《数字信号处理》教学参考资料（Numerical Recipes in C，The Art of Scientific Computing Second Edition）Chapter 06.5 Special Functions 6.5 Bessel Functions of Integer Order

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230 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). 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. In §6.7 we deal with fractional order. In fact, the more complicated routines for fractional order work fine for integer order too. For integer order, however, the routines in this section (and §6.6) are simpler and faster. Their only drawback is that they are limited by the precision of the underlying rational approximations. For full double precision, it is best to work with the routines for fractional order in §6.7. For any real ν, the Bessel function Jν(x) can be defined by the series representation Jν (x) = 1 2 x ν ∞ k=0 (−1 4x2)k k!Γ(ν + k + 1) (6.5.1) The series converges for all x, but it is not computationally very useful for x 1. For ν not an integer the Bessel function Yν (x) is given by Yν(x) = Jν(x) cos(νπ) − J−ν(x) sin(νπ) (6.5.2) The right-hand side goes to the correct limiting value Y n(x) as ν goes to some integer n, but this is also not computationally useful. For arguments x 0 (6.5.3) For x>ν, both Bessel functions look qualitatively like sine or cosine waves whose amplitude decays as x−1/2. The asymptotic forms for x ν are Jν(x) ∼ 2 πx cos x − 1 2 νπ − 1 4 π Yν(x) ∼ 2 πx sin x − 1 2 νπ − 1 4 π (6.5.4) In the transition region where x ∼ ν, the typical amplitudes of the Bessel functions are on the order Jν (ν) ∼ 21/3 32/3Γ( 2 3 ) 1 ν1/3 ∼ 0.4473 ν1/3 Yν (ν) ∼ − 21/3 31/6Γ( 2 3 ) 1 ν1/3 ∼ −0.7748 ν1/3 (6.5.5)

6.5 Bessel Functions of Integer Order 231 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). Bessel functions 1 .5 0 −.5 −1 −1.5 −2 0 2 4 6 8 10 Y0 Y1 Y2 J0 J1 J2 J3 x Figure 6.5.1. Bessel functions J0(x) through J3(x) and Y0(x) through Y2(x). which holds asymptotically for large ν. Figure 6.5.1 plots the first few Bessel functions of each kind. The Bessel functions satisfy the recurrence relations Jn+1(x) = 2n x Jn(x) − Jn−1(x) (6.5.6) and Yn+1(x) = 2n x Yn(x) − Yn−1(x) (6.5.7) As already mentioned in §5.5, only the second of these (6.5.7) is stable in the direction of increasing n for x<n. The reason that (6.5.6) is unstable in the direction of increasing n is simply that it is the same recurrence as (6.5.7): A small amount of “polluting” Yn introduced by roundoff error will quickly come to swamp the desired Jn, according to equation (6.5.3). A practical strategy for computing the Bessel functions of integer order divides into two tasks: first, how to compute J0, J1, Y0, and Y1, and second, how to use the recurrence relations stably to find other J’s and Y ’s. We treat the first task first: For x between zero and some arbitrary value (we will use the value 8), approximate J0(x) and J1(x) by rational functions in x. Likewise approximate by rational functions the “regular part” of Y0(x) and Y1(x), defined as Y0(x) − 2 π J0(x) ln(x) and Y1(x) − 2 π J1(x) ln(x) − 1 x (6.5.8) For 8 <x< ∞, use the approximating forms (n = 0, 1) Jn(x) = 2 πx Pn 8 x cos(Xn) − Qn 8 x sin(Xn) (6.5.9)

234 Chapter 6.Special Functions ans=(ans1/ans2)+0.636619772*(bessj1(x)*1og(x)-1.0/x); else Fitting function (6.5.10). z=8.0/x; y=z*z; xx=x-2.356194491: ans1=1.0+y*(0.183105e-2+y*(-0.3516396496e-4 +y*(0.2457520174e-5+y*(-0.240337019a-6))); ans2=0.04687499995+y*(-0.2002690873e-3 +y*(0.8449199096e-5+y*(-0.88228987e-6 +y*0.105787412e-6))); ans=sqrt(0.636619772/x)*(sin(xx)*ans1+z*cos(xx)*ans2); return ans; 83 鱼 19881992 We now turn to the second task,namely how to use the recurrence formulas (6.5.6)and(6.5.7)to get the Bessel functions n(x)and Yn(z)for n 2.The latter of these is straightforward,since its upward recurrence is always stable: from NUMERICAL RECIPESI float bessy(int n,float x) Returns the Bessel function Yn(x)for positive x and n 2. float bessy0(float x); float bessy1(float x); (North America 州bMe se to make one paper University Press. THE void nrerror(char error_text []) int ji ART float by,bym,byp,tox; if (n 2)nrerror("Index n less than 2 in bessy"); Programs tox=2.0/x; by=bessy1(x); Starting values for the recurrence. bym=bessy0(x); for (j=1;j<n;j++){ Recurrence (6.5.7). byp=j*tox*by-bym; to dir bym=by; by=byp; 1988199200 OF SCIENTIFIC COMPUTING(ISBN return by; v@cam 10-:6211 The cost of this algorithm is the call to bessy1 and bessyo(which generate a call to each of bessj1 and bessjo),plus O(n)operations in the recurrence. Numerical Recipes 43108 As for.().things are a bit more complicated.We can start the recurrence upward on n from.Jo and/1.but it will remain stable only while n does not exceed z.This is,however,just fine for calls with large z and small n,a case which (outside occurs frequently in practice. North Software. The harder case to provide for is that with x<n.The best thing to do here is to use Miller's algorithm(see discussion preceding equation 5.5.16),applying the recurrence dowmard from some arbitrary starting value and making use of the upward-unstable nature of the recurrence to put us onto the correct solution.When machine we finally arrive atJo orJ we are able to normalize the solution with the sum (5.5.16)accumulated along the way. The only subtlety is in deciding at how large an n we need start the downward recurrence so as to obtain a desired accuracy by the time we reach the n that we really want.If you play with the asymptotic forms(6.5.3)and(6.5.5),you should be able to convince yourself that the answer is to start larger than the desired n by

