cbknu.f
466 lines
| 13.9 KiB
| text/x-fortran
|
FortranFixedLexer
r1601 | *DECK CBKNU | |||
SUBROUTINE CBKNU (Z, FNU, KODE, N, Y, NZ, TOL, ELIM, ALIM) | ||||
C***BEGIN PROLOGUE CBKNU | ||||
C***SUBSIDIARY | ||||
C***PURPOSE Subsidiary to CAIRY, CBESH, CBESI and CBESK | ||||
C***LIBRARY SLATEC | ||||
C***TYPE ALL (CBKNU-A, ZBKNU-A) | ||||
C***AUTHOR Amos, D. E., (SNL) | ||||
C***DESCRIPTION | ||||
C | ||||
C CBKNU COMPUTES THE K BESSEL FUNCTION IN THE RIGHT HALF Z PLANE | ||||
C | ||||
C***SEE ALSO CAIRY, CBESH, CBESI, CBESK | ||||
C***ROUTINES CALLED CKSCL, CSHCH, CUCHK, GAMLN, I1MACH, R1MACH | ||||
C***REVISION HISTORY (YYMMDD) | ||||
C 830501 DATE WRITTEN | ||||
C 910415 Prologue converted to Version 4.0 format. (BAB) | ||||
C***END PROLOGUE CBKNU | ||||
C | ||||
COMPLEX CCH, CK, COEF, CONE, CRSC, CS, CSCL, CSH, CSR, CSS, CTWO, | ||||
* CZ, CZERO, F, FMU, P, PT, P1, P2, Q, RZ, SMU, ST, S1, S2, Y, Z, | ||||
* ZD, CELM, CY | ||||
REAL AA, AK, ALIM, ASCLE, A1, A2, BB, BK, BRY, CAZ, CC, DNU, | ||||
* DNU2, ELIM, ETEST, FC, FHS, FK, FKS, FNU, FPI, G1, G2, HPI, PI, | ||||
* P2I, P2M, P2R, RK, RTHPI, R1, S, SPI, TM, TOL, TTH, T1, T2, XX, | ||||
* YY, GAMLN, R1MACH, HELIM, ELM, XD, YD, ALAS, AS | ||||
INTEGER I, IDUM, IFLAG, INU, K, KFLAG, KK, KMAX, KODE, KODED, N, | ||||
* NZ, I1MACH, NW, J, IC, INUB | ||||
DIMENSION BRY(3), CC(8), CSS(3), CSR(3), Y(N), CY(2) | ||||
C | ||||
DATA KMAX / 30 / | ||||
DATA R1 / 2.0E0 / | ||||
DATA CZERO,CONE,CTWO /(0.0E0,0.0E0),(1.0E0,0.0E0),(2.0E0,0.0E0)/ | ||||
C | ||||
DATA PI, RTHPI, SPI ,HPI, FPI, TTH / | ||||
1 3.14159265358979324E0, 1.25331413731550025E0, | ||||
2 1.90985931710274403E0, 1.57079632679489662E0, | ||||
3 1.89769999331517738E0, 6.66666666666666666E-01/ | ||||
C | ||||
DATA CC(1), CC(2), CC(3), CC(4), CC(5), CC(6), CC(7), CC(8)/ | ||||
1 5.77215664901532861E-01, -4.20026350340952355E-02, | ||||
2 -4.21977345555443367E-02, 7.21894324666309954E-03, | ||||
3 -2.15241674114950973E-04, -2.01348547807882387E-05, | ||||
4 1.13302723198169588E-06, 6.11609510448141582E-09/ | ||||
C | ||||
C***FIRST EXECUTABLE STATEMENT CBKNU | ||||
XX = REAL(Z) | ||||
YY = AIMAG(Z) | ||||
CAZ = ABS(Z) | ||||
CSCL = CMPLX(1.0E0/TOL,0.0E0) | ||||
CRSC = CMPLX(TOL,0.0E0) | ||||
CSS(1) = CSCL | ||||
CSS(2) = CONE | ||||
CSS(3) = CRSC | ||||
CSR(1) = CRSC | ||||
CSR(2) = CONE | ||||
CSR(3) = CSCL | ||||
BRY(1) = 1.0E+3*R1MACH(1)/TOL | ||||
BRY(2) = 1.0E0/BRY(1) | ||||
BRY(3) = R1MACH(2) | ||||
NZ = 0 | ||||
IFLAG = 0 | ||||
KODED = KODE | ||||
RZ = CTWO/Z | ||||
