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SUBROUTINE RIBESL(X,ALPHA,NB,IZE,B,NCALC)
C-------------------------------------------------------------------
C
C This routine calculates Bessel functions I SUB(N+ALPHA) (X)
C for non-negative argument X, and non-negative order N+ALPHA,
C with or without exponential scaling.
C
C
C Explanation of variables in the calling sequence
C
C X - Working precision non-negative real argument for which
C I's or exponentially scaled I's (I*EXP(-X))
C are to be calculated. If I's are to be calculated,
C X must be less than EXPARG (see below).
C ALPHA - Working precision fractional part of order for which
C I's or exponentially scaled I's (I*EXP(-X)) are
C to be calculated. 0 .LE. ALPHA .LT. 1.0.
C NB - Integer number of functions to be calculated, NB .GT. 0.
C The first function calculated is of order ALPHA, and the
C last is of order (NB - 1 + ALPHA).
C IZE - Integer type. IZE = 1 if unscaled I's are to calculated,
C and 2 if exponentially scaled I's are to be calculated.
C B - Working precision output vector of length NB. If the routine
C terminates normally (NCALC=NB), the vector B contains the
C functions I(ALPHA,X) through I(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 beta = Radix for the floating-point system
C minexp = Smallest representable power of beta
C maxexp = Smallest power of beta that overflows
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 when IZE=2. Bear
C in mind that if ABS(X)=N, then at least N iterations
C of the backward recursion will be executed. The value
C of 10.0 ** 4 is used on every machine.
C EXPARG = Largest working precision argument that the library
C EXP routine can handle and upper limit on the
C magnitude of X when IZE=1; approximately
C LOG(beta**maxexp)
C
C
C Approximate values for some important machines are:
C
C beta minexp maxexp it
C
C CRAY-1 (S.P.) 2 -8193 8191 48
C Cyber 180/855
C under NOS (S.P.) 2 -975 1070 48
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 2 -126 128 24
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 2 -1022 1024 53
C IBM 3033 (D.P.) 16 -65 63 14
C VAX (S.P.) 2 -128 127 24
C VAX D-Format (D.P.) 2 -128 127 56
C VAX G-Format (D.P.) 2 -1024 1023 53
C
C
C NSIG ENTEN ENSIG RTNSIG
C
C CRAY-1 (S.P.) 15 1.0E+2465 1.0E+15 1.0E-4
C Cyber 180/855
C under NOS (S.P.) 15 1.0E+322 1.0E+15 1.0E-4
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 8 1.0E+38 1.0E+8 1.0E-2
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 16 1.0D+308 1.0D+16 1.0D-4
C IBM 3033 (D.P.) 5 1.0D+75 1.0D+5 1.0D-2
C VAX (S.P.) 8 1.0E+38 1.0E+8 1.0E-2
C VAX D-Format (D.P.) 17 1.0D+38 1.0D+17 1.0D-5
C VAX G-Format (D.P.) 16 1.0D+307 1.0D+16 1.0D-4
C
C
C ENMTEN XLARGE EXPARG
C
C CRAY-1 (S.P.) 1.84E-2466 1.0E+4 5677
C Cyber 180/855
C under NOS (S.P.) 1.25E-293 1.0E+4 741
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 4.70E-38 1.0E+4 88
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 8.90D-308 1.0D+4 709
C IBM 3033 (D.P.) 2.16D-78 1.0D+4 174
C VAX (S.P.) 1.17E-38 1.0E+4 88
C VAX D-Format (D.P.) 1.17D-38 1.0D+4 88
C VAX G-Format (D.P.) 2.22D-308 1.0D+4 709
C
C*******************************************************************
C*******************************************************************
C
C Error returns
C
C In case of an error, NCALC .NE. NB, and not all I's are
C calculated to the desired accuracy.
C
C NCALC .LT. 0: An argument is out of range. For example,
C NB .LE. 0, IZE is not 1 or 2, or IZE=1 and ABS(X) .GE. EXPARG.
C In this case, the B-vector is not calculated, and NCALC is
C set to MIN0(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 functions required are:
C
C DBLE, EXP, DGAMMA, GAMMA, INT, MAX, MIN, REAL, SQRT
C
C
C Acknowledgement
C
C This program is based on a program written by David J.
C Sookne (2) that computes values of the Bessel functions J or
C I of real argument and integer order. Modifications include
C the restriction of the computation to the I Bessel function
C of non-negative real argument, the extension of the computation
C to arbitrary positive order, the inclusion of optional
C exponential scaling, and the elimination of most underflow.
C An earlier version was published in (3).
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 "ALGORITHM 597, Sequence of Modified Bessel Functions
C of the First Kind," Cody, W. J., Trans. Math. Soft.,
C 1983, pp. 242-245.
