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rjbesl.f
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rjbesl.f
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SUBROUTINE RJBESL(X, ALPHA, NB, B, NCALC)
C---------------------------------------------------------------------
C This routine calculates Bessel functions J sub(N+ALPHA) (X)
C for non-negative argument X, and non-negative order N+ALPHA.
C
C
C Explanation of variables in the calling sequence.
C
C X - working precision non-negative real argument for which
C J's are to be calculated.
C ALPHA - working precision fractional part of order for which
C J's or exponentially scaled J'r (J*exp(X)) are
C to be calculated. 0 <= ALPHA < 1.0.
C NB - integer number of functions to be calculated, NB > 0.
C The first function calculated is of order ALPHA, and the
C last is of order (NB - 1 + ALPHA).
C B - working precision output vector of length NB. If RJBESL
C terminates normally (NCALC=NB), the vector B contains the
C functions J/ALPHA/(X) through J/NB-1+ALPHA/(X), or the
C corresponding exponentially scaled functions.
C NCALC - integer output variable indicating possible errors.
C Before using the vector B, the user should check that
C NCALC=NB, i.e., all orders have been calculated to
C the desired accuracy. See Error Returns below.
C
C
C*******************************************************************
C*******************************************************************
C
C Explanation of machine-dependent constants
C
C it = Number of bits in the mantissa of a working precision
C variable
C NSIG = Decimal significance desired. Should be set to
C INT(LOG10(2)*it+1). Setting NSIG lower will result
C in decreased accuracy while setting NSIG higher will
C increase CPU time without increasing accuracy. The
C truncation error is limited to a relative error of
C T=.5*10**(-NSIG).
C ENTEN = 10.0 ** K, where K is the largest integer such that
C ENTEN is machine-representable in working precision
C ENSIG = 10.0 ** NSIG
C RTNSIG = 10.0 ** (-K) for the smallest integer K such that
C K .GE. NSIG/4
C ENMTEN = Smallest ABS(X) such that X/4 does not underflow
C XLARGE = Upper limit on the magnitude of X. If ABS(X)=N,
C then at least N iterations of the backward recursion
C will be executed. The value of 10.0 ** 4 is used on
C every machine.
C
C
C Approximate values for some important machines are:
C
C
C it NSIG ENTEN ENSIG
C
C CRAY-1 (S.P.) 48 15 1.0E+2465 1.0E+15
C Cyber 180/855
C under NOS (S.P.) 48 15 1.0E+322 1.0E+15
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 24 8 1.0E+38 1.0E+8
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 53 16 1.0D+308 1.0D+16
C IBM 3033 (D.P.) 14 5 1.0D+75 1.0D+5
C VAX (S.P.) 24 8 1.0E+38 1.0E+8
C VAX D-Format (D.P.) 56 17 1.0D+38 1.0D+17
C VAX G-Format (D.P.) 53 16 1.0D+307 1.0D+16
C
C
C RTNSIG ENMTEN XLARGE
C
C CRAY-1 (S.P.) 1.0E-4 1.84E-2466 1.0E+4
C Cyber 180/855
C under NOS (S.P.) 1.0E-4 1.25E-293 1.0E+4
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 1.0E-2 4.70E-38 1.0E+4
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 1.0E-4 8.90D-308 1.0D+4
C IBM 3033 (D.P.) 1.0E-2 2.16D-78 1.0D+4
C VAX (S.P.) 1.0E-2 1.17E-38 1.0E+4
C VAX D-Format (D.P.) 1.0E-5 1.17D-38 1.0D+4
C VAX G-Format (D.P.) 1.0E-4 2.22D-308 1.0D+4
C
C*******************************************************************
C*******************************************************************
C
C Error returns
C
C In case of an error, NCALC .NE. NB, and not all J's are
C calculated to the desired accuracy.
C
C NCALC .LT. 0: An argument is out of range. For example,
C NBES .LE. 0, ALPHA .LT. 0 or .GT. 1, or X is too large.
C In this case, B(1) is set to zero, the remainder of the
C B-vector is not calculated, and NCALC is set to
C MIN(NB,0)-1 so that NCALC .NE. NB.
C
C NB .GT. NCALC .GT. 0: Not all requested function values could
C be calculated accurately. This usually occurs because NB is
C much larger than ABS(X). In this case, B(N) is calculated
C to the desired accuracy for N .LE. NCALC, but precision
C is lost for NCALC .LT. N .LE. NB. If B(N) does not vanish
C for N .GT. NCALC (because it is too small to be represented),
C and B(N)/B(NCALC) = 10**(-K), then only the first NSIG-K
C significant figures of B(N) can be trusted.