INU = FNU+0.5E0 | ||||
DNU = FNU - INU | ||||
IF (ABS(DNU).EQ.0.5E0) GO TO 110 | ||||
DNU2 = 0.0E0 | ||||
IF (ABS(DNU).GT.TOL) DNU2 = DNU*DNU | ||||
IF (CAZ.GT.R1) GO TO 110 | ||||
C----------------------------------------------------------------------- | ||||
C SERIES FOR ABS(Z).LE.R1 | ||||
C----------------------------------------------------------------------- | ||||
FC = 1.0E0 | ||||
SMU = CLOG(RZ) | ||||
FMU = SMU*CMPLX(DNU,0.0E0) | ||||
CALL CSHCH(FMU, CSH, CCH) | ||||
IF (DNU.EQ.0.0E0) GO TO 10 | ||||
FC = DNU*PI | ||||
FC = FC/SIN(FC) | ||||
SMU = CSH*CMPLX(1.0E0/DNU,0.0E0) | ||||
10 CONTINUE | ||||
A2 = 1.0E0 + DNU | ||||
C----------------------------------------------------------------------- | ||||
C GAM(1-Z)*GAM(1+Z)=PI*Z/SIN(PI*Z), T1=1/GAM(1-DNU), T2=1/GAM(1+DNU) | ||||
C----------------------------------------------------------------------- | ||||
T2 = EXP(-GAMLN(A2,IDUM)) | ||||
T1 = 1.0E0/(T2*FC) | ||||
IF (ABS(DNU).GT.0.1E0) GO TO 40 | ||||
C----------------------------------------------------------------------- | ||||
C SERIES FOR F0 TO RESOLVE INDETERMINACY FOR SMALL ABS(DNU) | ||||
C----------------------------------------------------------------------- | ||||
AK = 1.0E0 | ||||
S = CC(1) | ||||
DO 20 K=2,8 | ||||
AK = AK*DNU2 | ||||
TM = CC(K)*AK | ||||
S = S + TM | ||||
IF (ABS(TM).LT.TOL) GO TO 30 | ||||
20 CONTINUE | ||||
30 G1 = -S | ||||
GO TO 50 | ||||
40 CONTINUE | ||||
G1 = (T1-T2)/(DNU+DNU) | ||||
50 CONTINUE | ||||
G2 = 0.5E0*(T1+T2)*FC | ||||
G1 = G1*FC | ||||
F = CMPLX(G1,0.0E0)*CCH + SMU*CMPLX(G2,0.0E0) | ||||
PT = CEXP(FMU) | ||||
P = CMPLX(0.5E0/T2,0.0E0)*PT | ||||
Q = CMPLX(0.5E0/T1,0.0E0)/PT | ||||
S1 = F | ||||
S2 = P | ||||
AK = 1.0E0 | ||||
A1 = 1.0E0 | ||||
CK = CONE | ||||
BK = 1.0E0 - DNU2 | ||||
IF (INU.GT.0 .OR. N.GT.1) GO TO 80 | ||||
C----------------------------------------------------------------------- | ||||
C GENERATE K(FNU,Z), 0.0D0 .LE. FNU .LT. 0.5D0 AND N=1 | ||||
C----------------------------------------------------------------------- | ||||
IF (CAZ.LT.TOL) GO TO 70 | ||||
CZ = Z*Z*CMPLX(0.25E0,0.0E0) | ||||
T1 = 0.25E0*CAZ*CAZ | ||||
60 CONTINUE | ||||
F = (F*CMPLX(AK,0.0E0)+P+Q)*CMPLX(1.0E0/BK,0.0E0) | ||||
P = P*CMPLX(1.0E0/(AK-DNU),0.0E0) | ||||
Q = Q*CMPLX(1.0E0/(AK+DNU),0.0E0) | ||||
RK = 1.0E0/AK | ||||
CK = CK*CZ*CMPLX(RK,0.0) | ||||
S1 = S1 + CK*F | ||||
A1 = A1*T1*RK | ||||
BK = BK + AK + AK + 1.0E0 | ||||
AK = AK + 1.0E0 | ||||
IF (A1.GT.TOL) GO TO 60 | ||||
70 CONTINUE | ||||
Y(1) = S1 | ||||
