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liboctdyn.f
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************************ dwigner **************************************
subroutine dwigner(dj,jmax2,m2,mp2,beta,ndim)
implicit real*8(a-h,o-z)
* ***********************************************
* * Wigner matrizes: d^J_{M,M'} (cos(beta)) *
* * kept in dj(2*J) to consider *
* * halfinteger values of J *
* * input: *
* * jmax2 is 2 * jmax, jmax being *
* * the highest j required *
* * m2 and mp2: are (m * 2) and (mp *2) *
* * respectively *
* * beta: angle defined between 0 and Pi *
* ***********************************************
dimension dj(0:ndim)
if(jmax2.gt.ndim)then
write(6,*)' ** be carefull with dimensions in dwigner **'
write(6,*)' 2 jmax > ndim',jmax2,ndim
stop
endif
do j=0,jmax2
dj(j)=0.d0
enddo
jmin2 = max0(iabs(m2),iabs(mp2))
y=dcos(beta)
***> jmin=0 ---> M=M'=0 ---> d^J_{M,Mp} = P_J(cos(beta))
********************************************************
if(jmin2.eq.0)then
dj(jmin2)=1.d0
dj(jmin2+2)=y
do ind=jmin2+4,jmax2,2
xj=dble(ind)*0.5d0
indm1=ind-2
indm2=ind-4
dj(ind)=y*(2.d0*xj-1.d0)*dj(indm1)-(xj-1.d0)*dj(indm2)
dj(ind)=dj(ind)/xj
enddo
elseif(jmin2.gt.0)then
***> d^J_{M,M'} for J=jmin=max0(iabs(M),iabs(M')) > 0
*a) Normalization factor
facnum=0.d0
do i=1,jmin2
facnum=facnum+dlog(dble(i))
enddo
if(jmin2.eq.iabs(m2))then
mreal2=mp2
else
mreal2=m2
endif
facden1=0.d0
if(jmin2+mreal2.gt.0)then
do i=2,jmin2+mreal2,2
facden1=facden1+dlog(dble(i)*0.5d0)
enddo
endif
facden2=0.d0
if(jmin2-mreal2.gt.0)then
do i=2,jmin2-mreal2,2
facden2=facden2+dlog(dble(i)*0.5d0)
enddo
endif
fact=dexp(0.5d0*(facnum-facden1-facden2))
*b)
sign=1.d0
xsin=dsin(beta*0.5d0)
xcos=dcos(beta*0.5d0)
if(jmin2.eq.abs(m2))then
if(jmin2.eq.m2)then
if(mod((jmin2-mp2),2).eq.0)then
sign=(-1)**(int(dble(jmin2-mp2)*0.5d0+0.5d0))
a=xsin**(dble(jmin2-mp2)*0.5d0)
b=xcos**(dble(jmin2+mp2)*0.5d0)
else
write(6,*)' ** danger: complex phase **'
stop
endif
elseif(jmin2.eq.-m2)then
a=xsin**(dble(jmin2+mp2)*0.5d0)
b=xcos**(dble(jmin2-mp2)*0.5d0)
endif
elseif(jmin2.eq.abs(mp2))then
if(jmin2.eq.mp2)then
a=xsin**(dble(jmin2-m2)*0.5d0)
b=xcos**(dble(jmin2+m2)*0.5d0)
elseif(jmin2.eq.-mp2)then
if(mod((jmin2+m2),2).eq.0)then
sign=(-1)**int(dble(jmin2+m2)*0.5d0+0.5d0)
a=xsin**(dble(jmin2+m2)*0.5d0)
b=xcos**(dble(jmin2-m2)*0.5d0)
else
write(6,*)' ** danger: complex phase **'
stop
endif
endif
endif
*c)
dj(jmin2)=fact*sign*a*b
***> d^J_{M,M'} for J=jmin+1
xj=dble(jmin2)*0.5d0
xjp=xj+1.d0
xm=dble(m2)*0.5d0
xmp=dble(mp2)*0.5d0
facnum=xjp*(2.d0*xj+1.d0)
facden1=xjp*xjp-xm*xm
facden2=xjp*xjp-xmp*xmp
factot=facnum/dsqrt(facden1*facden2)
facj= y - xm * xmp /(xj*(xj+1.d0))
ind=jmin2
indp=ind+2
if(indp.le.jmax2)then
dj(indp)=factot*facj*dj(ind)
***> d^J_{M,M'} for J>jmin+1
do indp=jmin2+4,jmax2,2
ind=indp-2
indm=indp-4
xjp=dble(indp)*0.5d0
xj=xjp-1.d0
xjm=xjp-2.d0
facnum=xjp*(2.d0*xj+1.d0)
facden1=xjp*xjp-xm*xm