C
C
C Intrinsic and other functions required are:
C
C ABS, AINT, COS, DBLE, GAMMA (or DGAMMA), INT, MAX, MIN,
C
C REAL, SIN, SQRT
C
C
C Acknowledgement
C
C This program is based on a program written by David J. Sookne
C (2) that computes values of the Bessel functions J or I of real
C argument and integer order. Modifications include the restriction
C of the computation to the J Bessel function of non-negative real
C argument, the extension of the computation to arbitrary positive
C order, and the elimination of most underflow.
C
C References: "A Note on Backward Recurrence Algorithms," Olver,
C F. W. J., and Sookne, D. J., Math. Comp. 26, 1972,
C pp 941-947.
C
C "Bessel Functions of Real Argument and Integer Order,"
C Sookne, D. J., NBS Jour. of Res. B. 77B, 1973, pp
C 125-132.
C
C Latest modification: March 19, 1990
C
C Author: W. J. Cody
C Applied Mathematics Division
C Argonne National Laboratory
C Argonne, IL 60439
C
C---------------------------------------------------------------------
INTEGER I,J,K,L,M,MAGX,N,NB,NBMX,NCALC,NEND,NSTART
CS REAL GAMMA,
DOUBLE PRECISION DGAMMA,
1 ALPHA,ALPEM,ALP2EM,B,CAPP,CAPQ,CONV,EIGHTH,EM,EN,ENMTEN,ENSIG,
2 ENTEN,FACT,FOUR,FUNC,GNU,HALF,HALFX,ONE,ONE30,P,PI2,PLAST,
3 POLD,PSAVE,PSAVEL,RTNSIG,S,SUM,T,T1,TEMPA,TEMPB,TEMPC,TEST,
4 THREE,THREE5,TOVER,TWO,TWOFIV,TWOPI1,TWOPI2,X,XC,XIN,XK,XLARGE,
5 XM,VCOS,VSIN,Z,ZERO
DIMENSION B(NB), FACT(25)
C---------------------------------------------------------------------
C Mathematical constants
C
C PI2 - 2 / PI
C TWOPI1 - first few significant digits of 2 * PI
C TWOPI2 - (2*PI - TWOPI) to working precision, i.e.,
C TWOPI1 + TWOPI2 = 2 * PI to extra precision.
C---------------------------------------------------------------------
CS DATA PI2, TWOPI1, TWOPI2 /0.636619772367581343075535E0,6.28125E0,
CS 1 1.935307179586476925286767E-3/
CS DATA ZERO, EIGHTH, HALF, ONE /0.0E0,0.125E0,0.5E0,1.0E0/
CS DATA TWO, THREE, FOUR, TWOFIV /2.0E0,3.0E0,4.0E0,25.0E0/
CS DATA ONE30, THREE5 /130.0E0,35.0E0/
DATA PI2, TWOPI1, TWOPI2 /0.636619772367581343075535D0,6.28125D0,
1 1.935307179586476925286767D-3/
DATA ZERO, EIGHTH, HALF, ONE /0.0D0,0.125D0,0.5D0,1.0D0/
DATA TWO, THREE, FOUR, TWOFIV /2.0D0,3.0D0,4.0D0,25.0D0/
DATA ONE30, THREE5 /130.0D0,35.0D0/
C---------------------------------------------------------------------
C Machine-dependent parameters
C---------------------------------------------------------------------
CS DATA ENTEN, ENSIG, RTNSIG /1.0E38,1.0E8,1.0E-2/
CS DATA ENMTEN, XLARGE /1.2E-37,1.0E4/
DATA ENTEN, ENSIG, RTNSIG /1.0D308,1.0D16,1.0D-4/
DATA ENMTEN, XLARGE /8.9D-308,1.0D4/
C---------------------------------------------------------------------
C Factorial(N)
C---------------------------------------------------------------------
CS DATA FACT /1.0E0,1.0E0,2.0E0,6.0E0,24.0E0,1.2E2,7.2E2,5.04E3,
CS 1 4.032E4,3.6288E5,3.6288E6,3.99168E7,4.790016E8,6.2270208E9,
CS 2 8.71782912E10,1.307674368E12,2.0922789888E13,3.55687428096E14,
CS 3 6.402373705728E15,1.21645100408832E17,2.43290200817664E18,
CS 4 5.109094217170944E19,1.12400072777760768E21,
CS 5 2.585201673888497664E22,6.2044840173323943936E23/
DATA FACT /1.0D0,1.0D0,2.0D0,6.0D0,24.0D0,1.2D2,7.2D2,5.04D3,
1 4.032D4,3.6288D5,3.6288D6,3.99168D7,4.790016D8,6.2270208D9,
2 8.71782912D10,1.307674368D12,2.0922789888D13,3.55687428096D14,
3 6.402373705728D15,1.21645100408832D17,2.43290200817664D18,
4 5.109094217170944D19,1.12400072777760768D21,
5 2.585201673888497664D22,6.2044840173323943936D23/
C---------------------------------------------------------------------
C Statement functions for conversion and the gamma function.