IF (KODED.EQ.1) RETURN | ||||
Y(1) = S1*CEXP(Z) | ||||
RETURN | ||||
C----------------------------------------------------------------------- | ||||
C GENERATE K(DNU,Z) AND K(DNU+1,Z) FOR FORWARD RECURRENCE | ||||
C----------------------------------------------------------------------- | ||||
80 CONTINUE | ||||
IF (CAZ.LT.TOL) GO TO 100 | ||||
CZ = Z*Z*CMPLX(0.25E0,0.0E0) | ||||
T1 = 0.25E0*CAZ*CAZ | ||||
90 CONTINUE | ||||
F = (F*CMPLX(AK,0.0E0)+P+Q)*CMPLX(1.0E0/BK,0.0E0) | ||||
P = P*CMPLX(1.0E0/(AK-DNU),0.0E0) | ||||
Q = Q*CMPLX(1.0E0/(AK+DNU),0.0E0) | ||||
RK = 1.0E0/AK | ||||
CK = CK*CZ*CMPLX(RK,0.0E0) | ||||
S1 = S1 + CK*F | ||||
S2 = S2 + CK*(P-F*CMPLX(AK,0.0E0)) | ||||
A1 = A1*T1*RK | ||||
BK = BK + AK + AK + 1.0E0 | ||||
AK = AK + 1.0E0 | ||||
IF (A1.GT.TOL) GO TO 90 | ||||
100 CONTINUE | ||||
KFLAG = 2 | ||||
BK = REAL(SMU) | ||||
A1 = FNU + 1.0E0 | ||||
AK = A1*ABS(BK) | ||||
IF (AK.GT.ALIM) KFLAG = 3 | ||||
P2 = S2*CSS(KFLAG) | ||||
S2 = P2*RZ | ||||
S1 = S1*CSS(KFLAG) | ||||
IF (KODED.EQ.1) GO TO 210 | ||||
F = CEXP(Z) | ||||
S1 = S1*F | ||||
S2 = S2*F | ||||
GO TO 210 | ||||
C----------------------------------------------------------------------- | ||||
C IFLAG=0 MEANS NO UNDERFLOW OCCURRED | ||||
C IFLAG=1 MEANS AN UNDERFLOW OCCURRED- COMPUTATION PROCEEDS WITH | ||||
C KODED=2 AND A TEST FOR ON SCALE VALUES IS MADE DURING FORWARD | ||||
C RECURSION | ||||
C----------------------------------------------------------------------- | ||||
110 CONTINUE | ||||
COEF = CMPLX(RTHPI,0.0E0)/CSQRT(Z) | ||||
KFLAG = 2 | ||||
IF (KODED.EQ.2) GO TO 120 | ||||
IF (XX.GT.ALIM) GO TO 290 | ||||
C BLANK LINE | ||||
A1 = EXP(-XX)*REAL(CSS(KFLAG)) | ||||
PT = CMPLX(A1,0.0E0)*CMPLX(COS(YY),-SIN(YY)) | ||||
COEF = COEF*PT | ||||
120 CONTINUE | ||||
IF (ABS(DNU).EQ.0.5E0) GO TO 300 | ||||
C----------------------------------------------------------------------- | ||||
C MILLER ALGORITHM FOR ABS(Z).GT.R1 | ||||
C----------------------------------------------------------------------- | ||||
AK = COS(PI*DNU) | ||||
AK = ABS(AK) | ||||
IF (AK.EQ.0.0E0) GO TO 300 | ||||
FHS = ABS(0.25E0-DNU2) | ||||
IF (FHS.EQ.0.0E0) GO TO 300 | ||||
C----------------------------------------------------------------------- | ||||
C COMPUTE R2=F(E). IF ABS(Z).GE.R2, USE FORWARD RECURRENCE TO | ||||
C DETERMINE THE BACKWARD INDEX K. R2=F(E) IS A STRAIGHT LINE ON | ||||
C 12.LE.E.LE.60. E IS COMPUTED FROM 2**(-E)=B**(1-I1MACH(11))= | ||||
C TOL WHERE B IS THE BASE OF THE ARITHMETIC. | ||||
