Thompson - Computing for Scientists and Engineers (523188), страница 79
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V., 285, 313Noumerov algorithm, 285-287program for, 294-299numerical derivatives,see derivatives, numericalnumerical integrals,see integrals, numericalnumerical mathematicscompared to pure mathematics, 111numerical noise, 111-122computer versus hand calculator, 116in quadratic equation solution, 118roundoff errors, 112truncation errors, 112Numerical Recipes, 10, 421numerical recipes, 3and numerical noise, 116unsavory dishes, 241nuptial arch, 279Nussbaumer, H. J., 401, 420Nyquist criterion, 321objective function:definition of, 185dependence on weights, 185for normalization factorsby least squares, 200Ohm’s law resistance, 259Oppenheim, A. V., 334, 345, 375, 376order of a differential equation, 224ordinary differential equations, 224ordinary least squares (OLS), 191orthogonal functions, 185-189and linear least squares, 188for Fourier expansions, 3 19Legendre polynomials, 187Schmidt orthogonalization, 186outliers in least-squares fitting, 184overdamping, 263overshoot, see Wilbraham-Gibbs overshootPaeth, A.
W., 80, 98parameter bias:dependence on error model, 212for Gaussian distribution, 212for small samples, 2 13for uniform distribution, 213Gaussian distribution divergence, 213in slope from log transformation, 211Monte Carlo simulation, 214origin of in log transformation, 209parametric integrationfor Gaussian moments, 183Parseval’s theorem:interpretation of, 320relation to Wiener-Khinchin theorem, 320statement of, 320partial differential equations, 224Pascal:and C,6translating between Pascal and C, 423-428patient MAC, 366pedagogues, modem, 269Peters, R. D., 333, 376phase angle, 25in complex plane, 4 1of forced motion, 266phase-space plot, 264phasor diagram, 42pi to computer accuracy, 46Pippard, A.
B.. 40, 43, 49, 151,258, 264, 269, 313, 386, 420plane-polar coordinatesand complex numbers, 28Plauger, P. J., 8, 15Poincaré map, 264Poisson distribution, 185and normalization by least squares, 202for error model, 2 10Polya, G., 84, 98polynomial wiggle problem, 154potential from a charged wire, 145-151analytical integral, 146integral for, 145numerical integrals, 148-151power series, see Taylor seriespower spectrum of EEG, 372Pratt, W. K., 343, 345, 376predictor formulas:Adams methods, 245Adams trapezoid predictorfor first-order equation, 246Adams-Simpson predictorfor first-order equation, 246central Euler predictorfor first-order equation, 243Euler methods, 242, 280Euler method 1for second-order equation, 280INDEXEuler method 2for second-order equation, 28 1Euler method 3for second-order equation, 281Euler methods summarized, 28 1example of Euler methodsfor second-order equation, 282forward Euler predictorfor first-order equation, 243testing Euler predictorsfor first-order equations, 244testing Euler predictorsfor second-order equations, 29 l-294predictor-corrector methods, 247Press, W.
H., 7, 10, 15, 144, 151,304, 313, 360, 375, 376, 421, 422probability distribution, 184Gaussian, 182moments of, 182,211Poisson, 185uniform, 212program:complex-arithmetic functions, 21conjugate and modulusof complex functions, 25convert between coordinates, 45convoluting discretized functions, 398cosine and sine, 67DFT of oscillator, 326estimating significant digits, 112fast Fourier transform, 360first-order differential equations, 247Fourier analysis of EEG, 368Fourier series, 340geometric series, 54Horner polynomial, 104normalization by least squares, 204Noumerov algorithmfor second-order equations, 294potential from a charged wire, 148power-series expansion, 63, 93quantum harmonic oscillator, 301Riccati transformation, 308roots of quadratic equations, 119second-order Euler algorithms, 287Simpson integral formula, 137spline fitting, 161straight-line least squares, 214trapezoid integral formula, 136Voigt profile, 4 13working function value, derivatives,and integral, 107441world-record sprints, 229programs:caveat emptor, 10error messages from, 11programs and functions, index of, 429-430project, 9-10, 14computing derivatives numerically, 130convert between coordinates, 45electrostatic potential of a wire, 145fast Fourier transform program, 360Fourier analysis of EEG, 365Noumerov methodfor second-order equations, 294second-order Euler methods, 287solving first-orderdifferential equations, 247spline fitting, 161straight-line least-squares fits, 214testing the convergence of series, 85Voigt profile, 4 11proportional errors,2 10Protter, M.
H., 51, 59, 98, 376pure mathematics comparedto numerical mathematics, 111Pythagoras, theorem of, 33testing numerically, 97quadrants, sign of circular functions in, 45quadratic equation solution:program for stable method, 119subtractive cancellation in, 118quantum harmonic oscillator, 299-304Noumerov solution, 301-304quantum leap, 303radians to degrees, 66Rasala, R., 178, 179raster, 347real number, 19recurrence, 84recursion, 84logical problems with, 85Reed, B. C., 191, 219relative errors in differential equationsolutions, 242relaxation time in world-record sprints, 226repetition in mathematics and computing, 83iteration, 84recurrence, 84recursion, 84resonance, 265-269amplitude of, 266442INDEXresonance (continued)described by Lorentzian function, 389in Fourier transform of oscillator, 382maximum amplitude, 267phase of, 266Riccati transformation, 306-311example of, 307program for, 308-311Richardson extrapolation, 133Riemann zeta function, 353ringing, 348Roberts, R.
