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bessel.f
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bessel.f
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c Following is from the brambilla LH and FW code
ctk--calculation of Bessel function with a complex argument (x,y)
c br(nb),bi(nb)
c y=0,ize=0 >J_0(x)=br(1),J_1(x)=br(2)
c y=0,ize=1 >I_0(x)=br(1),I_1(x)=br(2),... ,I_n(x)=br(n+1),
c n=0,1,2,...; nb.ge.1
C x,y,br,bi - real
subroutine beslci(x,y,nb,ize,br,bi,ncalc)
c
c***** ***** ***** ***** ***** ***** *****
c
c journal of research national bureau of standards
c b-mathematical sciences vol.77-b nos.3-4 july-dec. 1973 p118
c routine to calculate bessel functions j and i
c of complex argument and integer order
c input variables
c x real part of complex argument
c y imaginary part of complex argument
c nb a positive integer designating highest order to be calculated
c ize zero for j*s one for i*s
c br for normal exit,br contains real part of solution vector
c bi for normal exit,bi contains imaginary part of solution vector
c normal exit if ncalc=nb
c br-bi-ncalc need not be initialized
c machine dependent constants
c nsig decimal significance desired. set to ifix(alog10(2)*nbit+1)
c nbit number of bits in the mantissa
c the relative truncation error is limited to t=.5*10**-nsig for
c order greater than abs(z).
c for order less than abs(z) (general test),the relative error is
c limited to t for function values of magnitude at least 1.
c the absolute error is limited to t for smaller values.
c nten largest integer k such that 10**k is machine representable.
c largez upper limit on the magnitude of z. if abs(z)=n, at least
c n iterations of the backward recursion will be executed.
c exparg largest argument that the library exp routine can handle.
c error returns
c let g denote either i or j.
c in case of an error, ncalc.ne.nb and not all g*s are calculated
c to the desired accuracy.
c if ncalc.lt.0, an argument is out of range. nb.le.0 or ize is
c neither 0 nor 1 or ize=0 and abs(y).gt.exparg, or ize=1 and
c abs(x).gt.exparg. in this case,the vectors br and bi are not
c calculated, and ncalc is set to min0(nb,0)-1 so ncalc.ne.nb.
c nb.gt.ncalc.gt.0 will occur if nb.gt.magz and abs(g-sub-nb-of-z/g
c -sub-magx-of-z).lt.10.**(nten/2), i.e. nb is much greater than
c magz. in this case, br(n) and bi(n) are calculated to the desired
c accuracy for n.le.ncalc, but for ncalc.lt.n.le.nb, precision is
c lost. if n.gt.ncalc and abs(g(ncalc-1)/g(n-1)).eq.10**-k, then
c the last k significant figures of g(n-1) (=br(n)+i*bi(n)) are
c erroneous. if the user wishes to calculate g(n-1) to higher
c accuracy, he should use an asymptotic formula for large order.
c
c***** ***** ***** ***** ***** ***** *****
c
dimension br(*),bi(*)
c data nsig,nten,largez,exparg/15,307,10000,700./
data nsig,nten,largez,exparg/15,38,10000,81./
tempar=sqrt(x*x+y*y)
magz=ifix(tempar)
if(nb.gt.0.and.magz.le.largez.and.((ize.eq.0.and.abs(y).le.exparg)
c.or.(ize.eq.1.and.abs(x).le.exparg))) go to 1
c error return z,nb,or ize is out of range
ncalc=min0(nb,0)-1
return
1 sign1=1-2*ize
ncalc=nb
c use 2-term ascending series for small z
if(tempar**4.lt.0.1**nsig) go to 50
c initialize the calculation of the p*s
nbmz=nb-magz
n=magz+1
if(abs(x).lt.abs(y)) go to 2
zinvr=1.0/(x+y*y/x)
zinvi=-y*zinvr/x
go to 3
2 zinvi=-1.0/(y+x*x/y)
zinvr=-x*zinvi/y
3 plastr=1.0
plasti=0.0
pr=sign1*(n+n)*zinvr
pi=sign1*(n+n)*zinvi
test=2.0*(10.0)**nsig
m=0
if(nbmz.lt.3) go to 6
c calculate p*s until n=nb-1. check for possible overflow.
tover=10.0**(nten-nsig)
nstart=magz+2
nend=nb-1
do 5 n=nstart,nend
poldr=plastr
poldi=plasti
plastr=pr
plasti=pi
pr=sign1*((n+n)*(plastr*zinvr-plasti*zinvi)-poldr)
pi=sign1*((n+n)*(plasti*zinvr+plastr*zinvi)-poldi)
if((pr/tover)**2+(pi/tover)**2-1.0) 5,5,7
5 continue
n=nend
c calculate special significance test for nbmz.gt.2.
tempbi=amax1(abs(pr),abs(pi))
tempbi=tempbi*sqrt(2.0*(10.0)**nsig*sqrt(((pr/tempbi)**2+(pi
c/tempbi)**2)*((plastr/tempbi)**2+(plasti/tempbi)**2)))
test=amax1(test,tempbi)
c calculate p*s until significance test is passed
6 n=n+1
poldr=plastr
poldi=plasti
plastr=pr
plasti=pi
pr=sign1*((n+n)*(plastr*zinvr-plasti*zinvi)-poldr)
pi=sign1*((n+n)*(plasti*zinvr+plastr*zinvi)-poldi)
if((pr/test)**2+(pi/test)**2.lt.1.0) go to 6
if(m.eq.1) go to 12
c calculate strict variant of significance test,and
c calculate p*s until this test is passed.
