Thompson - Computing for Scientists and Engineers (523188), страница 78
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D., 12, 14, 178, 179forced motion and resonances, 265-269Fortran:and C,7translating between Fortran and C,423428Fourier analysis of EEG, 365-375Fourier expansions:overview of, 3 16-3 18,types and nomenclature, 3 16-3 18Fourier integral transforms:analytical convolutions, 401406and Dirac delta distributions, 379convolution of, 393-411examples of, 380-393for exponential decay, 380-382for harmonic oscillation, 382from Fourier series. 377-379of Gaussian distribution, 386-389of Lorentzian function, 391-393of square-pulse function, 383of wedge function, 384overview, 3 18Fourier series, 334-349and Fourier integral transforms, 377-379and symmetry conditions, 339derived from DFT, 334-336examples of, 337-349for arbitrary intervals, 336generalized sawtooth, 350-353generalized sawtooth properties, 352in complex-exponential form, 335in cosine and sine form, 335interpreting coefficients, 336overview, 317program for, 340-343sawtooth function, 347-349sawtooth related to square pulse, 347square-pulse function, 338-340wedge function, 343-345wedge related to square pulse, 345window function, 345-347window related to square pulse, 346free motion:critical damping, 263damping parameter for, 262natural frequency for, 262second-order differential equations for, 261Full Width at Half Maximum (FWHM), 183function in C language:ANALYT, 251,290,297ANALYT for Schrödinger equation, 301ANALYTRb, 310ANALYTy, 310INDEXfunction in C language (continued)bitrev, 363CAdd, 22CConjugate, 26CDiv, 22CModulus, 26CMult, 22convolve_arrays, 400CosPoly, 68CSub, 22Der_1C, 109Der_1F, 109Der_23CD, 110Der_25CD, 110Der_2CCD, 109DIST, 233FDawson, 415FFT, 362FSsawtooth, 342FSsquare, 341FSwedge, 341FSwindow, 342FUNC, 251,290,296FUNC for Schrödinger equation, 301FUNCRb, 310FUNCy, 310Hone, 416Horner4Poly, 166Horner_Poly, 105Homer_Poly_2, 109Hthree, 417Htwo, 416Hzero, 416LeastSquares, 216MakeCartesian, 48MakePolar, 48Norrr0bj, 206PSarccos, 95PSarcsin, 96PScos, 94PSexp, 94PSln, 96PSsin, 95quadroots, 120SimpInt, 138,149SinPoly, 69SplineFit, 163SplineInt, 165SplineInterp, 165trap, 417TrapInt, 138,149Vs, 150y, 139, 150437yw, 166YWInt for trapezoid and Simpson, 139ywInt for spline, 166functions, index of, 429-430fundamental frequency in Fourier series, 336fusion-energy devices and Voigt profile, 419FWHM:additivity for Lorentzian convolutions, 404of Gaussian distribution, 183of Lorentzian function, 389Galilei, Galileo, 273, 279, 313Gauss, C.
F., 182Gauss plane, see complex planeGaussian distribution, 182-183, 210, 386and Heisenberg Uncertainty Relations,388and maximum likelihood. 182as Dirac delta distribution, 389compared with Lorentzian function, 390convolution of, 402convolution with Lorentzian, 406-411FFT of discretized, 387Fourier integral transform of, 386-389fourth moment of, 212Gaussian elimination in spline fits, 158Gaussian quadrature, 144Gear, C. W., 312, 313Gehani, N., 8, 15 ,423generalized logistic equation, 239generalized logistic growth, 239-24 1curves of, 240generalized sawtooth,Fourier series for, 350-353geometric series, 52-56definition and properties, 52for deriving DFT, 3 19for DFT of general exponential, 323integral to derive logarithm series, 57program for, 53Gibbs, J. W., 349, 376Gibbs phenomenon,see Wilbraham-Gibbs overshootGilbert, D., 277, 313Gleick.
J., 239, 255Gould, H., 3, 14graphics, 11-12and Mathematica, 12Great Fire of Londonand rebuilding St. Paul’s, 279Haberman, R., 239, 255, 264, 313Hamming, R. W., 375, 376, 397, 420438INDEXHarbison, S. P., 8, 15harmonic analyzer, 349harmonic expansion, see Fourier seriesharmonic oscillation:discrete Fourier transform of, 325Fourier integral transform, 382harmonics in Fourier series, 336Hartley transform, 334harvesting effects on logistic growth, 238Hasse, R. W., 236, 255Heisenberg Uncertainty Relations, 388Hermite polynomials, 300Hofstadter, D.
R., 85, 98homogeneous differential equation, 224Honig, E., 278, 279, 313Hooke, Robert, 279Hooke’s law, 259Homer polynomial:algorithm for, 103for cosine and sine, 67for spline integration, 176of limited use for series, 105program for, 104Hosmer, D. W., 241, 255Huygens, Christian, 279hyperbolic functions, 34-37cosh, 34graphs of, 37 1sign rule and circular functions, 35sinh, 34tanh, 36Taylor series for, 71imaginary number, 18, 19independent diagonal weighting model(IDWM), 192, 193, 196, 197with constant weights (IDWMC), 192induction:and recurrence, 84for convolution of Gaussians, 403for convolution of Lorentzians, 405for convolution of Voigt profiles, 408geometric series by, 52Taylor’s theorem proof by, 59infinite loop, see loop, infiniteIngard, K.