facden2=xjp*xjp-xmp*xmp
factot=facnum/dsqrt(facden1*facden2)
facj= y - xm*xmp/(xj*(xj+1.d0))
fac1jm=xj*xj-xm*xm
fac2jm=xj*xj-xmp*xmp
facden=xj*(2.d0*xj+1.d0)
facjm=dsqrt(fac1jm*fac2jm)/facden
dj(indp)=factot*( facj*dj(ind) -facjm*dj(indm) )
enddo
endif
****************************************
endif
return
end
*************************** gauleg **************************
SUBROUTINE GAULEG(W,XX,n)
IMPLICIT real*8 (A-H,O-Z)
dimension W(n),XX(n)
x1=-1.d0
x2=1.d0
YN=DFLOAT(N)
EPS=1.D-15
M=(N+1)/2
XM=0.5D0*(X1+X2)
XL=0.5D0*(X2-X1)
PI=DACOS(-1.D0)
DO 12 I=1,M
YI=DFLOAT(I)
Z=DCOS(PI*(YI-0.25D0)/(YN+0.5D0))
1 CONTINUE
P1=1.D0
P2=0.D0
DO 11 J=1,N
P3=P2
P2=P1
YJ=DFLOAT(J)
P1=((2.D0*YJ-1.D0)*Z*P2-(YJ-1.D0)*P3)/YJ
11 CONTINUE
PP=YN*(Z*P1-P2)/(Z*Z-1.D0)
Z1=Z
Z=Z1-P1/PP
IF(DABS(Z-Z1).GT.EPS)GO TO 1
XX(I)=XM-XL*Z
XX(N+1-I)=XM+XL*Z
W(I)=2.D0*XL/((1.D0-Z*Z)*PP*PP)
W(N+1-I)=W(I)
12 CONTINUE
RETURN
END
******************** FACTORIAL *********************************
subroutine factorial
IMPLICIT real*8(A-H,O-Z)
common/fct/fact(0:10000)
FACT(0)=0.D0
fact(1)=0.d0
n=10000
DO 10 I=2,N
FACT(i)=FACT(i-1)+DLOG(DFLOAT(I))
10 CONTINUE
RETURN
END
************************* tqli ****************************
SUBROUTINE TQLI(D,E,N,NP)
implicit real*8(a-h,o-z)
****************************************************
*** diagonalization of a tridiagonal matrix ***
*** from numerical recipies ***
****************************************************
DIMENSION D(NP),E(NP)
IF (N.GT.1) THEN
c overide of silly num recipes convention for off-diagonal
c elements. It is now assumed off diags are in E(1)..E(N-1)
c DO 11 I=2,N
c E(I-1)=E(I)
c11 CONTINUE
E(N)=0.d0
DO 15 L=1,N
ITER=0
1 DO 12 M=L,N-1
DD=dABS(D(M))+dABS(D(M+1))
IF (dABS(E(M))+DD.EQ.DD) GO TO 2
12 CONTINUE
M=N
2 IF(M.NE.L)THEN
IF(ITER.EQ.500)write(6,*) 'too many iterations'
ITER=ITER+1
G=(D(L+1)-D(L))/(2.d0*E(L))
R=dSQRT(G**2+1.d0)
G=D(M)-D(L)+E(L)/(G+dSIGN(R,G))
S=1.d0
C=1.d0
P=0.d0
DO 14 I=M-1,L,-1
F=S*E(I)
B=C*E(I)
IF(dABS(F).GE.dABS(G))THEN
C=G/F
R=dSQRT(C**2+1.d0)
E(I+1)=F*R
S=1.d0/R
C=C*S
ELSE
S=F/G
R=dSQRT(S**2+1.d0)
E(I+1)=G*R
C=1.d0/R
S=S*C
ENDIF
G=D(I+1)-P
R=(D(I)-G)*S+2.d0*C*B
P=S*R
D(I+1)=G+P
G=C*R-B
14 CONTINUE
D(L)=D(L)-P
E(L)=G
E(M)=0.d0
GO TO 1
ENDIF
15 CONTINUE
ENDIF
RETURN
END
************************** SPLIN **********************************
c FUNCTION SPLINQ(F,X,IOLD,NX,R,ndim)
subroutine SPLINQQ(F,X,IOLD,NX,R,ndim,spl)
IMPLICIT real*8 (A-H,O-Z)
C **********************************************************************
C USE IOLD TO SAVE TIME IF SPLINT IS GOING TO BE CALLED FOR INCREASING
C VALUES OF R:
C - FOR 1ST CALL, SET IOLD TO 2
C - FOR THE FOLLOWING CALLS, USE THE OUT VALUE OF IOLD
C IF THE VALUES OF R ARE NOT MONOTONICALLY INCREASING, USE SPLINT OR
C SPLINQQ OR SPLINP
C----------------------------------------------------------------------
C C. LEFORESTIER, UNIVERSITE PARIS-SUD, ORSAY, FRANCE.
C MODIFICATIONS BY N. HALBERSTADT, CNRS, ORSAY, FRANCE.