m=1
tempbi=amax1(abs(pr),abs(pi))
tempbr=sqrt(((pr/tempbi)**2+(pi/tempbi)**2)/
c((plastr/tempbi)**2+(plasti/tempbi)**2))
tempbi=(n+1)/tempar
if(tempbr+1.0/tempbr.gt.2.0*tempbi) tempbr=tempbi+sqrt(tempbi**2
c-1.0)
test=test/sqrt(tempbr-1.0/tempbr)
if((pr/test)**2+(pi/test)**2-1.0) 6,12,12
7 nstart=n+1
c to avoid overflow, normalize p*s by dividing by tover.
c calculate p*s until unnormalized p would overflow.
pr=pr/tover
pi=pi/tover
plastr=plastr/tover
plasti=plasti/tover
psaver=pr
psavei=pi
tempcr=plastr
tempci=plasti
test=10.0**(2*nsig)
8 n=n+1
poldr=plastr
poldi=plasti
plastr=pr
plasti=pi
pr=sign1*((n+n)*(plastr*zinvr-plasti*zinvi)-poldr)
pi=sign1*((n+n)*(plasti*zinvr+plastr*zinvi)-poldi)
if(pr**2+pi**2.le.test) go to 8
c calculate backward test,and find ncalc,the highest n
c such that the test is passed.
tempbr=sqrt((plastr**2+plasti**2)/(poldr**2+poldi**2))
tempbi=n/tempar
if(tempbr+1.0/tempbr.gt.2.0*tempbi) tempbr=tempbi+sqrt(tempbi**2
c-1.0)
test=0.5*(1.0-1.0/tempbr**2)/10.0**nsig
test=((plastr**2+plasti**2)*test)*((poldr**2+poldi**2)*test)
pr=plastr*tover
pi=plasti*tover
n=n-1
nend=min0(nb,n)
do 9 ncalc=nstart,nend
poldr=tempcr
poldi=tempci
tempcr=psaver
tempci=psavei
psaver=sign1*((n+n)*(tempcr*zinvr-tempci*zinvi)-poldr)
psavei=sign1*((n+n)*(tempci*zinvr+tempcr*zinvi)-poldi)
if((psaver**2+psavei**2)*(tempcr**2+tempci**2)-test) 9,9,10
9 continue
ncalc=nend+1
10 ncalc=ncalc-1
c the coefficient of b(n) in the normalized sum is
c m*sqrt(-1)**imag,where m=-2,0,or2,and imag is 0 or 1.
c calculate recursion rules for m and imag,and initialize them.
12 n=n+1
tempbr=ize*x+(1-ize)*y
ipos=0
if(tempbr) 13,14,13
13 ipos=ifix(1.1*tempbr/abs(tempbr))
14 mrecur=4*((2+ize+ipos)/2)-3-2*(ize+ipos)
k=2+ipos+2*ize*ipos**2-ize
l=n-4*(n/4)
mlast=2+8*((k*l)/4)-4*((k*l)/2)
if(ipos.eq.0.and.(l.eq.1.or.l.eq.3)) mlast=0
l=l+3-4*((l+3)/4)
m=2+8*((k*l)/4)-4*((k*l)/2)
if(ipos.eq.0.and.(l.eq.1.or.l.eq.3)) m=0
imrecr=(1-ize)*ipos**2
imag=imrecr*(l-2*(l/2))
c initialize the backward recursion and the normalization sum
tempbr=0.0
tempbi=0.0
if(abs(pi).gt.abs(pr)) go to 15
tempar=1.0/(pr+pi*(pi/pr))
tempai=-(pi*tempar)/pr
go to 16
15 tempai=-1.0/(pi+pr*(pr/pi))
tempar=-(pr*tempai)/pi
16 if(imag.ne.0) go to 17
sumr=m*tempar
sumi=m*tempai
go to 18
17 sumr=-m*tempai
sumi=m*tempar
18 nend=n-nb
if(nend) 26,22,19
c recur backward via difference equation calculating (but not
c storing) br(n) and bi(n) until n=nb
19 do 21 l=1,nend
n=n-1
tempcr=tempbr
tempci=tempbi
tempbr=tempar
tempbi=tempai
pr=(n+n)*zinvr
pi=(n+n)*zinvi
tempar=pr*tempbr-pi*tempbi-sign1*tempcr
tempai=pr*tempbi+pi*tempbr-sign1*tempci
imag=(1-imag)*imrecr
k=mlast
mlast=m
m=k*mrecur
if(imag.ne.0) go to 20
sumr=sumr+m*tempar
sumi=sumi+m*tempai
go to 21
20 sumr=sumr-m*tempai
sumi=sumi+m*tempar
21 continue
c store br(nb),bi(nb)
