MODELS CONSTANT RATIO OF WEIGHTS PROPERTIES OF THE LEAST-SQUARES SLOPES 6 4 LEAST-SQUARES NORMALIZATION FACTORS NORMALIZING FITTING-FUNCTION VALUES TO DATA NORMALIZING DATA TO FITTING VALUES THE BEST-FIT OBJECTIVE FUNCTION PROGRAM FOR NORMALIZING FACTORS LOGARITHMIC TRANSFORMATIONS AND PARAMETER BIASES THE ORIGIN OF BIAS PROBABILITY ANALYSIS FOR BIAS DEPENDENCE OF BIAS ON ERROR DISTRIBUTION PROJECT 6: PROGRAM FOR STRAIGHT-LINE LEAST-SQUARES FITS ORGANIZATION OF STRAIGHT-LINE LEAST SQUARES TESTING AND USING THE LEAST-SQUARES PROGRAM REFERENCES ON LEAST-SQUARES ANALYSIS DIFFERENTIAL EQUATIONS AND PHYSICAL SYSTEMS WHY ARE THERE DIFFERENTIAL EQUATIONS? NOTATION AND CLASSIFICATION HOMOGENEOUS AND LINEAR EQUATIONS NONLINEAR DIFFERENTIAL EQUATIONS 7 2 FIRST-ORDER LINEAR EQUATIONS: WORLD-RECORD SPRINTS KINEMATICS OF WORLD-RECORD SPRINTS WARMING UP TO THE PROBLEM PROGRAM FOR ANALYZING SPRINT DATA WOMEN SPRINTERS ARE GETTING FASTER 7 3 NONLINEAR DIFFERENTIAL EQUATIONS: LOGISTIC GROWTH THE LOGISTIC-GROWTH CURVE EXPLORING LOGISTIC-GROWTH CURVES GENERALIZED LOGISTIC GROWTH 7 4 NUMERICAL METHODS FOR FIRST-ORDER EQUATIONS PRESENTING ERROR VALUES EULER PREDICTOR FORMULAS TESTING THE EULER PREDICTORS ADAMS PREDICTOR FORMULAS 7 5 PROJECT 7: PROGRAM FOR SOLVING FIRST-ORDER EQUATIONS PROGRAMMING THE DIFFERENTIAL EQUATION SOLVER EXPLORING NUMERICAL FIRST-ORDER EQUATIONS REFERENCES ON FIRST-ORDER EQUATIONS 8 SECOND-ORDER DIFFERENTIAL EQUATIONS 8 1 FORCES, SECOND-ORDER EQUATIONS, RESONANCES FORCES AND SECOND-ORDER EQUATIONS MECHANICAL AND ELECTRICAL ANALOGS SOLVING AND INTERPRETING FREE-MOTION EQUATIONS FORCED MOTION AND RESONANCES 8 2 CATENARIES, CATHEDRALS, AND NUPTIAL ARCHES THE EQUATION OF THE CATENARY CATENARIES OF VARIOUS SHAPES AND STRENGTHS DEMONSTRATING ARCHES PRACTICAL ARCHES AND CATENARIES NUMERICAL METHODS FOR SECOND-ORDER DIFFERENTIAL EQUATIONS EULER-TYPE ALGORITHMS FOR SECOND-ORDER EQUATIONS REMOVING FIRST DERIVATIVES FROM SECOND-ORDER LINEAR EQUATIONS DERIVING THE NOUMEROV ALGORITHM FOR SECOND-ORDER EQUATIONS 2 8 8 4 PROJECT 8A: PROGAMMING SECOND-ORDER EULER METHODS PROGRAMMING THE EULER ALGORITHMS EULER ALGORITHMS AND THE EXPONENTIAL FUNCTION EULER ALGORITHMS AND THE COSINE FUNCTION 8 5 PROJECT 8B: NOUMEROV METHOD FOR LINEAR SECOND-ORDER EQUATIONS PROGRAMMING THE NOUMEROV METHOD TESTING NOUMEROV FOR EXPONENTIALS AND COSINES THE QUANTUM HARMONIC OSCILLATOR NOUMEROV SOLUTION OF THE QUANTUM OSCILLATOR 8 6 INTRODUCTION TO STIFF DIFFERENTIAL