C---------------------------------------------------------------------
CS CONV(I) = REAL(I)
CS FUNC(X) = GAMMA(X)
CONV(I) = DBLE(I)
FUNC(X) = DGAMMA(X)
C---------------------------------------------------------------------
C Check for out of range arguments.
C---------------------------------------------------------------------
MAGX = INT(X)
IF ((NB.GT.0) .AND. (X.GE.ZERO) .AND. (X.LE.XLARGE)
1 .AND. (ALPHA.GE.ZERO) .AND. (ALPHA.LT.ONE))
2 THEN
C---------------------------------------------------------------------
C Initialize result array to zero.
C---------------------------------------------------------------------
NCALC = NB
DO 20 I=1,NB
B(I) = ZERO
20 CONTINUE
C---------------------------------------------------------------------
C Branch to use 2-term ascending series for small X and asymptotic
C form for large X when NB is not too large.
C---------------------------------------------------------------------
IF (X.LT.RTNSIG) THEN
C---------------------------------------------------------------------
C Two-term ascending series for small X.
C---------------------------------------------------------------------
TEMPA = ONE
ALPEM = ONE + ALPHA
HALFX = ZERO
IF (X.GT.ENMTEN) HALFX = HALF*X
IF (ALPHA.NE.ZERO)
1 TEMPA = HALFX**ALPHA/(ALPHA*FUNC(ALPHA))
TEMPB = ZERO
IF ((X+ONE).GT.ONE) TEMPB = -HALFX*HALFX
B(1) = TEMPA + TEMPA*TEMPB/ALPEM
IF ((X.NE.ZERO) .AND. (B(1).EQ.ZERO)) NCALC = 0
IF (NB .NE. 1) THEN
IF (X .LE. ZERO) THEN
DO 30 N=2,NB
B(N) = ZERO
30 CONTINUE
ELSE
C---------------------------------------------------------------------
C Calculate higher order functions.
C---------------------------------------------------------------------
TEMPC = HALFX
TOVER = (ENMTEN+ENMTEN)/X
IF (TEMPB.NE.ZERO) TOVER = ENMTEN/TEMPB
DO 50 N=2,NB
TEMPA = TEMPA/ALPEM
ALPEM = ALPEM + ONE
TEMPA = TEMPA*TEMPC
IF (TEMPA.LE.TOVER*ALPEM) TEMPA = ZERO
B(N) = TEMPA + TEMPA*TEMPB/ALPEM
IF ((B(N).EQ.ZERO) .AND. (NCALC.GT.N))
1 NCALC = N-1
50 CONTINUE
END IF
END IF
ELSE IF ((X.GT.TWOFIV) .AND. (NB.LE.MAGX+1)) THEN
C---------------------------------------------------------------------
C Asymptotic series for X .GT. 21.0.
C---------------------------------------------------------------------
XC = SQRT(PI2/X)
XIN = (EIGHTH/X)**2
M = 11
IF (X.GE.THREE5) M = 8
IF (X.GE.ONE30) M = 4
XM = FOUR*CONV(M)
C---------------------------------------------------------------------
C Argument reduction for SIN and COS routines.