22 br(n)=tempar
bi(n)=tempai
if(n.gt.1) go to 23
c nb=1. since 2*tempar and 2*tempai were added to sumr and sumi
c respectively,we must subaxis(i)ract tempar and tempai
sumr=sumr-tempar
sumi=sumi-tempai
go to 35
c calculate and store br(nb-1),bi(nb-1)
23 n=n-1
pr=(n+n)*zinvr
pi=(n+n)*zinvi
br(n)=pr*tempar-pi*tempai-sign1*tempbr
bi(n)=pr*tempai+pi*tempar-sign1*tempbi
if(n.eq.1) go to 34
imag=(1-imag)*imrecr
k=mlast
mlast=m
m=k*mrecur
if(imag.ne.0) go to 24
sumr=sumr+m*br(n)
sumi=sumi+m*bi(n)
go to 30
24 sumr=sumr-m*bi(n)
sumi=sumi+m*br(n)
go to 30
c n.lt.nb,so store br(n),bi(n) and set higher orders zero
26 br(n)=tempar
bi(n)=tempai
nend=-nend
do 27 l=1,nend
br(n+l)=0.0
27 bi(n+l)=0.0
30 nend=n-2
if(nend.eq.0) go to 33
c calculate via difference equation and store br(n),bi(n)
c until n=2
do 32 l=1,nend
n=n-1
pr=(n+n)*zinvr
pi=(n+n)*zinvi
br(n)=pr*br(n+1)-pi*bi(n+1)-sign1*br(n+2)
bi(n)=pr*bi(n+1)+pi*br(n+1)-sign1*bi(n+2)
imag=(1-imag)*imrecr
k=mlast
mlast=m
m=k*mrecur
if(imag.ne.0) go to 31
sumr=sumr+m*br(n)
sumi=sumi+m*bi(n)
go to 32
31 sumr=sumr-m*bi(n)
sumi=sumi+m*br(n)
32 continue
c calculate and store br(1),bi(1)
33 br(1)=2.0*(br(2)*zinvr-bi(2)*zinvi)-sign1*br(3)
bi(1)=2.0*(br(2)*zinvi+bi(2)*zinvr)-sign1*bi(3)
34 sumr=sumr+br(1)
sumi=sumi+bi(1)
c calculate normalization factor. tempar+i*tempai
35 if(ize.eq.1) go to 36
tempcr=ipos*y
tempci=-ipos*x
go to 37
36 tempcr=ipos*x
tempci=ipos*y
37 tempcr=exp(tempcr)
tempbr=cos(tempci)
tempbi=sin(tempci)
if(abs(sumr).lt.abs(sumi)) go to 38
tempci=sumi/sumr
tempcr=(tempcr/sumr)/(1.0+tempci*tempci)
tempar=tempcr*(tempbr+tempbi*tempci)
tempai=tempcr*(tempbi-tempbr*tempci)
go to 39
38 tempci=sumr/sumi
tempcr=(tempcr/sumi)/(1.0+tempci*tempci)
tempar=tempcr*(tempbr*tempci+tempbi)
tempai=tempcr*(tempbi*tempci-tempbr)
c normalize
39 do 40 n=1,nb
tempbr=br(n)*tempar-bi(n)*tempai
bi(n)=br(n)*tempai+bi(n)*tempar
40 br(n)=tempbr
return
c two-term ascending series for small z
50 tempar=1.0
tempai=0.0
tempcr=0.25*(x*x-y*y)
tempci=0.5*x*y
br(1)=1.0-sign1*tempcr
bi(1)=-sign1*tempci
if(nb.eq.1) go to 52
do 51 n=2,nb
tempbr=(tempar*x-tempai*y)/(n+n-2)
tempai=(tempar*y+tempai*x)/(n+n-2)
tempar=tempbr
tempbr=n
br(n)=tempar*(1.0-sign1*tempcr/tempbr)+tempai*tempci/tempbr
51 bi(n)=tempai*(1.0-sign1*tempcr/tempbr)-tempar*tempci/tempbr
52 return
end
subroutine besj(x,n,bj,d,ier)
c-----------------------------------------------------------------------
c i.b.m. bessel function j
c x - argument
c n - order (integer .ge. 0)
c bj - j(x)
c d - relative allowable error (input)
c ier- error switch, if ier=0 all is ok
c-----------------------------------------------------------------------
bj=0.
if(n) 10,20,20
10 ier=1
return
20 if(x) 30,30,31
30 ier=2
bj=0.
if(n.eq.0) bj=1.
return
31 if(x-15.) 32,32,34
32 ntest=20.+10.*x-x*x/3
go to 36
34 ntest=90.+x/2.