EQUATIONS WHAT IS A STIFF DIFFERENTIAL EQUATION? THE RICCATI TRANSFORMATION PROGRAMMING THE RICCATI ALGORITHM MADELUNG’S TRANSFORMATION FOR STIFF EQUATIONS REFERENCES ON SECOND-ORDER EQUATIONS 9 DISCRETE FOURIER TRANSFORMS AND FOURIER SERIES 9 1 OVERVIEW OF FOURIER EXPANSIONS THE USES OF FOURIER EXPANSIONS TYPES AND NOMENCLATURE OF FOURIER EXPANSIONS DISCRETE FOURIER TRANSFORMS 3 DERIVATION OF THE DISCRETE TRANSFORM PROPERTIES OF THE DISCRETE TRANSFORM EXPONENTIAL DECAY AND HARMONIC OSCILLATION THE FAST FOURIER TRANSFORM ALGORITHM DERIVING THE FFT ALGORITHM BIT REVERSAL TO REORDER THE FFT COEFFICIENTS EFFICIENCY OF FFT AND CONVENTIONAL TRANSFORMS 9 4 FOURIER SERIES: HARMONIC APPROXIMATIONS FROM DISCRETE TRANSFORMS TO SERIES INTERPRETING FOURIER COEFFICIENTS FOURIER SERIES FOR ARBITRARY INTERVALS 9 5 SOME PRACTICAL FOURIER SERIES THE SQUARE-PULSE FUNCTION PROGRAM FOR FOURIER SERIES THE WEDGE FUNCTION THE WINDOW FUNCTION THE SAWTOOTH FUNCTION 9 6 DIVERSION: THE WILBRAHAM-GIBBS OVERSHOOT FOURIER SERIES FOR THE GENERALIZED SAWTOOTH THE WILBRAHAM-GIBBS PHENOMENON OVERSHOOT FOR THE SQUARE PULSE AND SAWTOOTH NUMERICAL METHODS FOR SUMMING TRIGONOMETRIC SERIES PROJECT 9A: PROGRAM FOR THE FAST FOURIER TRANSFORM BUILDING AND TESTING THE FFT FUNCTION SPEED TESTING THE FFT ALGORITHM PROJECT 9B: FOURIER ANALYSIS OF AN ELECTROENCEPHALOGRAM OVERVIEW OF EEGS AND THE CLINICAL RECORD PROGRAM FOR THE EEG ANALYSIS FREQUENCY SPECTRUM ANALYSIS OF THE EEG FILTERING THE EEG DATA: THE LANCZOS FILTER REFERENCES ON FOURIER EXPANSIONS 10 FOURIER INTEGRAL TRANSFORMS 10 1 FROM FOURIER SERIES TO FOURIER INTEGRALS THE TRANSITION FROM SERIES TO INTEGRALS WAVES AND FOURIER TRANSFORMS DIRAC DELTA DISTRIBUTIONS 10 2 EXAMPLES OF FOURIER TRANSFORMS FOURIER TRANSFORM OF THE WEDGE FUNCTION GAUSSIAN FUNCTIONS AND FOURIER TRANSFORMS LORENTZIAN FUNCTIONS AND THEIR PROPERTIES FOURIER INTEGRAL TRANSFORM OF A LORENTZIAN 10 3 CONVOLUTIONS AND FOURIER TRANSFORMS CONVOLUTIONS: DEFINITION AND INTERPRETATION CONVOLUTING A BOXCAR WITH A LORENTZIAN PROGRAM FOR CONVOLUTING DISCRETIZED FUNCTIONS FOURIER INTEGRAL TRANSFORMS AND CONVOLUTIONS CONVOLUTIONS OF GAUSSIANS AND OF LORENTZIANS CONVOLUTING GAUSSIANS WITH LORENTZIANS: VOIGT PROFILE 10 4 PROJECT 10: COMPUTING AND APPLYING THE VOIGT PROFILE THE NUMERICS OF DAWSON’S INTEGRAL PROGRAM FOR SERIES EXPANSION OF PROFILE PROGRAM FOR DIRECT INTEGRATION OF PROFILE APPLICATION TO STELLAR SPECTRA REFERENCES ON FOURIER INTEGRAL TRANSFORMS