C **********************************************************************
DIMENSION U(4)
DIMENSION F(ndim,2),X(ndim)
DATA UN/1D0/,TWO/2D0/,THREE/3D0/
C
IF(R.GE.X(NX)) GO TO 30
DO 10 IDOL = IOLD, NX
IF(R.LT.X(IDOL)) GOTO 20
10 CONTINUE
20 IOLD = IDOL
HI=X(IDOL)-X(IDOL-1)
XR=(R-X(IDOL-1))/HI
U(1)=XR*XR*(-TWO*XR+THREE)
U(3)=HI*XR*XR*(XR-UN)
U(2)=UN-U(1)
U(4)=HI*XR*((XR-TWO)*XR+UN)
SPLINQ=U(1)*F(IDOL,1)+U(2)*F(IDOL-1,1)
& +U(3)*F(IDOL,2)+U(4)*F(IDOL-1,2)
spl=splinq
RETURN
C >>> WARNING: THIS EXTENSION IS NOT VALID FOR EVERY POTENTIAL <<<
30 RO=X(NX)
YO=F(NX,1)
N2X=2*NX
YP=F(NX,2)
AIN=YO+YP*RO/6.D0
C8=-AIN*3.D0*RO**8
C6=YO*RO**6-C8/RO/RO
SPLINQ=C6/R**6+C8/R**8
spl=splinq
RETURN
END
SUBROUTINE SPLSET(F,X,NX,ndim)
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
C***********************************************************************
C* *
C* THIS ROUTINE SETS THE SPLINE INTERPOLATION ON THE GRID *
C* (X(I),I=1,NX) FOR THE FUNCTION (F(I),I=1,NX). *
C***********************************************************************
C
C
PARAMETER (NXMAX=20000)
C
DIMENSION F(ndim,2),X(ndim)
DIMENSION HX(NXMAX)
DIMENSION RLX(NXMAX-1),RMUX(NXMAX-1),XI(NXMAX-1),B(NXMAX-1)
DIMENSION AB(4),YZ(4),A(4)
C
DATA UN/1.0D0/,THREE/3.0D0/
C
IF (NX.GT.NXMAX) GO TO 999
C
NX2=NX-2
C COMPUTE THE HX'S
DO 10 I=2,NX
10 HX(I-1)=X(I)-X(I-1)
C COMPUTE LAMBDA'S & MU'S
DO 40 I=1,NX2
RLX(I)=HX(I+1)/(HX(I)+HX(I+1))
40 RMUX(I)=UN-RLX(I)
MAN=NX-3
DO 60 I=1,4
A(I)=X(MAN)
60 MAN=MAN+1
C
C SPLINE-FIT DE P(X)
DO 110 I=1,4
AB(I)=F(I,1)
110 YZ(I)=F(NX+I-4,1)
P0=DLAGRA(X,AB,4,1)
F(1,2)=P0
PN=DLAGRA(A,YZ,4,4)
F(NX,2)=PN
C CALCUL SECOND MEMBRE
DO 120 I=1,NX2
120 B(I)=THREE*RLX(I)/HX(I)*(F(I+1,1)-F(I,1))
& +THREE*RMUX(I)/HX(I+1)*(F(I+2,1)-F(I+1,1))
B(1)=B(1)-RLX(1)*P0
B(NX2)=B(NX2)-RMUX(NX2)*PN
CALL JORDAN(RMUX,RLX,XI,NX2,B)
DO 100 I=1,NX2
100 F(I+1,2)=XI(I)
RETURN
C
999 CONTINUE
PRINT 9000
PRINT 9999, NXMAX, NX
STOP
C
9000 FORMAT(//,2X,20('*'),' STOP IN SPLSET ',20('*'),/)
9999 FORMAT(2X,'DIMENSION PARAMETER NXMAX = ',I5,' TOO SMALL, '
& ,I5,' REQUIRED')
END
********************************* SCHR *************************
SUBROUTINE SCHR(E0,RMIN,RMAX,N,MAXIT,EPS,E2,KV,ITRY,v,p,npunt)
IMPLICIT real*8(A-H,O-Z)
DIMENSION Y(3),p(n),v(n)
ITRY=0
H=(RMAX-RMIN)/DFLOAT(N-1)
H2=H**2
HV=H2/12.d0
E=E0
TEST=-1.d0
DE=0.d0
DO 1 I=1,N
1 P(I)=0.d0
C BOUCLE DES ITERATIONS
12 DO 171 IT=1,MAXIT
XIT=IT
C INTEGRATION VERS L-INTERIEUR.PREMIERS PAS
P(N)=1.d-30
GN=V(N)-E
GI=V(N-1)-E
C E EST-IL TROP GRAND
IF(GI.GE.0.d0) GO TO 36
E=V(N-2)
GO TO 36
900 PRINT 899
899 FORMAT('LA TECHNIQUE UTILISEE EST EN DEFAUT')
ITRY=1
914 FORMAT(1H ,5(5X,E13.6))
RETURN
