6.6 Modified Bessel Functions of Integer Order
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1 6.6 Modified Bessel Functions of Integer Order 9 bjp=bjp*bigni bessj=bessj*bigni sum=sum*bigni if(jsum.ne.0)sum=sum+bj Accumulate the sum. jsum=1-jsum Change0to1orviceversa. if(j.eq.n)bessj=bjp Save the unnormalized answer. enddo 1 sum=.*sum-bj Compute (5.5.16) bessj=bessj/sum and use it to normalize the answer. if(.lt.0..and.mod(n,).eq.1)bessj=-bessj 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 Approimations (New York: Wiley), 6.8, p [1] 6.6 Modified Bessel Functions of Integer Order The modified Bessel functions I n () and K n () are equivalent to the usual Bessel functions J n and Y n evaluated for purely imaginary arguments. In detail, the relationship is I n () =( i) n J n (i) K n () = π in+1 [J n (i)+iy n (i)] (6.6.1) The particular choice of prefactor and of the linear combination of J n and Y n to form K n are simply choices that make the functions real-valued for real arguments. For small arguments n, both I n () and K n () become, asymptotically, simple powers of their argument I n () 1 ( ) n n 0 n! K 0 () ln() K n () (n 1)! ( ) n n>0 (6.6.) These epressions are virtually identical to those for J n () and Y n () in this region, ecept for the factor of /π difference between Y n () and K n (). In the region
2 30 Chapter 6. Special Functions 4 3 modified Bessel functions 1 K 0 K 1 K I 0 I Figure Modified Bessel functions I 0 () through I 3 (), K 0 () through K (). n, however, the modified functions have quite different behavior than the Bessel functions, I I n () 1 π ep() K n () π π ep( ) I 3 (6.6.3) The modified functions evidently have eponential rather than sinusoidal behavior for large arguments (see Figure 6.6.1). The smoothness of the modified Bessel functions, once the eponential factor is removed, makes a simple polynomial approimation of a few terms quite suitable for the functions I 0, I 1, K 0, and K 1. 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>1when >n. FUNCTION bessi0() REAL bessi0, Returns the modified Bessel function I 0 () for any real. REAL a DOUBLE PRECISION p1,p,p3,p4,p5,p6,p7,q1,q,q3,q4,q5,q6,q7, * q8,q9,y Accumulate polynomials in double precision. SAVE p1,p,p3,p4,p5,p6,p7,q1,q,q3,q4,q5,q6,q7,q8,q9 DATA p1,p,p3,p4,p5,p6,p7/1.0d0, d0, d0, d0, * d0, d-1, d-/ DATA q1,q,q3,q4,q5,q6,q7,q8,q9/ d0, d-1, * d-, d-, d-, d-1, * d-1, d-1, d-/
3 6.6 Modified Bessel Functions of Integer Order 31 if (abs().lt.3.75) then y=(/3.75)** bessi0=p1+y*(p+y*(p3+y*(p4+y*(p5+y*(p6+y*p7))))) a=abs() y=3.75/a bessi0=(ep(a)/sqrt(a))*(q1+y*(q+y*(q3+y*(q4 * +y*(q5+y*(q6+y*(q7+y*(q8+y*q9)))))))) FUNCTION bessk0() REAL bessk0, C USES bessi0 Returns the modified Bessel function K 0 () for positive real. REAL bessi0 DOUBLE PRECISION p1,p,p3,p4,p5,p6,p7,q1, * q,q3,q4,q5,q6,q7,y Accumulate polynomials in double precision. SAVE p1,p,p3,p4,p5,p6,p7,q1,q,q3,q4,q5,q6,q7 DATA p1,p,p3,p4,p5,p6,p7/ d0, d0, d0, * d-1,0.6698d-, d-3,0.74d-5/ DATA q1,q,q3,q4,q5,q6,q7/ d0, d-1, d-1, * d-1, d-, d-,0.5308d-3/ if (.le..0) then y=*/4.0 bessk0=(-log(/.0)*bessi0())+(p1+y*(p+y*(p3+ * y*(p4+y*(p5+y*(p6+y*p7)))))) y=(.0/) bessk0=(ep(-)/sqrt())*(q1+y*(q+y*(q3+ * y*(q4+y*(q5+y*(q6+y*q7)))))) FUNCTION bessi1() REAL bessi1, Returns the modified Bessel function I 1 () for any real. REAL a DOUBLE PRECISION p1,p,p3,p4,p5,p6,p7,q1,q,q3,q4,q5,q6,q7, * q8,q9,y Accumulate polynomials in double precision. SAVE p1,p,p3,p4,p5,p6,p7,q1,q,q3,q4,q5,q6,q7,q8,q9 DATA p1,p,p3,p4,p5,p6,p7/0.5d0, d0, d0, * d0, d-1, d-,0.3411d-3/ DATA q1,q,q3,q4,q5,q6,q7,q8,q9/ d0, d-1, * d-, d-, d-1,0.8967d-1, * d-1, d-1, d-/ if (abs().lt.3.75) then y=(/3.75)** bessi1=*(p1+y*(p+y*(p3+y*(p4+y*(p5+y*(p6+y*p7)))))) a=abs() y=3.75/a bessi1=(ep(a)/sqrt(a))*(q1+y*(q+y*(q3+y*(q4+ * y*(q5+y*(q6+y*(q7+y*(q8+y*q9)))))))) if(.lt.0.)bessi1=-bessi1