A., 334,376Romberg integration, 144roundoff errors, 112Rubinstein, R., 178, 179Rucker, R., 77, 98sample programs, 9simple example (geometric series), 55sawtooth function Fourier series, 347-349Schildt, H., 7, 15Schmidt orthogonalization, 186Schrödinger equation, 43, 225for harmonic oscillator, 299Schultz, M. H., 178, 179Schumaker, L.
L., 158, 178, 179second-order differential equations, 258and forced motion, 265complex-number representation, 265for free motion, 261-264for mechanical and electrical systems, 260removing first derivatives, 284-285separable differential equationfor world-record sprints, 228Shammas, N., 8, 15, 423Shannon sampling theorem, 321Siegel, A.
F., 181, 219sigmoid function, see logistic growthsignificant digits, program for, 112Simpson formula:comparison with trapezoid formula, 142composite formula, 141derivation of, 140error estimate, 141for numerical integrals, 140-142graphical interpretation, 141program for, 137, 141testing with cosines, 143sine function:programming series for, 86series for small angles. 66Taylor series for, 65Snell, J. L., 181, 182, 211, 219Solomon, F., 184, 185, 219, 397, 420source term in second-orderdifferential equation, 265Spehlmann, R., 366, 376spline:definition of, 154knot of, 155natural, 160-161order of, 155spline derivatives, 17 1and unstable problems, 174for cosines, 173for working function, 171spline endpoint conditions, 155, 159-161for working function, 160natural spline, 155spline fitting:algorithm for, 158and computer graphics, 178and computer-aided design, 178and type fonts, 178combined with least-squares fitting, 218compared to least squares, 153efficiency of, 175equations, 156-159exact fit to data, 153history of, 178of working function, 155program for, 161properties of fit, 156versus polynomial fit, 153spline integration, 175-177compared to trapezoidand Simpson rules, 175, 177derivation, 175for cosine, 177for working function, 177function for, 176testing the program, 176spline interpolation, 168-174first derivative from, 168for cosines, 173for working function, 170second derivative from, 168third derivative from, 169value from, 168square root, linearized approximations, 78square-pulse function:Fourier integral transform of, 383Fourier series for, 338-340stability and chaos in logistic growth, 239INDEXstandard deviation, 182in proportional-errors model, 210of independent Gaussian errors, 403star notation for convolution, 393Steinmetz, C.
P., 43stellar spectra, Voigt profile for, 419stiff differential equations, 304-312Madelung transformation, 311nature of, 305numerical example of, 306origin of term, 305Riccati transformation, 306stochastic process, 184stopping criterion for first-orderdifferential equations, 247straight-line least squares, 190-199errors in both variables, 190program for, 214-217testing and using program, 217weighting models, 190straight-line least squareswith errors in both variables:error estimates, 199minimum objective function, 198subtractive cancellation, 116-119,in quadratic equation solution 118in spline fitting, 174in variance calculations.
117symmetry conditions for Fourier series, 339system identification, see least-squaresfittingTaylor, A. E., 4, 14, 51, 57, 59, 98Taylor, J. R., 181, 209, 219Taylor series, 65for exponentials, 61for hyperbolic functions, 71for inverse circular functions, 70for logarithms, 72interpreting, 59remainder, 60Taylor’s theorem, 58, 59statement and proof, 58tensor, 406Thompson, D’Arcy W., 235, 255Thompson, W. J., 4, 14,185, 202, 219, 350, 376tomography, 397trajectories in complex plane, 38-41circular, 40translating between C, Fortran.and Pascal, 423-428443trapezoid formula:comparison with Simpson formula, 142composite formula, 136error estimate, 135for numerical integrals, 135-140graphical interpretation, 136program for, 136testing with cosines, 143trigonometric series,summing numerically, 359-360trigonometric series for overshoot, 355truncation errors, 112truncation filtering, 373Tuchinsky, P. M., 235, 255Tufte, E.
R., 11, 15type fonts and spline fitting, 178uniform distribution, 212unstable methods, 114-116unstable problems, 114-116and ill-conditioned matrices, 115and linear equation solution, 114and spline derivatives, 174relation to unstable equilibriumand chaos, 116stiff differential equations are, 305Vandergraft, J.S., 144, 151, 247, 255vanilla ice cream, 3variance calculations,subtractive cancellation in, 117Varner, R. L., 218, 219Verhulst equation, see logistic growthvibrations and waves, 42differential equations for, 269Fourier integral transforms and, 384Voigt profile, 406-411applied to stellar spectra, 4 19by direct integration, 418computation of, 411-419expansion in H functions for, 408-411program for series expansion, 413-418Voigt, Waldemar, 406Wagon, S., 11, 12, 15Weaver, H.
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