C
C Latest modification: May 30, 1989
C
C Modified by: W. J. Cody and L. Stoltz
C Applied Mathematics Division
C Argonne National Laboratory
C Argonne, IL 60439
C
C-------------------------------------------------------------------
INTEGER IZE,K,L,MAGX,N,NB,NBMX,NCALC,NEND,NSIG,NSTART
CS REAL GAMMA,
DOUBLE PRECISION DGAMMA,
1 ALPHA,B,CONST,CONV,EM,EMPAL,EMP2AL,EN,ENMTEN,ENSIG,
2 ENTEN,EXPARG,FUNC,HALF,HALFX,ONE,P,PLAST,POLD,PSAVE,PSAVEL,
3 RTNSIG,SUM,TEMPA,TEMPB,TEMPC,TEST,TOVER,TWO,X,XLARGE,ZERO
DIMENSION B(NB)
C-------------------------------------------------------------------
C Mathematical constants
C-------------------------------------------------------------------
CS DATA ONE,TWO,ZERO,HALF,CONST/1.0E0,2.0E0,0.0E0,0.5E0,1.585E0/
DATA ONE,TWO,ZERO,HALF,CONST/1.0D0,2.0D0,0.0D0,0.5D0,1.585D0/
C-------------------------------------------------------------------
C Machine-dependent parameters
C-------------------------------------------------------------------
CS DATA NSIG,XLARGE,EXPARG /8,1.0E4,88.0E0/
CS DATA ENTEN,ENSIG,RTNSIG/1.0E38,1.0E8,1.0E-2/
CS DATA ENMTEN/4.7E-38/
DATA NSIG,XLARGE,EXPARG /16,1.0D4,709.0D0/
DATA ENTEN,ENSIG,RTNSIG/1.0D308,1.0D16,1.0D-4/
DATA ENMTEN/8.9D-308/
C-------------------------------------------------------------------
C Statement functions for conversion
C-------------------------------------------------------------------
CS CONV(N) = REAL(N)
CS FUNC(X) = GAMMA(X)
CONV(N) = DBLE(N)
FUNC(X) = DGAMMA(X)
C-------------------------------------------------------------------
C Check for X, NB, OR IZE out of range.
C-------------------------------------------------------------------
IF ((NB.GT.0) .AND. (X .GE. ZERO) .AND.
1 (ALPHA .GE. ZERO) .AND. (ALPHA .LT. ONE) .AND.
2 (((IZE .EQ. 1) .AND. (X .LE. EXPARG)) .OR.
3 ((IZE .EQ. 2) .AND. (X .LE. XLARGE)))) THEN
C-------------------------------------------------------------------
C Use 2-term ascending series for small X
C-------------------------------------------------------------------
NCALC = NB
MAGX = INT(X)
IF (X .GE. RTNSIG) THEN
C-------------------------------------------------------------------
C Initialize the forward sweep, the P-sequence of Olver
C-------------------------------------------------------------------
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 (2*MAGX .GT. 5*NSIG) THEN
TEST = SQRT(TEST*P)
ELSE
TEST = TEST / CONST**MAGX
END IF
IF (NBMX .GE. 3) THEN
C-------------------------------------------------------------------
C Calculate P-sequence until N = NB-1. Check for possible overflow.
C-------------------------------------------------------------------
TOVER = ENTEN / ENSIG
NSTART = MAGX+2
NEND = NB - 1
DO 100 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-sequence by TOVER. Calculate
C P-sequence until ABS(P) .GT. 1.
C-------------------------------------------------------------------
TOVER = ENTEN
P = P / TOVER
PLAST = PLAST / TOVER
PSAVE = P
PSAVEL = PLAST
NSTART = N + 1
60 N = N + 1
EN = EN + TWO
POLD = PLAST
PLAST = P
P = EN * PLAST/X + POLD
IF (P .LE. ONE) GO TO 60
TEMPB = EN / X
C-------------------------------------------------------------------
C Calculate backward test, and find NCALC, the highest N
C such that the test is passed.
C-------------------------------------------------------------------
TEST = POLD*PLAST / ENSIG
TEST = TEST*(HALF-HALF/(TEMPB*TEMPB))
P = PLAST * TOVER
N = N - 1
EN = EN - TWO
NEND = MIN0(NB,N)
DO 80 L = NSTART, NEND
NCALC = L
POLD = PSAVEL
PSAVEL = PSAVE
PSAVE = EN * PSAVEL/X + POLD
IF (PSAVE*PSAVEL .GT. TEST) GO TO 90
80 CONTINUE
NCALC = NEND + 1
90 NCALC = NCALC - 1
GO TO 120
END IF
100 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-sequence until significance test passed.
C-------------------------------------------------------------------
110 N = N + 1
EN = EN + TWO
POLD = PLAST
PLAST = P
P = EN * PLAST/X + POLD
IF (P .LT. TEST) GO TO 110
C-------------------------------------------------------------------
C Initialize the backward recursion and the normalization sum.
