Có 220+ tài liệu thuộc chủ đề "tư liệu lập trình"
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Evaluation of Functions. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call North America only),or send email to [email protected] (outside North America).. You evaluate a rational function like R(x. q ν x ν...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. Economize NMANY power series coefficients e[0..NMANY-1] in the range (a, b) into NFEW coefficients d[0..NFEW-1].. Equivalent to economizing the series, we could instead have evaluated f (x) at enough points to construct its Chebyshev approximation...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. In § 5.8 and § 5.10 we learned how to find good polynomial approximations to a given function f (x) in a given interval a ≤ x ≤ b. q k x k ≈ f(x)...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. cos x/(1 + e x ) in the interval (0, π). The iterations do not converge in the order that the figure suggests: In fact, it is the second iteration that is best (has smallest...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. Midy, P., and Yakovlev, Y. 2 of The Art of Computer Programming (Reading, MA: Addison-Wesley) [see solutions to exercises 4.2.1.16 and 4.6.4.41].. 5.5 Recurrence Relations and Clenshaw’s Recurrence Formula. e − x − xE n...
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If the coefficients a, b, c, are complex rather than real, then the above formulas still hold, except that in equation (5.6.4) the sign of the square root should be chosen so as to make. Equation (5.6.7) is numerically robust for x ≥ 0. Equation (5.6.8) is of course valid only for x ≥ 1.. There are two sources of...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. Easy, right? The definition of the derivative, the limit as h → 0 of. There are two sources of error in equation (5.7.1), truncation error and roundoff error. The truncation error comes from higher terms...
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The Chebyshev polynomial of degree n is denoted T n (x), and is given by the explicit formula. (There also exist inverse formulas for the powers of x in terms of the T n ’s — see equations . The Chebyshev polynomials are orthogonal in the interval. The polynomial T n (x) has n zeros in the interval. At all...
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If you have obtained the Chebyshev coefficients that approximate a function in a certain range (e.g., from chebft in § 5.8), then it is a simple matter to transform them to Chebyshev coefficients corresponding to the derivative or integral of the function. m − 1 are the coefficients that approximate a function f in equation (5.8.9), C i are the...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call North America only),or send email to [email protected] (outside North America).. Special Functions. Special functions are sometimes called higher transcendental functions (higher than what?) or...
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The gamma function is defined by the integral. When the argument z is an integer, the gamma function is just the familiar factorial function, but offset by one,. The gamma function satisfies the recurrence relation. This scheme is entirely specific to the gamma function, seemingly plucked from thin air. c N , the gamma function is given by. Often the...
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which is related to the gamma function by B(z, w. Returns the value of the beta function B(z, w).. Abramowitz, M., and Stegun, I.A. 6.2 Incomplete Gamma Function, Error. The incomplete gamma function is defined by P (a, x. 1 (6.2.2) The incomplete gamma function P (a, x) is monotonic and (for a greater than one or so) rises from...
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is Euler’s constant. #define EULER Euler’s constant γ.. if (x <= 0.0) nrerror("Bad argument in ei");
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1976, Journal of Research of the National Bureau of Standards , vol. 6.10 Dawson’s Integral. e t 2 dt (6.10.1). It is first convenient to shift the summation index to center it approximately on the maximum of the exponential term. Advantage is also taken of the symmetry of the coefficients e − (n 0 h) 2 by breaking the summation...
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Cody, W.J., Pociorek, K.A., and Thatcher, H.C. Elliptic integrals occur in many applications, because any integral of the form Z. R(t, s) dt (6.11.1). (a 1 + b 1 t)(a 2 + b 2 t)(a 3 + b 3 t)(a 4 + b 4 t) (6.11.2) where we have written the quartic s 2 in factored form. one of the...
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6.12 Hypergeometric Functions 271. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call North America only),or send email to [email protected] (outside North America).. Erd ´elyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G....
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. incomplete beta function I x(a,b). The incomplete beta function I x (a, b) for five different pairs of (a, b). 6.4 Incomplete Beta Function, Student’s Distribution, F-Distribution, Cumulative Binomial Distribution. The incomplete beta function is...
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6.5 Bessel Functions of Integer Order. This section and the next one present practical algorithms for computing various kinds of Bessel functions of integer order. For integer order, however, the routines in this section (and § 6.6) are simpler and faster. For any real ν, the Bessel function J ν (x) can be defined by the series representation. For ν...
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6.6 Modified Bessel Functions of Integer Order. The modified Bessel functions I n (x) and K n (x) are equivalent to the usual Bessel functions J n and Y n evaluated for purely imaginary arguments. In the region x n, however, the modified functions have quite different behavior than the Bessel functions,. The modified functions evidently have exponential rather than...
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if (n <. Abramowitz, M., and Stegun, I.A. 6.7 Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions. Many algorithms have been proposed for computing Bessel functions of fractional order numerically. The routines given here are rather complicated, but they can be recommended wholeheartedly.. Ordinary Bessel Functions. Three of the relations come from two continued fractions, one of which...