36 APR=(RMAX-H)*DSQRT(GI)-RMAX*DSQRT(GN)
IF(APR.GT.50.d0) APR=50.d0
P(N-1)=P(N)*DEXP(-APR)
38 Y(1)=(1.d0-HV*GN)*P(N)
40 Y(2)=(1.d0-HV*GI)*P(N-1)
C INTEGRATION
44 M=N-2
46 Y(3)=Y(2)+((Y(2)-Y(1))+H2*GI*P(M+1))
48 GI=V(M)-E
50 P(M)=Y(3)/(1.d0-HV*GI)
52 IF(DABS(P(M)).LT.1.d+34) GO TO 70
C DEPASSEMENT DE LA LIMITE
M1=M+1
PM=P(M1)
179 FORMAT(2X,'PM = ',E16.8/)
55 DO 56 J=M1,N
56 P(J)=P(J)/PM
58 Y(1)=Y(1)/PM
C
C NOUVEAU DEPART
60 Y(2)=Y(2)/PM
62 Y(3)=Y(3)/PM
GI=V(M+1)-E
GO TO 46
C L-INTEGRATION VERS L-INTERIEUR EST-ELLE TERMINEE
70 IF((DABS(P(M)).LE.DABS(P(M+1))).OR.(M.LE.2))GO TO 90
81 Y(1)=Y(2)
82 Y(2)=Y(3)
84 M=M-1
GO TO 46
C L-INTEGRATION VERS L-INTERIEUR EST TERMINEE
90 PM=P(M)
MSAVE=M
92 YIN=Y(2)/PM
94 DO 96 J=M,N
96 P(J)=P(J)/PM
C INTEGRATION VERS L-EXTERIEUR.PREMIERS PAS
100 P(1)=1.d0
102 Y(1)=0.d0
104 GI=V(1)-E
106 Y(2)=(1.D0-HV*GI)*P(1)
C
C INTEGRATION
108 DO 132 I=2,M
110 Y(3)=Y(2)+((Y(2)-Y(1))+H2*GI*P(I-1))
112 GI=V(I)-E
114 P(I)=Y(3)/(1.d0-HV*GI)
116 IF(DABS(P(I)).LT.1.d+34) GO TO 130
C LA LIMITE A ETE DEPASSEE
118 I1=I-1
PM=P(I1)
DO 120 J=1,I1
120 P(J)=P(J)/PM
122 Y(1)=Y(1)/PM
124 Y(2)=Y(2)/PM
126 Y(3)=Y(3)/PM
GI=V(I1)-E
GO TO 110
130 Y(1)=Y(2)
132 Y(2)=Y(3)
C L-INTEGRATION VERS L-EXTERIEUR EST TERMINEE
C
PM=P(M)
IF(PM) 135,149,135
135 YOUT=Y(1)/PM
136 YM=Y(3)/PM
138 DO 140 J=1,M
140 P(J)=P(J)/PM
C LES DEUX BRANCHES SONT MAINTENANT RACCORDEES
C CORRECTION
142 DF=0.d0
144 DO 146 J=1,N
146 DF=DF-P(J)**2
148 F=(-YOUT-YIN+2.d0*YM)/H2+(V(M)-E)
DOLD=DE
IF(DABS(F).LT.1.d+37) GO TO 150
149 F=9.99999d+29
DF=-F
DE=DABS(0.0001d0*E)
GO TO 152
150 DE=-F/DF
152 CONTINUE
156 FORMAT(I2,5X,E16.8,5X,E16.8,5X,E16.8,5X,E16.8)
164 EOLD=E
E=E+DE
TEST=DMAX1((DABS(DOLD)-DABS(DE)),TEST)
IF(TEST.LT.0.d0) GO TO 171
IF(DABS(E-EOLD).LT.EPS)GO TO 172
171 CONTINUE
SCHROD=1.d0
GO TO 173
C LES ITERATIONS SONT TERMINEES
172 SCHROD=0.d0
C LES ITERATIONS ONT CONVERGE
C COMPTAGE DES NOEUDS
173 KV=0
NL=N-2
174 DO 192 J=3,NL
176 IF(P(J))178,177,177
177 IF(P(J-1))180,192,192
178 IF(P(J-1))192,270,184
C NOEUD POSITIF
180 IF(P(J+1))192,182,182
182 IF(P(J-2))190,192,192
C NOEUD NEGATIF
184 IF(P(J+1))186,192,192
186 IF(P(J-2))192,190,190
C LE NOEUD EST-IL DU A UN SOUS-DEPASSEMENT
270 IF(P(J+1))280,192,192
280 IF(P(J-2))192,192,190
190 KV=KV+1
192 CONTINUE
E2=E
C NORMALISATION
200 SN=DSQRT(-H*DF)
202 DO 204 J=1,N
204 P(J)=P(J)/SN
250 FORMAT(10X,'SCHR=',I1,/)
C ECRITURE DE LA SOLUTION
214 FORMAT('V=',I3,5X,' E=',E16.8/)
228 FORMAT(10X,E16.8,10X,E16.8)
RETURN
END
SUBROUTINE BESJOT (L,X,F,DF,R)
C
C THIS PROGRAM GENERATES THE STANDARD AND MODIFIED VERSIONS OF THE SPHERICAL
C BESSEL-FUNCTIONS OF FIRST(BESJOT)- AND SECOND(BESSEN)-KIND RESPECTIVELY.