36 if(n-ntest) 40,38,38
38 ier=4
return
40 ier=0
n1=n+1
bprev=0.
if(x-5.) 50,60,60
50 ma=x+6.
go to 70
60 ma=1.4*x+60./x
70 mb=n+ifix(x)/4+2
mzero=max0(ma,mb)
mmax=ntest
100 do 190 m=mzero,mmax,3
fm1=1.e-28
fm=0.
alpha=0.
if(m-(m/2)*2)120,110,120
110 jt=-1
go to 130
120 jt=1
130 m2=m-2
do 160 k=1,m2
mk=m-k
bmk=2.*float(mk)*fm1/x-fm
fm=fm1
fm1=bmk
if(mk-n-1) 150,140,150
140 bj=bmk
150 jt=-jt
s=1+jt
160 alpha=alpha+bmk*s
bmk=2.*fm1/x-fm
if(n) 180,170,180
170 bj=bmk
180 alpha=alpha+bmk
bj=bj/alpha
if(abs(bj-bprev)-abs(d*bj))200,200,190
190 bprev=bj
ier=3
200 return
end
c
c
c88888888888888888888888888888888888888888888888888888888888888888888888888
SUBROUTINE DBESJ(X,ALPHA,N,Y,NZ)
C***BEGIN PROLOGUE DBESJ
C***DATE WRITTEN 750101 (YYMMDD)
C***REVISION DATE 851111 (YYMMDD)
C***CATEGORY NO. C10A3
C***KEYWORDS BESSEL FUNCTION,DOUBLE PRECISION,J BESSEL FUNCTION,
C SPECIAL FUNCTION
C***AUTHOR AMOS, D. E., (Sandia National Laboratories, Albuquerque)
C DANIEL, S. L., (Sandia National Laboratories, Albuquerque)
C WESTON, M. K., (Sandia National Laboratories, Albuquerque)
C***PURPOSE Compute an N member sequence of J Bessel functions
C J/SUB(ALPHA+K-1)/(X), K=1,...,N for non-negative ALPHA
C and X. (At most 14 digits.)
C***DESCRIPTION
C
C Written by D. E. Amos, S. L. Daniel and M. K. Weston, January 1975
C
C References
C SAND-75-0147
C
C CDC 6600 Subroutines IBESS and JBESS for Bessel Functions
C I(NU,X) and J(NU,X), X .GE. 0, NU .GE. 0 by D. E. Amos, S. L.
C Daniel, M. K. Weston. ACM Trans Math Software,3,pp 76-92
C (1977)
C
C Tables of Bessel Functions of Moderate or Large Orders,
C NPL Mathematical Tables, Vol. 6, by F. W. J. Olver, Her
C Majesty's Stationery Office, London, 1962.
C
C Abstract **** a double precision routine ****
C DBESJ computes an N member sequence of J Bessel functions
C J/sub(ALPHA+K-1)/(X), K=1,...,N for non-negative ALPHA and X.
C A combination of the power series, the asymptotic expansion
C for X to infinity and the uniform asymptotic expansion for
C NU to infinity are applied over subdivisions of the (NU,X)
C plane. For values of (NU,X) not covered by one of these
C formulae, the order is incremented or decremented by integer
C values into a region where one of the formulae apply. Backward
C recursion is applied to reduce orders by integer values except
C where the entire sequence lies in the oscillatory region. In
C this case forward recursion is stable and values from the
C asymptotic expansion for X to infinity start the recursion
C when it is efficient to do so. Leading terms of the series and
C uniform expansion are tested for underflow. If a sequence is
C requested and the last member would underflow, the result is
C set to zero and the next lower order tried, etc., until a
C member comes on scale or all members are set to zero.
C Overflow cannot occur.
C
C The maximum number of significant digits obtainable
C is the smaller of 14 and the number of digits carried in
C double precision arithmetic.
C
C DBESJ calls DASYJY, DJAIRY, DLNGAM, D1MACH, I1MACH, XERROR
C
C Description of Arguments
C
C Input X,ALPHA are double precision
C X - X .GE. 0.0D0
C ALPHA - order of first member of the sequence,
C ALPHA .GE. 0.0D0
C N - number of members in the sequence, N .GE. 1
C
C Output Y is double precision
C Y - a vector whose first N components contain
C values for J/sub(ALPHA+K-1)/(X), K=1,...,N
C NZ - number of components of Y set to zero due to
C underflow,
C NZ=0 , normal return, computation completed
C NZ .NE. 0, last NZ components of Y set to zero,
C Y(K)=0.0D0, K=N-NZ+1,...,N.
C
C Error Conditions
C Improper input arguments - a fatal error
C Underflow - a non-fatal error (NZ .NE. 0)
C***REFERENCES CDC 6600 SUBROUTINES IBESS AND JBESS FOR BESSEL
C FUNCTIONS I(NU,X) AND J(NU,X), X .GE. 0, NU .GE. 0,
C BY D. E. AMOS, S. L.DANIEL, M. K. WESTON, ACM
C TRANSACTIONS ON MATHEMATICALSOFTWARE, VOL. 3,
C PP. 76-92 (1977).
