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Lập Trình C# all Chap "NUMERICAL RECIPES IN C" part 25

Tham khảo tài liệu 'lập trình c# all chap "numerical recipes in c" part 25', công nghệ thông tin phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 6.9 Fresnel Integrals Cosine and Sine Integrals 255 6.9 Fresnel Integrals Cosine and Sine Integrals Fresnel Integrals The two Fresnel integrals are defined by C x fx cos t2 dt Jo V2 7 x dt o 6.9.1 The most convenient way of evaluating these functions to arbitrary precision is to use power series for small x and a continued fraction for large x. The series are n2 x5 m4 x9 C x x - U 5 2 U 9 s x i vxL - i V _xl i V -xiL 27 3 1 W 7-3 2 11 -5 6.9.2 There is a complex continued fraction that yields both S x and C x simultaneously 1 i C x iS x - erf z z 1 - i x 6.9.3 where Z erfc z -LIZ -L -L. y y z z z z z 2z 7 1 1.2 3.4 p V 2z2 1 - 2z2 5 - 2z2 9 - 6.9.4 In the last line we have converted the standard form of the continued fraction to its even form see 5.2 which converges twice as fast. We must be careful not to evaluate the alternating series 6.9.2 at too large a value of x inspection of the terms shows that x 1.5 is a good point to switch over to the continued fraction. Note that for large x C x I---sin x2 2 x 2 S x 2 - 1 cos x 6.9.5 Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 Thus the precision of the routine frenel may be limited by the precision of the library routines for sine and cosine for large x. 256 Chapter 6. Special Functions include math.h include complex.h define EPS 6.0e-8 define MAXIT 100 define FPMIN 1.0e-30 define XMIN 1.5 define PI 3.1415927 define PIBY2 PI 2.0 Here EPS is the relative error MAXIT is the maximum number of iterations allowed FPMIN is a number near the smallest representable floating-point number XMIN is the dividing line between using the series and continued fraction. define TRUE 1 define ONE Complex 1.0 0.0 void frenel float x float s float c Computes the Fresnel integrals S x and C x for all real x. void nrerror char error_text int k n odd float a ax fact pix2 sign sum sumc sums term test fcomplex b cc d h del cs ax fabs x if ax sqrt FPMIN Special case avoid failure of convergence s 0.0