C FOR DEFINITIONS COMPARE: 'NBS-H1NDBOOK OF MATHEMATICAL FUNCTIONS',
C (ABRAMOWITZ+STEGUN,EDS./N.Y.:1964), SECTIONS 10.1.1 ON PAGE 437 FOR
C STANDARD VERSIONS AND SS.10.2.2 + 10.2.3 ON P.443 FOR THE MODIFIED ONES.
C L=INDEX(NATURAL NUMBERS INCLUDING ZERO), X=ARGUMENT(REAL,D.P.), F=OUTPUT.
C
C THE SIGN OF THE ARGUMENT IS USED TO DETERMINE THE VERSIONS:
C THE OUTCOMES F(=FIRST-KIND-FUNCTIONS) AND G(=SECOND-KIND-F.) MUST BE
C DIVIDED (RESP. MULTIPLIED) BY THE L-TH POWER OF THE REDUCTION LOGFAC
C TO GET THE MODIFIED VERSIONS, USE ARGUMENT WITH NEGATIVE SIGN }
C
C BY FORMULAS 10.1.31 ON PAGE 439 LOC.CIT. AND 10.2.7 ON P.443 IBID.,
C SOLUTIONS HAVE BEEN TESTED TO BE CORRECT TO TWELVE PLACES AT LEAST IN THE
C RANGE COMBINING X=1...441 AND L=0...340 .
C
C BESJOT IS DIVIDED INTO THREE PARTS, CORRESPONDING TO WETHER X > L, O
C WHILE X < L, BEEING 0.5*X*X < 2*L OR 0.5*X*X > 2*L RESPECTIVELY .
C
C
C MODIF. POUR X PLUS GRAND QUE L DANS BESJOT : R=1
C QQ SOIT X DANS BESSEN : R=1
C POUR EVITER LES OVERFLOWS OU UNDERFLOWS DANS LE PROG. APPELE POUR 50
C VERSION JAN. 77
C
C J.M.LAUNAY, MEUDON, FRANCE
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
C
F = 0.D0
R = 1.D0
IF(X) 51,50,52
50 IF(L.EQ.0) F=1.D0
DF = 0.D0
RETURN
C
51 SINIX = DSINH(-X)
COSIX = DCOSH(-X)
W = -1.D0
GOTO 53
52 PI = 6.283185307179586D0
XR = DMOD(X,PI)
SINIX = DSIN(XR)
COSIX = DCOS(XR)
W = +1.D0
53 Z = 1.D0 / DABS(X)
A = DBLE(L)
R = A * Z
IF(DABS(X)-A) 2,2,1
2 IF(0.5D0*X*X-2.D0*A) 4,4,3
C
C FOR THE FOLLOWING VERSION SEE PAGE 439, SECTION 10.1.19 LOC.CIT., AN
C SECTION 10.1.11 ON PAGE 438 IBIDEM}
C FOR THE MODIFIED CASE LOOK UP SS.10.2.18 AND 10.2.13 ON PS.444 AND 4
C THIS VERSION IS USED, IF X > L .
C
1 F0 = SINIX * Z
G0 =-COSIX * Z
R=1.D0
IF(L) 11,11,12
11 F = F0
DF =-G0 - F0*Z
RETURN
12 IF(L-1) 13,13,14
13 F = W * (F0-COSIX) * Z
DF = F0 - 2.0D0*F*Z
RETURN
14 F1 = W * (F0-COSIX) * Z
IF(L.EQ.2) GOTO 15
J = L-2
DO 10 I=1,J
F2 = W * (F1*DBLE(2*I+1)*Z - F0 )
F0 = F1
10 F1 = F2
15 F = W * (F1*DBLE(2*L-1)*Z - F0 )
DF = F1 - (L+1)*F*Z
RETURN
C
C FOR THE FOLLOWING VERSION SEE PAGE 453, EXAMPLE 2 LOC.CIT.}
C THIS VERSION IS USED, IF X < L AND 0.5*X*X > 2*L .