C-------------------------------------------------------------------
120 N = N + 1
EN = EN + TWO
TEMPB = ZERO
TEMPA = ONE / P
EM = CONV(N) - ONE
EMPAL = EM + ALPHA
EMP2AL = (EM - ONE) + (ALPHA + ALPHA)
SUM = TEMPA * EMPAL * EMP2AL / EM
NEND = N - NB
IF (NEND .LT. 0) THEN
C-------------------------------------------------------------------
C N .LT. NB, so store B(N) and set higher orders to zero.
C-------------------------------------------------------------------
B(N) = TEMPA
NEND = -NEND
DO 130 L = 1, NEND
130 B(N+L) = ZERO
ELSE
IF (NEND .GT. 0) THEN
C-------------------------------------------------------------------
C Recur backward via difference equation, calculating (but
C not storing) B(N), until N = NB.
C-------------------------------------------------------------------
DO 140 L = 1, NEND
N = N - 1
EN = EN - TWO
TEMPC = TEMPB
TEMPB = TEMPA
TEMPA = (EN*TEMPB) / X + TEMPC
EM = EM - ONE
EMP2AL = EMP2AL - ONE
IF (N .EQ. 1) GO TO 150
IF (N .EQ. 2) EMP2AL = ONE
EMPAL = EMPAL - ONE
SUM = (SUM + TEMPA*EMPAL) * EMP2AL / EM
140 CONTINUE
END IF
C-------------------------------------------------------------------
C Store B(NB)
C-------------------------------------------------------------------
150 B(N) = TEMPA
IF (NB .LE. 1) THEN
SUM = (SUM + SUM) + TEMPA
GO TO 230
END IF
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 220
EM = EM - ONE
EMP2AL = EMP2AL - ONE
IF (N .EQ. 2) EMP2AL = ONE
EMPAL = EMPAL - ONE
SUM = (SUM + B(N)*EMPAL) * EMP2AL / EM
END IF
NEND = N - 2
IF (NEND .GT. 0) THEN
C-------------------------------------------------------------------
C Calculate via difference equation and store B(N), until N = 2.
C-------------------------------------------------------------------
DO 200 L = 1, NEND
N = N - 1
EN = EN - TWO
B(N) = (EN*B(N+1)) / X +B(N+2)
EM = EM - ONE
EMP2AL = EMP2AL - ONE
IF (N .EQ. 2) EMP2AL = ONE
EMPAL = EMPAL - ONE
SUM = (SUM + B(N)*EMPAL) * EMP2AL / EM
200 CONTINUE
END IF
C-------------------------------------------------------------------
C Calculate B(1)
C-------------------------------------------------------------------
B(1) = TWO*EMPAL*B(2) / X + B(3)
220 SUM = (SUM + SUM) + B(1)
C-------------------------------------------------------------------
C Normalize. Divide all B(N) by sum.
C-------------------------------------------------------------------
230 IF (ALPHA .NE. ZERO)
1 SUM = SUM * FUNC(ONE+ALPHA) * (X*HALF)**(-ALPHA)
IF (IZE .EQ. 1) SUM = SUM * EXP(-X)
TEMPA = ENMTEN
IF (SUM .GT. ONE) TEMPA = TEMPA * SUM
DO 260 N = 1, NB
IF (B(N) .LT. TEMPA) B(N) = ZERO
B(N) = B(N) / SUM
260 CONTINUE
RETURN
C-------------------------------------------------------------------
C Two-term ascending series for small X.
C-------------------------------------------------------------------
ELSE
TEMPA = ONE
EMPAL = ONE + ALPHA
HALFX = ZERO
IF (X .GT. ENMTEN) HALFX = HALF * X
IF (ALPHA .NE. ZERO) TEMPA = HALFX**ALPHA /FUNC(EMPAL)
IF (IZE .EQ. 2) TEMPA = TEMPA * EXP(-X)
TEMPB = ZERO
IF ((X+ONE) .GT. ONE) TEMPB = HALFX * HALFX
B(1) = TEMPA + TEMPA*TEMPB / EMPAL
IF ((X .NE. ZERO) .AND. (B(1) .EQ. ZERO)) NCALC = 0
IF (NB .GT. 1) THEN
IF (X .EQ. ZERO) THEN
DO 310 N = 2, NB
B(N) = ZERO
310 CONTINUE
ELSE
C-------------------------------------------------------------------
C Calculate higher-order functions.
C-------------------------------------------------------------------
TEMPC = HALFX
TOVER = (ENMTEN + ENMTEN) / X
IF (TEMPB .NE. ZERO) TOVER = ENMTEN / TEMPB
DO 340 N = 2, NB
TEMPA = TEMPA / EMPAL
EMPAL = EMPAL + ONE
TEMPA = TEMPA * TEMPC
IF (TEMPA .LE. TOVER*EMPAL) TEMPA = ZERO
B(N) = TEMPA + TEMPA*TEMPB / EMPAL
IF ((B(N) .EQ. ZERO) .AND. (NCALC .GT. N))
1 NCALC = N-1
340 CONTINUE
END IF
END IF
END IF
ELSE
NCALC = MIN0(NB,0)-1
END IF
RETURN
C---------- Last line of RIBESL ----------
END
|