C***ROUTINES CALLED D1MACH,DASYJY,DJAIRY,DLNGAM,I1MACH,XERROR
C***END PROLOGUE DBESJ
EXTERNAL DJAIRY
INTEGER I,IALP,IDALP,IFLW,IN,INLIM,IS,I1,I2,K,KK,KM,KT,N,NN,
1 NS,NZ
INTEGER I1MACH
DOUBLE PRECISION AK,AKM,ALPHA,ANS,AP,ARG,COEF,DALPHA,DFN,DTM,
1 EARG,ELIM1,ETX,FIDAL,FLGJY,FN,FNF,FNI,FNP1,FNU,
2 FNULIM,GLN,PDF,PIDT,PP,RDEN,RELB,RTTP,RTWO,RTX,RZDEN,
3 S,SA,SB,SXO2,S1,S2,T,TA,TAU,TB,TEMP,TFN,TM,TOL,
4 TOLLN,TRX,TX,T1,T2,WK,X,XO2,XO2L,Y,SLIM,RTOL
DOUBLE PRECISION D1MACH, DLNGAM
DIMENSION Y(*), TEMP(3), FNULIM(2), PP(4), WK(7)
DATA RTWO,PDF,RTTP,PIDT / 1.34839972492648D+00,
1 7.85398163397448D-01, 7.97884560802865D-01, 1.57079632679490D+00/
DATA PP(1), PP(2), PP(3), PP(4) / 8.72909153935547D+00,
1 2.65693932265030D-01, 1.24578576865586D-01, 7.70133747430388D-04/
DATA INLIM / 150 /
DATA FNULIM(1), FNULIM(2) / 100.0D0, 60.0D0 /
C***FIRST EXECUTABLE STATEMENT DBESJ
cBH write(*,*)'DBESJ:x,alpha,n,y(1),nzero',x,alpha,n,y(1),nz
cBH
cBH write(*,*)'DBESJ:D1MACH(1:5)',D1MACH(1),D1MACH(2),D1MACH(3),
cBH + D1MACH(4),D1MACH(5)
NZ = 0
KT = 1
NS=0
C I1MACH(14) REPLACES I1MACH(11) IN A DOUBLE PRECISION CODE
C I1MACH(15) REPLACES I1MACH(12) IN A DOUBLE PRECISION CODE
TA = D1MACH(3)
TOL = DMAX1(TA,1.0D-15)
I1 = I1MACH(14) + 1
I2 = I1MACH(15)
TB = D1MACH(5)
cBH write(*,*)'DBESJ:I1,I2,TB',I1,I2,TB
ELIM1 = 2.303D0*(DBLE(FLOAT(-I2))*TB-3.0D0)
RTOL=1.0D0/TOL
SLIM=D1MACH(1)*RTOL*1.0D+3
C TOLLN = -LN(TOL)
TOLLN = 2.303D0*TB*DBLE(FLOAT(I1))
TOLLN = DMIN1(TOLLN,34.5388D0)
IF (N-1) 720, 10, 20
10 KT = 2
20 NN = N
IF (X) 730, 30, 80
30 IF (ALPHA) 710, 40, 50
40 Y(1) = 1.0D0
IF (N.EQ.1) RETURN
I1 = 2
GO TO 60
50 I1 = 1
60 DO 70 I=I1,N
Y(I) = 0.0D0
70 CONTINUE
RETURN
80 CONTINUE
IF (ALPHA.LT.0.0D0) GO TO 710
C
IALP = INT(SNGL(ALPHA))
FNI = DBLE(FLOAT(IALP+N-1))
FNF = ALPHA - DBLE(FLOAT(IALP))
DFN = FNI + FNF
FNU = DFN
XO2 = X*0.5D0
SXO2 = XO2*XO2
C
C DECISION TREE FOR REGION WHERE SERIES, ASYMPTOTIC EXPANSION FOR X
C TO INFINITY AND ASYMPTOTIC EXPANSION FOR NU TO INFINITY ARE
C APPLIED.
C
IF (SXO2.LE.(FNU+1.0D0)) GO TO 90
TA = DMAX1(20.0D0,FNU)
IF (X.GT.TA) GO TO 120
IF (X.GT.12.0D0) GO TO 110
XO2L = DLOG(XO2)
NS = INT(SNGL(SXO2-FNU)) + 1
GO TO 100
90 FN = FNU
FNP1 = FN + 1.0D0
XO2L = DLOG(XO2)
IS = KT
IF (X.LE.0.50D0) GO TO 330
NS = 0
100 FNI = FNI + DBLE(FLOAT(NS))
DFN = FNI + FNF
FN = DFN
FNP1 = FN + 1.0D0
IS = KT
IF (N-1+NS.GT.0) IS = 3
GO TO 330
110 ANS = DMAX1(36.0D0-FNU,0.0D0)
NS = INT(SNGL(ANS))
FNI = FNI + DBLE(FLOAT(NS))
DFN = FNI + FNF
FN = DFN
IS = KT
IF (N-1+NS.GT.0) IS = 3
GO TO 130
120 CONTINUE
RTX = DSQRT(X)
TAU = RTWO*RTX
TA = TAU + FNULIM(KT)
IF (FNU.LE.TA) GO TO 480
FN = FNU
IS = KT
C
C UNIFORM ASYMPTOTIC EXPANSION FOR NU TO INFINITY
C
130 CONTINUE
I1 = IABS(3-IS)
I1 = MAX0(I1,1)
FLGJY = 1.0D0
CALL DASYJY(DJAIRY,X,FN,FLGJY,I1,TEMP(IS),WK,IFLW)
IF(IFLW.NE.0) GO TO 380
GO TO (320, 450, 620), IS
310 TEMP(1) = TEMP(3)
KT = 1
320 IS = 2
FNI = FNI - 1.0D0
DFN = FNI + FNF
FN = DFN
IF(I1.EQ.2) GO TO 450
GO TO 130
C
C SERIES FOR (X/2)**2.LE.NU+1
C
330 CONTINUE
GLN = DLNGAM(FNP1)
ARG = FN*XO2L - GLN
IF (ARG.LT.(-ELIM1)) GO TO 400
EARG = DEXP(ARG)
340 CONTINUE
S = 1.0D0
IF (X.LT.TOL) GO TO 360
AK = 3.0D0
T2 = 1.0D0
T = 1.0D0
S1 = FN
DO 350 K=1,17
S2 = T2 + S1
T = -T*SXO2/S2
S = S + T
IF (DABS(T).LT.TOL) GO TO 360
T2 = T2 + AK
AK = AK + 2.0D0
S1 = S1 + FN
350 CONTINUE
360 CONTINUE
TEMP(IS) = S*EARG
GO TO (370, 450, 610), IS
370 EARG = EARG*FN/XO2
FNI = FNI - 1.0D0
DFN = FNI + FNF
FN = DFN
IS = 2
GO TO 340
C
C SET UNDERFLOW VALUE AND UPDATE PARAMETERS
C UNDERFLOW CAN ONLY OCCRFOR NS=0 SINCE THE ORDER MUST BE LARGER
C THAN 36. THEREFORE, NS NEE NOT BE TESTED.