C---------------------------------------------------------------------
T = AINT(X/(TWOPI1+TWOPI2)+HALF)
Z = ((X-T*TWOPI1)-T*TWOPI2) - (ALPHA+HALF)/PI2
VSIN = SIN(Z)
VCOS = COS(Z)
GNU = ALPHA + ALPHA
DO 80 I=1,2
S = ((XM-ONE)-GNU)*((XM-ONE)+GNU)*XIN*HALF
T = (GNU-(XM-THREE))*(GNU+(XM-THREE))
CAPP = S*T/FACT(2*M+1)
T1 = (GNU-(XM+ONE))*(GNU+(XM+ONE))
CAPQ = S*T1/FACT(2*M+2)
XK = XM
K = M + M
T1 = T
DO 70 J=2,M
XK = XK - FOUR
S = ((XK-ONE)-GNU)*((XK-ONE)+GNU)
T = (GNU-(XK-THREE))*(GNU+(XK-THREE))
CAPP = (CAPP+ONE/FACT(K-1))*S*T*XIN
CAPQ = (CAPQ+ONE/FACT(K))*S*T1*XIN
K = K - 2
T1 = T
70 CONTINUE
CAPP = CAPP + ONE
CAPQ = (CAPQ+ONE)*(GNU*GNU-ONE)*(EIGHTH/X)
B(I) = XC*(CAPP*VCOS-CAPQ*VSIN)
IF (NB.EQ.1) GO TO 300
T = VSIN
VSIN = -VCOS
VCOS = T
GNU = GNU + TWO
80 CONTINUE
C---------------------------------------------------------------------
C If NB .GT. 2, compute J(X,ORDER+I) I = 2, NB-1
C---------------------------------------------------------------------
IF (NB .GT. 2) THEN
GNU = ALPHA + ALPHA + TWO
DO 90 J=3,NB
B(J) = GNU*B(J-1)/X - B(J-2)
GNU = GNU + TWO
90 CONTINUE
END IF
C---------------------------------------------------------------------
C Use recurrence to generate results. First initialize the
C calculation of P*S.
C---------------------------------------------------------------------
ELSE
NBMX = NB - MAGX
N = MAGX + 1
EN = CONV(N+N) + (ALPHA+ALPHA)
PLAST = ONE
P = EN/X
C---------------------------------------------------------------------
C Calculate general significance test.
C---------------------------------------------------------------------
TEST = ENSIG + ENSIG
IF (NBMX .GE. 3) THEN
C---------------------------------------------------------------------
C Calculate P*S until N = NB-1. Check for possible overflow.
C---------------------------------------------------------------------
TOVER = ENTEN/ENSIG
NSTART = MAGX + 2
NEND = NB - 1
EN = CONV(NSTART+NSTART) - TWO + (ALPHA+ALPHA)
DO 130 K=NSTART,NEND
N = K
EN = EN + TWO
POLD = PLAST
PLAST = P
P = EN*PLAST/X - POLD
IF (P.GT.TOVER) THEN
C---------------------------------------------------------------------
C To avoid overflow, divide P*S by TOVER. Calculate P*S until
C ABS(P) .GT. 1.
C---------------------------------------------------------------------
TOVER = ENTEN
P = P/TOVER
PLAST = PLAST/TOVER
PSAVE = P
PSAVEL = PLAST
NSTART = N + 1
100 N = N + 1
EN = EN + TWO
POLD = PLAST
PLAST = P
P = EN*PLAST/X - POLD
IF (P.LE.ONE) GO TO 100
TEMPB = EN/X
C---------------------------------------------------------------------
C Calculate backward test and find NCALC, the highest N such that
C the test is passed.
C---------------------------------------------------------------------
TEST = POLD*PLAST*(HALF-HALF/(TEMPB*TEMPB))
TEST = TEST/ENSIG
P = PLAST*TOVER
N = N - 1
EN = EN - TWO
NEND = MIN(NB,N)
DO 110 L=NSTART,NEND
POLD = PSAVEL
PSAVEL = PSAVE
PSAVE = EN*PSAVEL/X - POLD
IF (PSAVE*PSAVEL.GT.TEST) THEN
NCALC = L - 1
GO TO 190
END IF
110 CONTINUE
NCALC = NEND
GO TO 190
END IF
130 CONTINUE
N = NEND
EN = CONV(N+N) + (ALPHA+ALPHA)
C---------------------------------------------------------------------
C Calculate special significance test for NBMX .GT. 2.
C---------------------------------------------------------------------
TEST = MAX(TEST,SQRT(PLAST*ENSIG)*SQRT(P+P))
END IF
C---------------------------------------------------------------------
C Calculate P*S until significance test passes.
C---------------------------------------------------------------------
140 N = N + 1
EN = EN + TWO
POLD = PLAST
PLAST = P
P = EN*PLAST/X - POLD
IF (P.LT.TEST) GO TO 140
C---------------------------------------------------------------------
C Initialize the backward recursion and the normalization sum.
