《数字信号处理》教学参考资料（Numerical Recipes in C，The Art of Scientific Computing Second Edition）Chapter 06.3 Special Functions 6.3 Exponential Integrals

团购合买资源类别：文库，文档格式：PDF，文档页数：5，文件大小：63.07KB

222 Chapter 6. Special Functions Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). CITED REFERENCES AND FURTHER READING: Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications, New York), Chapters 6, 7, and 26. Pearson, K. (ed.) 1951, Tables of the Incomplete Gamma Function (Cambridge: Cambridge University Press). 6.3 Exponential Integrals The standard definition of the exponential integral is En(x) = ∞ 1 e−xt tn dt, x > 0, n = 0, 1,... (6.3.1) The function defined by the principal value of the integral Ei(x) = − ∞ −x e−t t dt = x −∞ et t dt, x > 0 (6.3.2) is also called an exponential integral. Note that Ei(−x) is related to −E1(x) by analytic continuation. The function En(x) is a special case of the incomplete gamma function En(x) = xn−1Γ(1 − n, x) (6.3.3) We can therefore use a similar strategy for evaluating it. The continued fraction — just equation (6.2.6) rewritten — converges for all x > 0: En(x) = e−x 1 x + n 1 + 1 x + n + 1 1 + 2 x + ··· (6.3.4) We use it in its more rapidly converging even form, En(x) = e−x 1 x + n − 1 · n x + n + 2 − 2(n + 1) x + n + 4 − ··· (6.3.5) The continued fraction only really converges fast enough to be useful for x >∼ 1. For 0 < x <∼ 1, we can use the series representation En(x) = (−x)n−1 (n − 1)! [− ln x + ψ(n)] − ∞ m=0 m=n−1 (−x)m (m − n + 1)m! (6.3.6) The quantity ψ(n) here is the digamma function, given for integer arguments by ψ(1) = −γ, ψ(n) = −γ + n −1 m=1 1 m (6.3.7)

6.3 Exponential Integrals 223 where y=0.5772156649...is Euler's constant.We evaluate the expression(6.3.6) in order of ascending powers of x: 1 En(x)=- (-x)n-2 1-n) -2-m)1+-n1②-…+ (-1)(n-21 a-血x+j--y+-n+ 1·n +2n++ (6.3.8) The first square bracket is omitted when n=1.This method of evaluation has the 83g advantage that for large n the series converges before reaching the term containing (n).Accordingly,one needs an algorithm for evaluating (n)only for small n, n s20-40.We use equation (6.3.7),although a table look-up would improve efficiency slightly. Amos [1]presents a careful discussion of the truncation error in evaluating equation(6.3.8),and gives a fairly elaborate termination criterion.We have found that simply stopping when the last term added is smaller than the required tolerance (Nort server works about as well. Two special cases have to be handled separately: Americ computer, University Press. THE ART e-z Eo(x)= (6.3.9) 9 Program En(0)= n-1 n>1 The routine expint allows fast evaluation of En()to any accuracy EPS within the reach of your machine's word length for floating-point numbers. The to dir only modification required for increased accuracy is to supply Euler's constant with enough significant digits.Wrench [2]can provide you with the first 328 digits if OF SCIENTIFIC COMPUTING(ISBN 1988-19920 necessary! #include 10-521 #define MAXIT 100 Maximum allowed number of iterations. #define EULER 0.5772156649 Euler's constant #define FPMIN 1.0e-30 Close to smallest representable floating-point number. Numerical Recipes 43108 #define EPS 1.0e-7 Desired relative error,not smaller than the machine pre- cision. float expint(int n,float x) (outside 膜 Evaluates the exponential integral En(z). Software. void nrerror(char error_text[]); int i,ii,nm1; float a,b,c,d,del,fact,h,psi,ansi nm1=n-1; if(n<0Ilx<0.0Il(x==0.02数(n=0I川n==1))) nrerror("bad arguments in expint"); else if (n ==0)ans=exp(-x)/x; Special case. else if(x==0.0)ans=1.0/nm1; Another special case. else

6.3 Exponential Integrals 223 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). where γ = 0.5772156649 ... is Euler’s constant. We evaluate the expression (6.3.6) in order of ascending powers of x: En(x) = − 1 (1 − n) − x (2 − n) · 1 + x2 (3 − n)(1 · 2) −··· + (−x)n−2 (−1)(n − 2)! + (−x)n−1 (n − 1)! [− ln x + ψ(n)] − (−x)n 1 · n! + (−x)n+1 2 · (n + 1)! + ··· (6.3.8) The first square bracket is omitted when n = 1. This method of evaluation has the advantage that for large n the series converges before reaching the term containing ψ(n). Accordingly, one needs an algorithm for evaluating ψ(n) only for small n, n 1 (6.3.9) The routine expint allows fast evaluation of En(x) to any accuracy EPS within the reach of your machine’s word length for floating-point numbers. The only modification required for increased accuracy is to supply Euler’s constant with enough significant digits. Wrench [2] can provide you with the first 328 digits if necessary! #include #define MAXIT 100 Maximum allowed number of iterations. #define EULER 0.5772156649 Euler’s constant γ. #define FPMIN 1.0e-30 Close to smallest representable floating-point number. #define EPS 1.0e-7 Desired relative error, not smaller than the machine precision. float expint(int n, float x) Evaluates the exponential integral En(x). { void nrerror(char error_text[]); int i,ii,nm1; float a,b,c,d,del,fact,h,psi,ans; nm1=n-1; if (n < 0 || x < 0.0 || (x==0.0 && (n==0 || n==1))) nrerror("bad arguments in expint"); else { if (n == 0) ans=exp(-x)/x; Special case. else { if (x == 0.0) ans=1.0/nm1; Another special case. else {

6.3 Exponential Integrals 225 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). #include #define EULER 0.57721566 Euler’s constant γ. #define MAXIT 100 Maximum number of iterations allowed. #define FPMIN 1.0e-30 Close to smallest representable floating-point number. #define EPS 6.0e-8 Relative error, or absolute error near the zero of Ei at x = 0.3725. float ei(float x) Computes the exponential integral Ei(x) for x > 0. { void nrerror(char error_text[]); int k; float fact,prev,sum,term; if (x MAXIT) nrerror("Series failed in ei"); return sum+log(x)+EULER; } else { Use asymptotic series. sum=0.0; Start with second term. term=1.0; for (k=1;k<=MAXIT;k++) { prev=term; term *= k/x; if (term < EPS) break; Since final sum is greater than one, term itself approximates the relative error. if (term < prev) sum += term; Still converging: add new term. else { sum -= prev; Diverging: subtract previous term and break; exit. } } return exp(x)*(1.0+sum)/x; } } CITED REFERENCES AND FURTHER READING: Stegun, I.A., and Zucker, R. 1974, Journal of Research of the National Bureau of Standards, vol. 78B, pp. 199–216; 1976, op. cit., vol. 80B, pp. 291–311. Amos D.E. 1980, ACM Transactions on Mathematical Software, vol. 6, pp. 365–377 [1]; also vol. 6, pp. 420–428. Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications, New York), Chapter 5. Wrench J.W. 1952, Mathematical Tables and Other Aids to Computation, vol. 6, p. 255. [2]

点击进入文档下载页（PDF格式）

已到末页，全文结束

点击下载（PDF格式）

浏览记录