U., 43, 49, 269, 313, 386, 420initial conditions of differential equations,223integrals, analytical, by Mathematica, 99integrals, numerical, 133-144comparison of trapezoidand Simpson rules, 142for potential from a charged wire, 145higher-order methods, 144Simpson formula, 140testing with cosines, 143trapezoid formula, 135with constant stepsize, 134-135integration by splines, see spline integrationinterpolation by splines,see spline interpolationinverse circular functions,Taylor series for, 70Isobe, T., 199, 218iteration, 84Jaffe, A.
J., 181, 218Jain, M. K., 247, 255 , 304, 312, 313Keller, J. B., 226, 255Keller model of world-record sprints, 226Kellison, S. G., 83, 98Kernighan, B. W., 11, 15, 85, 98Kerrigan, J. F., 8, 15, 423kinematics of sprinting,see world-record sprintsKinsella, A., 218Kirchhoff’s second law, 260knot, see splineKoenig, A., 8, 15Koonin, S. E., 3, 14Körner, T.
W., 318, 350, 376Kronecker delta and Diracdelta distributions. 380L’Hôpital’s rule:used for DFT of exponential decay, 323used for DFT of harmonic oscillation, 325Lancaster, P., 178, 179Lanczos filter, 373-375Landau, R. H., 421, 422least-squares criterion, 182-185least-squares fitting,see also linear least squares:and maximum likelihood, 182combined with splines, 218compared to splines, 181disadvantages of, 184outlier influence, 184weights in, 185least-squares normalization factors,see normalization factors by leastsquaresLegendre and least-squares criterion, 182Legendre polynomials, 187Lewis, Carl.
233INDEXLichten, W., 181, 218Lichtenberg, D. B., 235, 255Lindstrom, P. A., 83, 98linear least squares,see also straight-line least squares:and discrete Fourier transforms, 3 19and orthogonal functions, 185, 188equations for coefficients, 188links between chapters, 13logarithm functions:programming series for, 90Taylor series for, 72-75logarithmic transformation:of Lorentzian function, 393parameter bias in, 208logarithmic transformationand parameter bias, 214logistic distribution, logistic growth,235-24 1and ecology, 235curve of, 236differential equation for, 235dimensionless form, 237equilibrium condition, 236generalized, 239-24 1graphs of, 237harvesting effects, 238origins of, 235point of inflexion, 238predicting extinction, 239properties of, 238-239stability and chaos, 239various terms for, 236loop, infinite, see infinite loopLorentzian trajectory in complex plane, 39Lorentzian function for resonance, 268Lorentzian functions, 389-393binning effects on, 396compared with Gaussian distribution, 390convolution of, 403convolution with Gaussian, 406411convolution with window function, 394definition of, 389Fourier integral transform of, 391-393FWHM of, 389normalized, 405resonance frequency, 389lumped-parameter analysisof world-record sprints, 226Lyons, L., 181, 219Macdonald, J.
R., 190, 191, 219Maclaurin series, see Taylor series439Madelung transformationfor stiff differential equations, 311mandala for FFT algorithm, 330Maron, M. J., 117, 144, 151Mathematica, 5, 20, 49, 52, 421:for ordinary differential equations, 224mathematical modeling, 2maximum likelihoodand least-squares fitting, 182, 185mechanical-electrical analogs, 259-261Menai Strait, 277Meredith, D. C., 3, 14Mesterton-Gibbons, M., 2, 14Meyer, W. J., 2, 14Michelson, Albert, 349Mihalas, D., 404, 419, 420Miller, W., 69, 98Modi, J.
J., 5, 14modulus, program for, 25moment of inertiaand least-squares straight line, 190Monte Carlo simulation:for error estimating, 218for parameter bias, 214Müldner, T., 8, 15, 423Nakamura, S., 3, 14, 144, 151, 304, 313natural frequency for free motion, 262natural spline, see also spline endpointconditions, 160-161compared to exact endpoint conditions,161minimal curvature property of, 160negative feedback and logistic growth, 235Newton’s equationas pair of first-order equations, 258Newton, Isaac:inventing differential equations, 22 1and uniform-density catenary, 279as Keeper of the Mint, 80nonlinear differential equations,225, 235-24 1catenary equation, 270normal distribution,see Gaussian distributionnormalization factors by least squares,199-208best-fit objective function, 203dependence on weighting, 202for world-record sprints, 229normalizing data to fitting function, 201normalizing fitting function to data, 200Poisson distribution of errors, 202440INDEXnormalization factors by least squares(continued)program for, 204-208normalization of Lorentzian functions, 405notations for derivatives, 223, 241Noumerov, B.
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