4 3 Chapter 6. Special Functions FUNCTION bessk1() REAL bessk1, C USES bessi1 Returns the modified Bessel function K 1 () for positive real. REAL bessi1 DOUBLE PRECISION p1,p,p3,p4,p5,p6,p7,q1, * q,q3,q4,q5,q6,q7,y Accumulate polynomials in double precision. SAVE p1,p,p3,p4,p5,p6,p7,q1,q,q3,q4,q5,q6,q7 DATA p1,p,p3,p4,p5,p6,p7/1.0d0, d0, d0, * d0, d-1, d-, d-4/ DATA q1,q,q3,q4,q5,q6,q7/ d0, d0, d-1, * d-1, d-, d-, d-3/ if (.le..0) then y=*/4.0 bessk1=(log(/.0)*bessi1())+(1.0/)*(p1+y*(p+ * y*(p3+y*(p4+y*(p5+y*(p6+y*p7)))))) y=.0/ bessk1=(ep(-)/sqrt())*(q1+y*(q+y*(q3+ * y*(q4+y*(q5+y*(q6+y*q7)))))) C The recurrence relation for I n () and K n () is the same as that for J n () and Y n () provided that i is substituted for. This has the effect of changing a sign in the relation, ( ) n I n+1 () = I n ()+I n 1 () ( ) (6.6.4) n K n+1 () =+ K n ()+K n 1 () These relations are always unstable for upward recurrence. For K n, itself growing, this presents no problem. For I n, however, the strategy of downward recursion is therefore required once again, and the starting point for the recursion may be chosen in the same manner as for the routine bessj. The only fundamental difference is that the normalization formula for I n () has an alternating minus sign in successive terms, which again arises from the substitution of i for in the formula used previously for J n 1=I 0 () I ()+I 4 () I 6 ()+ (6.6.5) In fact, we prefer simply to normalize with a call to bessi0. With this simple modification, the recursion routines bessj and bessy become the new routines bessi and bessk: FUNCTION bessk(n,) INTEGER n REAL bessk, USES bessk0,bessk1 Returns the modified Bessel function Kn() for positive and n. INTEGER j REAL bk,bkm,bkp,to,bessk0,bessk1 if (n.lt.) pause bad argument n in bessk to=.0/
5 6.6 Modified Bessel Functions of Integer Order 33 C bkm=bessk0() Upward recurrence for all... bk=bessk1() do 11 j=1,n-1...and here it is. bkp=bkm+j*to*bk bkm=bk bk=bkp enddo 11 bessk=bk FUNCTION bessi(n,) INTEGER n,iacc REAL bessi,,bigno,bigni PARAMETER (IACC=40,BIGNO=1.0e10,BIGNI=1.0e-10) USES bessi0 Returns the modified Bessel function In() for any real and n. INTEGER j,m REAL bi,bim,bip,to,bessi0 if (n.lt.) pause bad argument n in bessi if (.eq.0.) then bessi=0. to=.0/abs() bip=0.0 bi=1.0 bessi=0. m=*((n+int(sqrt(float(iacc*n))))) Downward recurrence from even m. do 11 j=m,1,-1 Make IACC larger to increase accuracy. bim=bip+float(j)*to*bi The downward recurrence. bip=bi bi=bim if (abs(bi).gt.bigno) then Renormalize to prevent overflows. bessi=bessi*bigni bi=bi*bigni bip=bip*bigni if (j.eq.n) bessi=bip enddo 11 bessi=bessi*bessi0()/bi Normalize with bessi0. if (.lt.0..and.mod(n,).eq.1) bessi=-bessi 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), 9.8. [1] Carrier, G.F., Krook, M. and Pearson, C.E. 1966, Functions of a Comple Variable (New York: McGraw-Hill), pp. 0ff.
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