C
3 N = A + 25.D0 + DSQRT(A)
B0 = 0.D0
B1 = 1.D0
DO 20 J=1,N
B2 = W * (B1*DBLE(2*(N-J)+3)*Z - B0/R) / R
B0 = B1
IF(N-L-J) 22,21,22
21 F = B2
GOTO 20
22 IF(N-L+1-J) 20,23,20
23 DF = B2
20 B1 = B2
DF = W**(L-1) * (DF/B1) * SINIX*Z
F = W**L * (F/B1) * SINIX*Z
DF = DF*R - (L+1)*F*Z
RETURN
C
C FOR THE FOLLOWING VERSION SEE P1GE 437, FORMULA 10.1.2 LOC.CIT.}
C FOR THE MODIFIED CASE FORMULA 10.2.5 ON PAGE 443 IS VALID.
C THIS VERSION IS USED, IF X < L AND 0.5*X*X < 2*L .
C
C
4 Y = -W * 0.5D0 * X * X
S0 = DBLE(2*L-1)
S1 = DBLE(2*L+1)
P0 = 1.D0
P1 = 1.D0
C0 = 1.D0
C1 = 1.D0
DO 30 I=1,15
S0 = S0 + 2.D0
S1 = S1 + 2.D0
P0 = Y*P0/(S0*DBLE(I))
P1 = Y*P1/(S1*DBLE(I))
C0 = C0 + P0
30 C1 = C1 + P1
Q = 1.D0
IF(L.EQ.1) GOTO 32
J = L - 1
DO 31 I=1,J
31 Q = Q * A/DBLE(2*I+1)
F = Q * A/DBLE(2*L+1) * C1
DF = Q*C0*R - DBLE(L+1)*F*Z
RETURN
32 F = C1 / 3.D0
DF = C0*R - 2.D0*F*Z
RETURN
C
C
ENTRY BESSEN(L,X,G,DG,R)
C
C SPHERICAL BESSEL-(AND MODIFIED BESSEL-) FUNCTIONS OF THE SECOND KIND
C THIS VERSION IS VALID FOR ALL INDICES AND ARGUMENTS.
C
G = 0.D0
A = DBLE(L)
R = 1.D0
IF(X) 61,60,62
60 WRITE(6,300)
300 FORMAT(1H0,'******* ARGUMENT OF SPHERICAL BESSEL-FUNCTION OF SECO
1ND KIND SHOULD NOT BE ZERO }')
RETURN
61 SINIX = DSINH(-X)
COSIX = DCOSH(-X)
W = -1.D0
GOTO 63
62 PI = 6.283185307179586D0
XR = DMOD(X,PI)
SINIX = DSIN(XR)
COSIX = DCOS(XR)
W = +1.D0
63 Z = 1.D0 / DABS(X)
G0 =-W * COSIX * Z
F0 = W * SINIX * Z
IF(L) 41,41,42
41 G = G0
DG = W*F0 - G0*Z
RETURN
42 R=1.D0
IF(L-1) 43,43,44
43 G = W * (G0-SINIX)*Z
DG = G0 - 2.0D0*G*Z
RETURN
44 G1 = W * (G0-SINIX)*Z
IF(L.EQ.2) GOTO 45
J = L-2
DO 40 I=1,J
G2 = W * (G1*DBLE(2*I+1)*Z - G0 )
G0 = G1
40 G1 = G2
45 G = W * (G1*DBLE(2*L-1)*Z - G0 )
DG = G1 - DBLE(L+1)*G*Z
RETURN
END
******************************* BESSEL ***********************************
SUBROUTINE BESPH2 (F,DF,G,DG,AA,ARG,KEY,IBUG)
C#######################################################################
C# CALCULATES LINEARLY INDEPENDANT SOLUTIONS OF THE EQUATION #
C# 2 2 2 #
C# ( D / DX - A / X + 1 ) Y(X) = 0 #
C# - #
C# WHERE A = L*(L+1) WITH L INTEGER .GE. 0 #
C# + (-) SIGN CORRESPONDS TO AN OPEN (CLOSED) CHANNEL #
C# THE SOLUTIONS ARE OBTAINED FROM BESSEL FUNCTIONS OBTAINED #
C# IN BESJOT, BESSEN, BESSIK SUBROUTINES #
C# SEE ABRAMOWITZ AND STEGUN (CHAP. 10) #
C# ASYMPTOTIC BEHAVIOUR IS : #
C# F # SIN (X-L*PI/2) ; G # -COS (X-L*PI/2) FOR OPEN CHANNELS #
C# F # SINH (X); G # EXP (-X) FOR CLOSED CHANNELS #
C#---------------------------------------------------------------------#
C# AA : L*(L+1) #
C# ARG : ARGUMENT VALUE #
C# IF POSITIVE THEN ARGUMENT Z IS REAL (Z = ARG) #
C# IF NEGATIVE THEN ARGUMENT Z IS IMAGINARY (Z = -I*ARG) #
C# WHERE I = (-1)**0.5 #
C# F,G,DF,DG : REGULAR AND IRREGULAR FUNCTIONS AND THEIR DERIVATIVES#
C# KEY : .LT.0 TO SUPPRESS EXPONENTIAL FACTORS IN BESSIK #