C
380 Y(NN) = 0.0D0
NN = NN - 1
FNI = FNI - 1.0D0
DFN = FNI + FNF
FN = DFN
IF (NN-1) 440, 390, 130
390 KT = 2
IS = 2
GO TO 130
400 Y(NN) = 0.0D0
NN = NN - 1
FNP1 = FN
FNI = FNI - 1.0D0
DFN = FNI + FNF
FN = DFN
IF (NN-1) 440, 410, 420
410 KT = 2
IS = 2
420 IF (SXO2.LE.FNP1) GO TO 430
GO TO 130
430 ARG = ARG - XO2L + DLOG(FNP1)
IF (ARG.LT.(-ELIM1)) GO TO 400
GO TO 330
440 NZ = N - NN
cBH write(*,*)'**0**DBESJ:Y(1)',Y(1)
RETURN
C
C BACKWARD RECURSION SECTION
C
450 CONTINUE
IF(NS.NE.0) GO TO 451
NZ = N - NN
IF (KT.EQ.2) GO TO 470
C BACKWARD RECUR FROM INDEX ALPHA+NN-1 TO ALPHA
Y(NN) = TEMP(1)
Y(NN-1) = TEMP(2)
cBH write(*,*)'**0.1**DBESJ:Y(1)',Y(1)
IF (NN.EQ.2) RETURN
451 CONTINUE
TRX = 2.0D0/X
DTM = FNI
TM = (DTM+FNF)*TRX
AK=1.0D0
TA=TEMP(1)
TB=TEMP(2)
IF(DABS(TA).GT.SLIM) GO TO 455
TA=TA*RTOL
TB=TB*RTOL
AK=TOL
455 CONTINUE
KK=2
IN=NS-1
IF(IN.EQ.0) GO TO 690
IF(NS.NE.0) GO TO 670
K=NN-2
DO 460 I=3,NN
S=TB
TB = TM*TB - TA
TA=S
Y(K)=TB*AK
DTM = DTM - 1.0D0
TM = (DTM+FNF)*TRX
K = K - 1
460 CONTINUE
RETURN
470 Y(1) = TEMP(2)
cBH write(*,*)'**0.2**DBESJ:Y(1)',Y(1)
RETURN
C
C ASYMPTOTIC EXPANSION FOR X TO INFINITY WITH FORWARD RECURSION IN
C OSCILLATORY REGION X.GT.MAX(20, NU), PROVIDED THE LAST MEMBER
C OF THE SEQUENCE IS ALSO IN THE REGION.