C----------------------------------------------------------------------- | ||||
T1 = (I1MACH(11)-1)*R1MACH(5)*3.321928094E0 | ||||
T1 = MAX(T1,12.0E0) | ||||
T1 = MIN(T1,60.0E0) | ||||
T2 = TTH*T1 - 6.0E0 | ||||
IF (XX.NE.0.0E0) GO TO 130 | ||||
T1 = HPI | ||||
GO TO 140 | ||||
130 CONTINUE | ||||
T1 = ATAN(YY/XX) | ||||
T1 = ABS(T1) | ||||
140 CONTINUE | ||||
IF (T2.GT.CAZ) GO TO 170 | ||||
C----------------------------------------------------------------------- | ||||
C FORWARD RECURRENCE LOOP WHEN ABS(Z).GE.R2 | ||||
C----------------------------------------------------------------------- | ||||
ETEST = AK/(PI*CAZ*TOL) | ||||
FK = 1.0E0 | ||||
IF (ETEST.LT.1.0E0) GO TO 180 | ||||
FKS = 2.0E0 | ||||
RK = CAZ + CAZ + 2.0E0 | ||||
A1 = 0.0E0 | ||||
A2 = 1.0E0 | ||||
DO 150 I=1,KMAX | ||||
AK = FHS/FKS | ||||
BK = RK/(FK+1.0E0) | ||||
TM = A2 | ||||
A2 = BK*A2 - AK*A1 | ||||
A1 = TM | ||||
RK = RK + 2.0E0 | ||||
FKS = FKS + FK + FK + 2.0E0 | ||||
FHS = FHS + FK + FK | ||||
FK = FK + 1.0E0 | ||||
TM = ABS(A2)*FK | ||||
IF (ETEST.LT.TM) GO TO 160 | ||||
150 CONTINUE | ||||
GO TO 310 | ||||
160 CONTINUE | ||||
FK = FK + SPI*T1*SQRT(T2/CAZ) | ||||
FHS = ABS(0.25E0-DNU2) | ||||
GO TO 180 | ||||
170 CONTINUE | ||||
C----------------------------------------------------------------------- | ||||
C COMPUTE BACKWARD INDEX K FOR ABS(Z).LT.R2 | ||||
C----------------------------------------------------------------------- | ||||
A2 = SQRT(CAZ) | ||||
AK = FPI*AK/(TOL*SQRT(A2)) | ||||
AA = 3.0E0*T1/(1.0E0+CAZ) | ||||
BB = 14.7E0*T1/(28.0E0+CAZ) | ||||
AK = (ALOG(AK)+CAZ*COS(AA)/(1.0E0+0.008E0*CAZ))/COS(BB) | ||||
FK = 0.12125E0*AK*AK/CAZ + 1.5E0 | ||||
180 CONTINUE | ||||
K = FK | ||||
C----------------------------------------------------------------------- | ||||
C BACKWARD RECURRENCE LOOP FOR MILLER ALGORITHM | ||||
C----------------------------------------------------------------------- | ||||
FK = K | ||||
FKS = FK*FK | ||||
P1 = CZERO | ||||
P2 = CMPLX(TOL,0.0E0) | ||||
CS = P2 | ||||
DO 190 I=1,K | ||||
A1 = FKS - FK | ||||
A2 = (FKS+FK)/(A1+FHS) | ||||
RK = 2.0E0/(FK+1.0E0) | ||||
T1 = (FK+XX)*RK | ||||
T2 = YY*RK | ||||
PT = P2 | ||||
P2 = (P2*CMPLX(T1,T2)-P1)*CMPLX(A2,0.0E0) | ||||
P1 = PT | ||||
CS = CS + P2 | ||||
FKS = A1 - FK + 1.0E0 | ||||
FK = FK - 1.0E0 | ||||
190 CONTINUE | ||||
C----------------------------------------------------------------------- | ||||
C COMPUTE (P2/CS)=(P2/ABS(CS))*(CONJG(CS)/ABS(CS)) FOR BETTER | ||||
C SCALING | ||||