C---------------------------------------------------------------------
190 N = N + 1
EN = EN + TWO
TEMPB = ZERO
TEMPA = ONE/P
M = 2*N - 4*(N/2)
SUM = ZERO
EM = CONV(N/2)
ALPEM = (EM-ONE) + ALPHA
ALP2EM = (EM+EM) + ALPHA
IF (M .NE. 0) SUM = TEMPA*ALPEM*ALP2EM/EM
NEND = N - NB
IF (NEND .GT. 0) THEN
C---------------------------------------------------------------------
C Recur backward via difference equation, calculating (but not
C storing) B(N), until N = NB.
C---------------------------------------------------------------------
DO 200 L=1,NEND
N = N - 1
EN = EN - TWO
TEMPC = TEMPB
TEMPB = TEMPA
TEMPA = (EN*TEMPB)/X - TEMPC
M = 2 - M
IF (M .NE. 0) THEN
EM = EM - ONE
ALP2EM = (EM+EM) + ALPHA
IF (N.EQ.1) GO TO 210
ALPEM = (EM-ONE) + ALPHA
IF (ALPEM.EQ.ZERO) ALPEM = ONE
SUM = (SUM+TEMPA*ALP2EM)*ALPEM/EM
END IF
200 CONTINUE
END IF
C---------------------------------------------------------------------
C Store B(NB).
C---------------------------------------------------------------------
210 B(N) = TEMPA
IF (NEND .GE. 0) THEN
IF (NB .LE. 1) THEN
ALP2EM = ALPHA
IF ((ALPHA+ONE).EQ.ONE) ALP2EM = ONE
SUM = SUM + B(1)*ALP2EM
GO TO 250
ELSE
C---------------------------------------------------------------------
C Calculate and store B(NB-1).
C---------------------------------------------------------------------
N = N - 1
EN = EN - TWO
B(N) = (EN*TEMPA)/X - TEMPB
IF (N.EQ.1) GO TO 240
M = 2 - M
IF (M .NE. 0) THEN
EM = EM - ONE
ALP2EM = (EM+EM) + ALPHA
ALPEM = (EM-ONE) + ALPHA
IF (ALPEM.EQ.ZERO) ALPEM = ONE
SUM = (SUM+B(N)*ALP2EM)*ALPEM/EM
END IF
END IF
END IF
NEND = N - 2
IF (NEND .NE. 0) THEN
C---------------------------------------------------------------------
C Calculate via difference equation and store B(N), until N = 2.
C---------------------------------------------------------------------
DO 230 L=1,NEND
N = N - 1
EN = EN - TWO
B(N) = (EN*B(N+1))/X - B(N+2)
M = 2 - M
IF (M .NE. 0) THEN
EM = EM - ONE
ALP2EM = (EM+EM) + ALPHA
ALPEM = (EM-ONE) + ALPHA
IF (ALPEM.EQ.ZERO) ALPEM = ONE
SUM = (SUM+B(N)*ALP2EM)*ALPEM/EM
END IF
230 CONTINUE
END IF
C---------------------------------------------------------------------
C Calculate B(1).
C---------------------------------------------------------------------
B(1) = TWO*(ALPHA+ONE)*B(2)/X - B(3)
240 EM = EM - ONE
ALP2EM = (EM+EM) + ALPHA
IF (ALP2EM.EQ.ZERO) ALP2EM = ONE
SUM = SUM + B(1)*ALP2EM
C---------------------------------------------------------------------
C Normalize. Divide all B(N) by sum.
C---------------------------------------------------------------------
250 IF ((ALPHA+ONE).NE.ONE)
1 SUM = SUM*FUNC(ALPHA)*(X*HALF)**(-ALPHA)
TEMPA = ENMTEN
IF (SUM.GT.ONE) TEMPA = TEMPA*SUM
DO 260 N=1,NB
IF (ABS(B(N)).LT.TEMPA) B(N) = ZERO
B(N) = B(N)/SUM
260 CONTINUE
END IF
C---------------------------------------------------------------------
C Error return -- X, NB, or ALPHA is out of range.
C---------------------------------------------------------------------
ELSE
B(1) = ZERO
NCALC = MIN(NB,0) - 1
END IF
C---------------------------------------------------------------------
C Exit
C---------------------------------------------------------------------
300 RETURN
C ---------- Last line of RJBESL ----------
END