C# IBUG : .GT. 0 TO PRINT THE OUTPUT #
C#---------------------------------------------------------------------#
C# HAVE BEEN TESTED ON WRONSKIAN RELATION W(F,G) = F*DG-DF*G = 1 #
C# FOR THE FOLLOWING VALUES OF THE ARGUMENT AND ORDERS : #
C# 0 =< L < 100 AND 0 =< Z < 100 (ERROR IN W LESS THAN 10**-12)#
C# 0 =< L < 30 AND 0 =< Z < 100*I (ERROR IN W LESS THAN 10**-10)#
C# OUTSIDE THIS RANGE CHECK FOR UNDERFLOWS, OVERFLOWS, DIVIDE CHECKS#
C# AND DEXP CAPACITY. #
C#---------------------------------------------------------------------#
C# J.M.L. 08/1981 ; UPDATE : 12/1981 #
C# J.M.LAUNAY, MEUDON, FRANCE #
C#######################################################################
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION FF(2),DFF(2)
COMMON /TBESP2/ TIME(2,2)
C DATA PI /3.1415 92653 58979 32384 D0/,TINY /1.D-10/
pi=dacos(-1.d0)
tiny=1.d-15
C
IF (AA .LT. -0.25D0) GO TO 10
FL = -0.5D0+DSQRT(AA+0.25D0)+TINY
L = FL
IF (DABS(FL-L) .LT. 2.D0*TINY) GO TO 20
10 PRINT 9010,AA
RETURN
C
20 X = DABS(ARG)
SK2 = -1.
IF (ARG .GT. 0.D0) GO TO 100
TWOPIM = 2.D0/PI
LP = L+1
LM = L-1
CALL BESSIK (L ,X,BI ,BK ,KEY)
CALL BESSIK (LM,X,BIM,BKM,KEY)
CALL BESSIK (LP,X,BIP,BKP,KEY)
F = -BI*X
DF = -( (L*BIM+LP*BIP)/(L+LP)*X+BI)
G = BK*X*TWOPIM
DG = (-(L*BKM+LP*BKP)/(L+LP)*X+BK)*TWOPIM
GO TO 1000
C
100 CALL BESJOT (L,X,BJ,DBJ,R)
CALL BESSEN (L,X,BN,DBN,RG)
SK2 = 1.
IF (L .EQ. 0) GO TO 200
DO 210 I = 1,L
BJ = BJ/R
DBJ = DBJ/R
BN = BN*RG
DBN = DBN*RG
210 CONTINUE
200 F = X*BJ
G = X*BN
DF = X*DBJ+BJ
DG = X*DBN+BN
C
C --- TIME SURFACE INTEGRALS CALCULATION
C
1000 FF(1) = F
FF(2) = G
DFF(1) = DF
DFF(2) = DG
VV = AA/(X*X)-SK2
C DO 1010 I = 1,2
C DO 1011 J = 1,2
C TIME(I,J) = -0.125*(FF(I)*DFF(J)+DFF(I)*FF(J)
C & +2.*X*(FF(I)*VV*FF(J)-DFF(I)*DFF(J)) )
1011 CONTINUE
1010 CONTINUE
IF (IBUG .LE. 0) RETURN
C
2000 WM1 = F*DG-DF*G-1.D0
C PRINT 9000,AA,ARG,F,DF,G,DG,WM1
RETURN
C
9000 FORMAT (' BESPH2 ',1F12.4,1F12.4,1P,4D20.12,1D9.1)
9010 FORMAT (' ****** BESPH2 ROUTINE; AA = ',F12.2,' IS NOT L*(L+1)'
& ,' WITH L INTEGER .GE. 0; REQUEST ABORTED ******')
END
SUBROUTINE BESSIK (L,X,BI,BK,KEY)
C#######################################################################
C# CALCULATION OF MODIFIED SPHERICAL BESSEL FUNCTIONS OF THE THIRD #
C# KIND : #
C# (PI/(2*X))**1/2 * I (X) BY FORMULA 10.2.5 #
C# L+1/2 FROM THE HANDBOOK (P.443) #
C# (PI/(2*X))**1/2 * K (X) BY FORMULA 10.2.15 #
C# L+1/2 FROM THE HANDBOOK (P.444) #
C#---------------------------------------------------------------------#
C# L : ORDER (INTEGER) #
C# X : ARGUMENT (POSITIVE REAL NUMBER) #
C# BI,BK : BESSEL FUNCTIONS ; #
C# KEY =0: NORMAL, >0: MULTIPLIES BI BY EXP(X), BK BY EXP(-X) #
C# <0: '' BI BY EXP(-X), BK BY EXP(X) #
C#######################################################################
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