C----------------------------------------------------------------------- | ||||
TM = ABS(CS) | ||||
PT = CMPLX(1.0E0/TM,0.0E0) | ||||
S1 = PT*P2 | ||||
CS = CONJG(CS)*PT | ||||
S1 = COEF*S1*CS | ||||
IF (INU.GT.0 .OR. N.GT.1) GO TO 200 | ||||
ZD = Z | ||||
IF(IFLAG.EQ.1) GO TO 270 | ||||
GO TO 240 | ||||
200 CONTINUE | ||||
C----------------------------------------------------------------------- | ||||
C COMPUTE P1/P2=(P1/ABS(P2)*CONJG(P2)/ABS(P2) FOR SCALING | ||||
C----------------------------------------------------------------------- | ||||
TM = ABS(P2) | ||||
PT = CMPLX(1.0E0/TM,0.0E0) | ||||
P1 = PT*P1 | ||||
P2 = CONJG(P2)*PT | ||||
PT = P1*P2 | ||||
S2 = S1*(CONE+(CMPLX(DNU+0.5E0,0.0E0)-PT)/Z) | ||||
C----------------------------------------------------------------------- | ||||
C FORWARD RECURSION ON THE THREE TERM RECURSION RELATION WITH | ||||
C SCALING NEAR EXPONENT EXTREMES ON KFLAG=1 OR KFLAG=3 | ||||
C----------------------------------------------------------------------- | ||||
210 CONTINUE | ||||
CK = CMPLX(DNU+1.0E0,0.0E0)*RZ | ||||
IF (N.EQ.1) INU = INU - 1 | ||||
IF (INU.GT.0) GO TO 220 | ||||
IF (N.EQ.1) S1=S2 | ||||
ZD = Z | ||||
IF(IFLAG.EQ.1) GO TO 270 | ||||
GO TO 240 | ||||
220 CONTINUE | ||||
INUB = 1 | ||||
IF (IFLAG.EQ.1) GO TO 261 | ||||
225 CONTINUE | ||||
P1 = CSR(KFLAG) | ||||
ASCLE = BRY(KFLAG) | ||||
DO 230 I=INUB,INU | ||||
ST = S2 | ||||
S2 = CK*S2 + S1 | ||||
S1 = ST | ||||
CK = CK + RZ | ||||
IF (KFLAG.GE.3) GO TO 230 | ||||
P2 = S2*P1 | ||||
P2R = REAL(P2) | ||||
P2I = AIMAG(P2) | ||||
P2R = ABS(P2R) | ||||
P2I = ABS(P2I) | ||||
P2M = MAX(P2R,P2I) | ||||
IF (P2M.LE.ASCLE) GO TO 230 | ||||
KFLAG = KFLAG + 1 | ||||
ASCLE = BRY(KFLAG) | ||||
S1 = S1*P1 | ||||
S2 = P2 | ||||
S1 = S1*CSS(KFLAG) | ||||
S2 = S2*CSS(KFLAG) | ||||
P1 = CSR(KFLAG) | ||||
230 CONTINUE | ||||
IF (N.EQ.1) S1 = S2 | ||||
240 CONTINUE | ||||
Y(1) = S1*CSR(KFLAG) | ||||
IF (N.EQ.1) RETURN | ||||
Y(2) = S2*CSR(KFLAG) | ||||
IF (N.EQ.2) RETURN | ||||
KK = 2 | ||||
250 CONTINUE | ||||
KK = KK + 1 | ||||
IF (KK.GT.N) RETURN | ||||
P1 = CSR(KFLAG) | ||||
ASCLE = BRY(KFLAG) | ||||
DO 260 I=KK,N | ||||
P2 = S2 | ||||
S2 = CK*S2 + S1 | ||||
S1 = P2 | ||||
CK = CK + RZ | ||||
P2 = S2*P1 | ||||
Y(I) = P2 | ||||
IF (KFLAG.GE.3) GO TO 260 | ||||
P2R = REAL(P2) | ||||
P2I = AIMAG(P2) | ||||
P2R = ABS(P2R) | ||||
P2I = ABS(P2I) | ||||
P2M = MAX(P2R,P2I) | ||||
IF (P2M.LE.ASCLE) GO TO 260 | ||||
KFLAG = KFLAG + 1 | ||||
ASCLE = BRY(KFLAG) | ||||
S1 = S1*P1 | ||||
S2 = P2 | ||||
S1 = S1*CSS(KFLAG) | ||||
