C--------------------------------------------------------------------
C Fortran 77 program to test BESY1
C
C Data required
C
C None
C
C Subprograms required from this package
C
C MACHAR - an environmental inquiry program providing
C information on the floating-point arithmetic
C system. Note that the call to MACHAR can
C be deleted provided the following six
C parameters are assigned the values indicated
C
C IBETA - the radix of the floating-point system
C IT - the number of base-IBETA digits in the
C significand of a floating-point number
C MAXEXP - the smallest integer such that
C FLOAT(IBETA)**MAXEXP causes overflow
C EPS - the smallest positive floating-point
C number such that 1.0+EPS .NE. 1.0
C XMIN - the smallest positive normalized
C floating-point power of the radix
C XMAX - the largest finite floating-point
C number
C
C REN(K) - a function subprogram returning random real
C numbers uniformly distributed over (0,1)
C
C
C Intrinsic functions required are:
C
C ABS, DBLE, FLOAT, LOG, MAX, SIGN, SQRT
C
C Reference: "Performance evaluation of programs for certain
C Bessel functions", W. J. Cody and L. Stoltz,
C ACM Trans. on Math. Software, Vol. 15, 1989,
C pp 41-48.
C
C Latest modification: March 27, 1990
C
C Authors: George Zazi, W. J. Cody
C Mathematics and Computer Science Division
C Argonne National Laboratory
C Argonne, IL 60439
C
C--------------------------------------------------------------------
LOGICAL SFLAG, TFLAG
INTEGER I,IBETA,IEXP,II,IND,IOUT,IRND,IT,J,JT,K1,K2,K3,
1 MACHEP,MAXEXP,MB,MBM1,MBP1,MINEXP,N,NDUM,NEGEP,NGRD
CS REAL
CD DOUBLE PRECISION
1 A,AIT,ALBETA,ALEPS,ALL9,B,BESY0,BESY1,BETA,C,CONV,
2 DEL,EIGHT,EPS,EPSNEG,FOUR,HALF,HUND,ONE,REN,R6,R7,
3 SGN,SIXTEN,SUM,T,TWENTY,TWO56,T1,U,W,X,XI,XL,
4 XLAM,XMAX,XMB,XMIN,X1,Y,YX,Z,ZERO,ZZ
DIMENSION U(560),XI(2),YX(2)
C--------------------------------------------------------------------
C Mathematical constants
C--------------------------------------------------------------------
DATA IOUT/6/
CS DATA ZERO,ONE,FOUR/0.0E0,1.0E0,4.0E0/,
CS 1 EIGHT,TWENTY,ALL9,TWO56/8.0E0,20.0E0,-999.0E0,256.0E0/,
CS 2 HALF,SIXTEN,XLAM,HUND/0.5E0,16.0E0,0.9375E0,100.0E0/
CD DATA ZERO,ONE,FOUR/0.0D0,1.0D0,4.0E0/,
CD 1 EIGHT,TWENTY,ALL9,TWO56/8.0D0,20.0D0,-999.0D0,256.0D0/,
CD 2 HALF,SIXTEN,XLAM,HUND/0.5D0,16.0D0,0.9375D0,100.0D0/
C--------------------------------------------------------------------
C Zeroes of Y1
C--------------------------------------------------------------------
CS DATA XI/562.0E0,1390.0E0/
CS DATA YX(1)/-9.5282 39309 77227 03132 E-4/
CS 1 YX(2)/-2.1981 83008 05980 02741 E-6/
CD DATA XI/562.0D0,1390.0D0/
CD DATA YX(1)/-9.5282 39309 77227 03132 D-4/,
CD 1 YX(2)/-2.1981 83008 05980 02741 D-6/
C--------------------------------------------------------------------
C Statement functions for conversion between integer and float
C--------------------------------------------------------------------
CS CONV(NDUM) = FLOAT(NDUM)
CD CONV(NDUM) = DBLE(FLOAT(NDUM))
C--------------------------------------------------------------------
C Determine machine parameters and set constants