234 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). ans=(ans1/ans2)+0.636619772*(bessj1(x)*log(x)-1.0/x); } else { Fitting function (6.5.10). z=8.0/x; y=z*z; xx=x-2.356194491; ans1=1.0+y*(0.183105e-2+y*(-0.3516396496e-4 +y*(0.2457520174e-5+y*(-0.240337019e-6)))); ans2=0.04687499995+y*(-0.2002690873e-3 +y*(0.8449199096e-5+y*(-0.88228987e-6 +y*0.105787412e-6))); ans=sqrt(0.636619772/x)*(sin(xx)*ans1+z*cos(xx)*ans2); } return ans; } We now turn to the second task, namely how to use the recurrence formulas (6.5.6) and (6.5.7) to get the Bessel functions J n(x) and Yn(x) for n ≥ 2. The latter of these is straightforward, since its upward recurrence is always stable: float bessy(int n, float x) Returns the Bessel function Yn(x) for positive x and n ≥ 2. { float bessy0(float x); float bessy1(float x); void nrerror(char error_text[]); int j; float by,bym,byp,tox; if (n < 2) nrerror("Index n less than 2 in bessy"); tox=2.0/x; by=bessy1(x); Starting values for the recurrence. bym=bessy0(x); for (j=1;j<n;j++) { Recurrence (6.5.7). byp=j*tox*by-bym; bym=by; by=byp; } return by; } The cost of this algorithm is the call to bessy1 and bessy0 (which generate a call to each of bessj1 and bessj0), plus O(n) operations in the recurrence. As for Jn(x), things are a bit more complicated. We can start the recurrence upward on n from J0 and J1, but it will remain stable only while n does not exceed x. This is, however, just fine for calls with large x and small n, a case which occurs frequently in practice. The harder case to provide for is that with x<n. The best thing to do here is to use Miller’s algorithm (see discussion preceding equation 5.5.16), applying the recurrence downward from some arbitrary starting value and making use of the upward-unstable nature of the recurrence to put us onto the correct solution. When we finally arrive at J0 or J1 we are able to normalize the solution with the sum (5.5.16) accumulated along the way. The only subtlety is in deciding at how large an n we need start the downward recurrence so as to obtain a desired accuracy by the time we reach the n that we really want. If you play with the asymptotic forms (6.5.3) and (6.5.5), you should be able to convince yourself that the answer is to start larger than the desired n by

236 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), Chapter 9. Hart, J.F., et al. 1968, Computer Approximations (New York: Wiley), §6.8, p. 141. [1] 6.6 Modified Bessel Functions of Integer Order The modified Bessel functions In(x) and Kn(x) are equivalent to the usual Bessel functions Jn and Yn evaluated for purely imaginary arguments. In detail, the relationship is In(x)=(−i) nJn(ix) Kn(x) = π 2 i n+1[Jn(ix) + iYn(ix)] (6.6.1) The particular choice of prefactor and of the linear combination of J n and Yn to form Kn are simply choices that make the functions real-valued for real arguments x. For small arguments x n, both In(x) and Kn(x) become, asymptotically, simple powers of their argument In(x) ≈ 1 n! x 2 n n ≥ 0 K0(x) ≈ − ln(x) Kn(x) ≈ (n − 1)! 2 x 2 −n n > 0 (6.6.2) These expressions are virtually identical to those for Jn(x) and Yn(x) in this region, except for the factor of −2/π difference between Y n(x) and Kn(x). In the region x n, however, the modified functions have quite different behavior than the Bessel functions, In(x) ≈ 1 √ 2πx exp(x) Kn(x) ≈ π √2πx exp(−x) (6.6.3) The modified functions evidently have exponential rather than sinusoidal behavior for large arguments (see Figure 6.6.1). The smoothness of the modified Bessel functions, once the exponential factor is removed, makes a simple polynomial approximation of a few terms quite suitable for the functions I 0, I1, K0, and K1. The following routines, based on polynomial coefficients given by Abramowitz and Stegun [1], evaluate these four functions, and will provide the basis for upward recursion for n > 1 when x>n.

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