S2 = S2*CSS(KFLAG) | ||||
P1 = CSR(KFLAG) | ||||
260 CONTINUE | ||||
RETURN | ||||
C----------------------------------------------------------------------- | ||||
C IFLAG=1 CASES, FORWARD RECURRENCE ON SCALED VALUES ON UNDERFLOW | ||||
C----------------------------------------------------------------------- | ||||
261 CONTINUE | ||||
HELIM = 0.5E0*ELIM | ||||
ELM = EXP(-ELIM) | ||||
CELM = CMPLX(ELM,0.0) | ||||
ASCLE = BRY(1) | ||||
ZD = Z | ||||
XD = XX | ||||
YD = YY | ||||
IC = -1 | ||||
J = 2 | ||||
DO 262 I=1,INU | ||||
ST = S2 | ||||
S2 = CK*S2+S1 | ||||
S1 = ST | ||||
CK = CK+RZ | ||||
AS = ABS(S2) | ||||
ALAS = ALOG(AS) | ||||
P2R = -XD+ALAS | ||||
IF(P2R.LT.(-ELIM)) GO TO 263 | ||||
P2 = -ZD+CLOG(S2) | ||||
P2R = REAL(P2) | ||||
P2I = AIMAG(P2) | ||||
P2M = EXP(P2R)/TOL | ||||
P1 = CMPLX(P2M,0.0E0)*CMPLX(COS(P2I),SIN(P2I)) | ||||
CALL CUCHK(P1,NW,ASCLE,TOL) | ||||
IF(NW.NE.0) GO TO 263 | ||||
J=3-J | ||||
CY(J) = P1 | ||||
IF(IC.EQ.(I-1)) GO TO 264 | ||||
IC = I | ||||
GO TO 262 | ||||
263 CONTINUE | ||||
IF(ALAS.LT.HELIM) GO TO 262 | ||||
XD = XD-ELIM | ||||
S1 = S1*CELM | ||||
S2 = S2*CELM | ||||
ZD = CMPLX(XD,YD) | ||||
262 CONTINUE | ||||
IF(N.EQ.1) S1 = S2 | ||||
GO TO 270 | ||||
264 CONTINUE | ||||
KFLAG = 1 | ||||
INUB = I+1 | ||||
S2 = CY(J) | ||||
J = 3 - J | ||||
S1 = CY(J) | ||||
IF(INUB.LE.INU) GO TO 225 | ||||
IF(N.EQ.1) S1 = S2 | ||||
GO TO 240 | ||||
270 CONTINUE | ||||
Y(1) = S1 | ||||
IF (N.EQ.1) GO TO 280 | ||||
Y(2) = S2 | ||||
280 CONTINUE | ||||
ASCLE = BRY(1) | ||||
CALL CKSCL(ZD, FNU, N, Y, NZ, RZ, ASCLE, TOL, ELIM) | ||||
INU = N - NZ | ||||
IF (INU.LE.0) RETURN | ||||
KK = NZ + 1 | ||||
S1 = Y(KK) | ||||
Y(KK) = S1*CSR(1) | ||||
IF (INU.EQ.1) RETURN | ||||
KK = NZ + 2 | ||||
S2 = Y(KK) | ||||
Y(KK) = S2*CSR(1) | ||||
IF (INU.EQ.2) RETURN | ||||
T2 = FNU + (KK-1) | ||||
CK = CMPLX(T2,0.0E0)*RZ | ||||
KFLAG = 1 | ||||
GO TO 250 | ||||
290 CONTINUE | ||||
C----------------------------------------------------------------------- | ||||
C SCALE BY EXP(Z), IFLAG = 1 CASES | ||||
C----------------------------------------------------------------------- | ||||
KODED = 2 | ||||
IFLAG = 1 | ||||
KFLAG = 2 | ||||
GO TO 120 | ||||
C----------------------------------------------------------------------- | ||||
C FNU=HALF ODD INTEGER CASE, DNU=-0.5 | ||||
C----------------------------------------------------------------------- | ||||
300 CONTINUE | ||||
S1 = COEF | ||||
S2 = COEF | ||||
GO TO 210 | ||||
310 CONTINUE | ||||
NZ=-2 | ||||
RETURN | ||||
END | ||||