C
480 CONTINUE
IN = INT(SNGL(ALPHA-TAU+2.0D0))
IF (IN.LE.0) GO TO 490
IDALP = IALP - IN - 1
KT = 1
GO TO 500
490 CONTINUE
IDALP = IALP
IN = 0
500 IS = KT
FIDAL = DBLE(FLOAT(IDALP))
DALPHA = FIDAL + FNF
ARG = X - PIDT*DALPHA - PDF
SA = DSIN(ARG)
SB = DCOS(ARG)
COEF = RTTP/RTX
ETX = 8.0D0*X
510 CONTINUE
DTM = FIDAL + FIDAL
DTM = DTM*DTM
TM = 0.0D0
IF (FIDAL.EQ.0.0D0 .AND. DABS(FNF).LT.TOL) GO TO 520
TM = 4.0D0*FNF*(FIDAL+FIDAL+FNF)
520 CONTINUE
TRX = DTM - 1.0D0
T2 = (TRX+TM)/ETX
S2 = T2
RELB = TOL*DABS(T2)
T1 = ETX
S1 = 1.0D0
FN = 1.0D0
AK = 8.0D0
DO 530 K=1,13
T1 = T1 + ETX
FN = FN + AK
TRX = DTM - FN
AP = TRX + TM
T2 = -T2*AP/T1
S1 = S1 + T2
T1 = T1 + ETX
AK = AK + 8.0D0
FN = FN + AK
TRX = DTM - FN
AP = TRX + TM
T2 = T2*AP/T1
S2 = S2 + T2
IF (DABS(T2).LE.RELB) GO TO 540
AK = AK + 8.0D0
530 CONTINUE
540 TEMP(IS) = COEF*(S1*SB-S2*SA)
IF(IS.EQ.2) GO TO 560
FIDAL = FIDAL + 1.0D0
DALPHA = FIDAL + FNF
IS = 2
TB = SA
SA = -SB
SB = TB
GO TO 510
C
C FORWARD RECURSION SECTION
C
560 IF (KT.EQ.2) GO TO 470
S1 = TEMP(1)
S2 = TEMP(2)
TX = 2.0D0/X
TM = DALPHA*TX
IF (IN.EQ.0) GO TO 580
C
C FORWARD RECUR TO INDEX ALPHA
C
DO 570 I=1,IN
S = S2
S2 = TM*S2 - S1
TM = TM + TX
S1 = S
570 CONTINUE
IF (NN.EQ.1) GO TO 600
S = S2
S2 = TM*S2 - S1
TM = TM + TX
S1 = S
580 CONTINUE
C
C FORWARD RECUR FROM INDEX ALPHA TO ALPHA+N-1
C
Y(1) = S1
Y(2) = S2
IF (NN.EQ.2) RETURN
DO 590 I=3,NN
Y(I) = TM*Y(I-1) - Y(I-2)
TM = TM + TX
590 CONTINUE
cBH write(*,*)'**0.3**DBESJ:Y(1)',Y(1)
RETURN
600 Y(1) = S2
cBH write(*,*)'**1**DBESJ:Y(1)',Y(1)
RETURN
C
C BACKWARD RECURSION WITH NORMALIZATION BY
C ASYMPTOTIC EXPANSION FOR NU TO INFINITY OR POWER SERIES.
C
610 CONTINUE
C COMPUTATION OF LAST ORDER FOR SERIES NORMALIZATION
AKM = DMAX1(3.0D0-FN,0.0D0)
KM = INT(SNGL(AKM))
TFN = FN + DBLE(FLOAT(KM))
TA = (GLN+TFN-0.9189385332D0-0.0833333333D0/TFN)/(TFN+0.5D0)
TA = XO2L - TA
TB = -(1.0D0-1.5D0/TFN)/TFN
AKM = TOLLN/(-TA+DSQRT(TA*TA-TOLLN*TB)) + 1.5D0
IN = KM + INT(SNGL(AKM))
GO TO 660
620 CONTINUE
C COMPUTATION OF LAST ORDER FOR ASYMPTOTIC EXPANSION NORMALIZATION
GLN = WK(3) + WK(2)
IF (WK(6).GT.30.0D0) GO TO 640
RDEN = (PP(4)*WK(6)+PP(3))*WK(6) + 1.0D0
RZDEN = PP(1) + PP(2)*WK(6)
TA = RZDEN/RDEN
IF (WK(1).LT.0.10D0) GO TO 630
TB = GLN/WK(5)
GO TO 650
630 TB=(1.259921049D0+(0.1679894730D0+0.0887944358D0*WK(1))*WK(1))
1 /WK(7)
GO TO 650
640 CONTINUE
TA = 0.5D0*TOLLN/WK(4)
TA=((0.0493827160D0*TA-0.1111111111D0)*TA+0.6666666667D0)*TA*WK(6)
IF (WK(1).LT.0.10D0) GO TO 630
TB = GLN/WK(5)
650 IN = INT(SNGL(TA/TB+1.5D0))
IF (IN.GT.INLIM) GO TO 310
660 CONTINUE
DTM = FNI + DBLE(FLOAT(IN))
TRX = 2.0D0/X
TM = (DTM+FNF)*TRX
TA = 0.0D0
TB = TOL
KK = 1
AK=1.0D0
670 CONTINUE
C
C BACKWARD RECUR UNINDEXED
C
DO 680 I=1,IN
S = TB
TB = TM*TB - TA
TA = S
DTM = DTM - 1.0D0
TM = (DTM+FNF)*TRX
680 CONTINUE
C NORMALIZATION
IF (KK.NE.1) GO TO 690
S=TEMP(3)
SA=TA/TB
TA=S
TB=S
IF(DABS(S).GT.SLIM) GO TO 685
TA=TA*RTOL
TB=TB*RTOL
AK=TOL
685 CONTINUE
TA=TA*SA
KK = 2
IN = NS
IF (NS.NE.0) GO TO 670
690 Y(NN) = TB*AK
NZ = N - NN
cBH write(*,*)'**1.1**DBESJ:Y(1)',Y(1)
IF (NN.EQ.1) RETURN
K = NN - 1
S=TB
TB = TM*TB - TA
TA=S
Y(K)=TB*AK
cBH write(*,*)'**1.2**DBESJ:Y(1)',Y(1)
IF (NN.EQ.2) RETURN
DTM = DTM - 1.0D0