COMMON /LGFAC/ FCT(50000)
DATA NTIME /0/
PI=DACOS(-1.D0)
FACTOR=DLOG(1.D-30)
C
NTIME = NTIME+1
IF (NTIME .EQ. 1) CALL FACLG
BI = 0.D0
BK = 0.D0
IF (L .LT. 0) RETURN
FL = L
S2 = DLOG(2.D0)
SX = DLOG(X)
HOX = 0.5D0/X
CI = FL*SX
SHX2 = DLOG(0.5D0)+2.D0*SX
ALFA=0.D0
IF(KEY.LT.0)ALFA=X
IF(KEY.GT.0)ALFA=-X
BETA=X-ALFA
C COMPUTATION OF BK
SUM = HOX*DEXP(-BETA)
IF (L .EQ. 0) GO TO 100
SHOX = DLOG(HOX)
SHOX1 = 0.D0
KMIN = 0
KMAX = L
SUM = 0.D0
C
DO 10 K = KMIN,KMAX
SHOX1 = SHOX+SHOX1
ST = FCT(L+K+1)-FCT(K+1)-FCT(L-K+1)+SHOX1
IF (ST .LT. -180.D0) GO TO 100
SUM = SUM+DEXP(ST-BETA)
10 CONTINUE
100 BK = PI*SUM
C COMPUTATION OF BI
SUM = 0.D0
K = 0
LK = L
200 CONTINUE
ST = K*SHX2-FCT(K+1)-( FCT(2*LK+2)-LK*S2-FCT(LK+1) )
K = K+1
LK = L+K
IF(2*LK+2.GT.5000)GO TO 9990
TERM=DEXP(ST-ALFA+CI)
SUM=SUM+TERM
SSUM=0.D0
IF(SUM.GT.0.D0)SSUM=DLOG(SUM)
IF(SSUM+ALFA.LT.FACTOR)GO TO 200
IF(SSUM.EQ.0.D0)GO TO 200
IF((TERM/SUM).GT.1.D-20)GO TO 200
BI=SUM
RETURN
9990 CONTINUE
PRINT *,'***> ERROR IN BESSIK: MAX. VALUE OF FACTORIAL=',2*LK+2
STOP
END
C***********************************************************************
SUBROUTINE JORDAN(MU,LAMBDA,X,N,B)
C***********************************************************************
IMPLICIT DOUBLE PRECISION(A-H,L-M,O-Z)
C
PARAMETER(NX2MAX=20000)
C
DIMENSION MU(N),LAMBDA(N),X(N),B(N)
DIMENSION PIV(NX2MAX)
C
IF(N.GT.NX2MAX) GO TO 999
C
C
C CALCUL DES PIVOTS
PIV(1)=2.D0
DO 10 I=2,N
PIV(I)=2.D0-LAMBDA(I)*MU(I-1)/PIV(I-1)
10 B(I)=B(I)-LAMBDA(I)/PIV(I-1)*B(I-1)
C
X(N)=B(N)/PIV(N)
I=N-1
20 X(I)=(B(I)-X(I+1)*MU(I))/PIV(I)
I=I-1
IF(I.GT.0) GOTO 20
RETURN
C
999 CONTINUE
PRINT 9000
PRINT 9999, NX2MAX, N
STOP
9000 FORMAT(//,2X,20('*'),' STOP IN JORDAN ',20('*'),/)
9999 FORMAT(2X,'DIMENSION PARAMETER NX2MAX TOO SMALL , ',I5,
& 'REQUIRED')
END
C***********************************************************************
DOUBLE PRECISION FUNCTION DLAGRA(X,Y,MIN,IP)
C***********************************************************************
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DIMENSION X(MIN),Y(MIN)
DLAGRA=0.D0
DO 10 I=1,MIN
IF(I.EQ.IP) GOTO 10
YP=Y(I)
DO 20 J=1,MIN
IF(J.EQ.IP) GOTO 20
IF(J.EQ.I) GOTO 20
YP=YP*(X(IP)-X(J))
20 CONTINUE
DO 30 J=1,MIN
IF(J.EQ.I) GOTO 30
YP=YP/(X(I)-X(J))
30 CONTINUE
DLAGRA=DLAGRA+YP
10 CONTINUE
DO 40 I=1,MIN
IF(I.EQ.IP) GOTO 40
DLAGRA=DLAGRA+Y(IP)/(X(IP)-X(I))
40 CONTINUE
RETURN
END
SUBROUTINE FACLG
C#######################################################################
C# INITIALISATION OF LOGARITHMS OF FACTORIALS ARRAY #
C#######################################################################
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
COMMON /LGFAC/ FCT(50000)
DATA NTIMES /0/
C
NTIMES = NTIMES+1
IF (NTIMES .GT. 1) RETURN
FCT(1) = 0.D0
DO 10 I = 1,4999
AI = I
FCT(I+1) = FCT(I)+DLOG(AI)
10 CONTINUE
C