TM = (DTM+FNF)*TRX
K=NN-2
C
C BACKWARD RECUR INDEXED
C
DO 700 I=3,NN
S=TB
TB = TM*TB - TA
TA=S
Y(K)=TB*AK
DTM = DTM - 1.0D0
TM = (DTM+FNF)*TRX
K = K - 1
700 CONTINUE
cBH write(*,*)'**2**DBESJ:Y(1)',Y(1)
RETURN
C
C
C
710 CONTINUE
CALL XERROR('DBESJ - ORDER, ALPHA, LESS THAN ZERO.', 37, 2, 1)
RETURN
720 CONTINUE
CALL XERROR('DBESJ - N LESS THAN ONE.', 24, 2, 1)
RETURN
730 CONTINUE
CALL XERROR('DBESJ - X LESS THAN ZERO.', 25, 2, 1)
RETURN
END
SUBROUTINE DJAIRY(X,RX,C,AI,DAI)
C***BEGIN PROLOGUE DJAIRY
C***REFER TO DBESJ,DBESY
C
C 1-2-74
C DJAIRY computes the Airy function AI(X)
C and its derivative DAI(X) for DASYJY
C
C INPUT
C
C X - Argument, computed by DASYJY, X unrestricted
C RX - RX=DSQRT(DABS(X)), computed by DASYJY
C C - C=2.*(DABS(X)**1.5)/3., computed by DASYJY
C
C OUTPUT
C
C AI - Value of function AI(X)
C DAI - Value of the derivative DAI(X)
C
C Written by
C
C D. E. Amos
C S. L. Daniel
C M. K. Weston
C***ROUTINES CALLED (NONE)
C***END PROLOGUE DJAIRY
C
INTEGER I, J, M1, M1D, M2, M2D, M3, M3D, M4, M4D, N1, N1D, N2,
1 N2D, N3, N3D, N4, N4D
DOUBLE PRECISION A,AI,AJN,AJP,AK1,AK2,AK3,B,C,CCV,CON1,CON2,
1 CON3, CON4, CON5, CV, DA, DAI, DAJN, DAJP, DAK1, DAK2, DAK3,
2 DB, EC, E1, E2, FPI12, F1, F2, RTRX, RX, SCV, T, TEMP1, TEMP2,
3 TT, X
DIMENSION AJP(19), AJN(19), A(15), B(15)
DIMENSION AK1(14), AK2(23), AK3(14)
DIMENSION DAJP(19), DAJN(19), DA(15), DB(15)
DIMENSION DAK1(14), DAK2(24), DAK3(14)
SAVE N1, N2, N3, N4, M1, M2, M3, M4, FPI12, CON1, CON2, CON3,
1 CON4, CON5, AK1, AK2, AK3, AJP, AJN, A, B,
2 N1D, N2D, N3D, N4D, M1D, M2D, M3D, M4D, DAK1, DAK2, DAK3,
3 DAJP, DAJN, DA, DB
DATA N1,N2,N3,N4/14,23,19,15/
DATA M1,M2,M3,M4/12,21,17,13/
DATA FPI12,CON1,CON2,CON3,CON4,CON5/
1 1.30899693899575D+00, 6.66666666666667D-01, 5.03154716196777D+00,
2 3.80004589867293D-01, 8.33333333333333D-01, 8.66025403784439D-01/
DATA AK1(1), AK1(2), AK1(3), AK1(4), AK1(5), AK1(6), AK1(7),
1 AK1(8), AK1(9), AK1(10),AK1(11),AK1(12),AK1(13),
2 AK1(14) / 2.20423090987793D-01,-1.25290242787700D-01,
3 1.03881163359194D-02, 8.22844152006343D-04,-2.34614345891226D-04,
4 1.63824280172116D-05, 3.06902589573189D-07,-1.29621999359332D-07,
5 8.22908158823668D-09, 1.53963968623298D-11,-3.39165465615682D-11,
6 2.03253257423626D-12,-1.10679546097884D-14,-5.16169497785080D-15/
DATA AK2(1), AK2(2), AK2(3), AK2(4), AK2(5), AK2(6), AK2(7),
1 AK2(8), AK2(9), AK2(10),AK2(11),AK2(12),AK2(13),AK2(14),
2 AK2(15),AK2(16),AK2(17),AK2(18),AK2(19),AK2(20),AK2(21),
3 AK2(22),AK2(23) / 2.74366150869598D-01, 5.39790969736903D-03,
4-1.57339220621190D-03, 4.27427528248750D-04,-1.12124917399925D-04,
5 2.88763171318904D-05,-7.36804225370554D-06, 1.87290209741024D-06,
6-4.75892793962291D-07, 1.21130416955909D-07,-3.09245374270614D-08,
7 7.92454705282654D-09,-2.03902447167914D-09, 5.26863056595742D-10,
8-1.36704767639569D-10, 3.56141039013708D-11,-9.31388296548430D-12,
9 2.44464450473635D-12,-6.43840261990955D-13, 1.70106030559349D-13,
1-4.50760104503281D-14, 1.19774799164811D-14,-3.19077040865066D-15/
DATA AK3(1), AK3(2), AK3(3), AK3(4), AK3(5), AK3(6), AK3(7),
1 AK3(8), AK3(9), AK3(10),AK3(11),AK3(12),AK3(13),
2 AK3(14) / 2.80271447340791D-01,-1.78127042844379D-03,