C--------------------------------------------------------------------
CALL MACHAR(IBETA,IT,IRND,NGRD,MACHEP,NEGEP,IEXP,MINEXP,
1 MAXEXP,EPS,EPSNEG,XMIN,XMAX)
BETA = CONV(IBETA)
ALBETA = LOG(BETA)
ALEPS = LOG(EPS)
AIT = CONV(IT)
JT = 0
B = EPS
C--------------------------------------------------------------------
C Random argument accuracy tests (based on the
C multiplication theorem)
C--------------------------------------------------------------------
DO 300 J = 1, 3
SFLAG = (J .EQ. 1) .AND. (MAXEXP/IT .LE. 5)
K1 = 0
K2 = 0
K3 = 0
X1 = ZERO
R6 = ZERO
R7 = ZERO
N = 2000
A = B
IF (J .EQ. 1) THEN
B = FOUR
ELSE IF (J .EQ. 2) THEN
B = EIGHT
ELSE
B = TWENTY
N = 500
END IF
SGN = ONE
DEL = (B - A) / CONV(N)
XL = A
DO 200 I = 1, N
X = DEL * REN(JT) + XL
C--------------------------------------------------------------------
C Carefully purify arguments and evaluate identities
C--------------------------------------------------------------------
Y = X/XLAM
W = SIXTEN * Y
Y = (W + Y) - W
X = Y * XLAM
U(1) = BESY0(Y)
U(2) = BESY1(Y)
TFLAG = SFLAG .AND. (Y .LT. HALF)
IF (TFLAG) THEN
U(1) = U(1) * EPS
U(2) = U(2) * EPS
END IF
MB = 1
XMB = ONE
Y = Y * HALF
W = (ONE-XLAM)*(ONE+XLAM)
C = W * Y
T = ABS(XLAM*U(2))
T1 = EPS/HUND
DO 110 II = 3, 60
U(II) = XMB/Y * U(II-1) - U(II-2)
T1 = XMB * T1 / C
XMB = XMB + ONE
MB = MB + 1
MBP1=MB + 1
Z = ABS(U(II))
IF (Z/T1 .LT. T) THEN
GO TO 120
ELSE IF (Y .LT. XMB) THEN
IF (Z .GT. XMAX*(Y/XMB)) THEN
A= X
XL=XL+DEL
GO TO 200
END IF
END IF
IF (T1 .GT. ONE/EPS) THEN
T = T * T1
T1 = ONE
END IF
110 CONTINUE
120 SUM = U(MBP1)
IND = MBP1
MBM1 =MB-1
DO 155 II = 1, MBM1
IND = IND - 1
XMB = XMB - ONE
SUM = SUM * W * Y / XMB + U(IND)
155 CONTINUE
ZZ = SUM*XLAM
IF (TFLAG) ZZ = ZZ / EPS
Z = BESY1(X)
Y = Z
IF (ABS(U(2)) .GT. ABS(Y)) Y = U(2)
C--------------------------------------------------------------------
C Accumulate results
C--------------------------------------------------------------------
W = (Z - ZZ) / Y
IF (W .GT. ZERO) THEN
K1 = K1 + 1
ELSE IF (W .LT. ZERO) THEN
K3 = K3 + 1
ELSE
K2 = K2 + 1
END IF
W = ABS(W)
IF (W .GT. R6) THEN
R6 = W
X1 = X
END IF
R7 = R7 + W * W
XL = XL + DEL
200 CONTINUE
C--------------------------------------------------------------------
C Process and output statistics
C--------------------------------------------------------------------
N = K1 + K2 + K3
R7 = SQRT(R7/CONV(N))
WRITE (IOUT,1000)
WRITE (IOUT,1010) N,A,B
WRITE (IOUT,1011) K1,K2,K3
WRITE (IOUT,1020) IT,IBETA
IF (R6 .NE. ZERO) THEN
W = LOG(ABS(R6))/ALBETA
ELSE
W = ALL9
END IF
IF (J .EQ. 4) THEN
WRITE (IOUT,1024) R6,IBETA,W,X1
ELSE
WRITE (IOUT,1021) R6,IBETA,W,X1
END IF
W = MAX(AIT+W,ZERO)
WRITE (IOUT,1022) IBETA,W
IF (R7 .NE. ZERO) THEN
W = LOG(ABS(R7))/ALBETA
ELSE
W = ALL9
END IF
IF (J .EQ. 4) THEN
WRITE (IOUT,1025) R7,IBETA,W
ELSE
WRITE (IOUT,1023) R7,IBETA,W
END IF
W = MAX(AIT+W,ZERO)
WRITE (IOUT,1022) IBETA,W
300 CONTINUE
C--------------------------------------------------------------------
C Special tests
C--------------------------------------------------------------------
WRITE (IOUT,1030)
WRITE (IOUT,1031) IBETA
DO 330 I = 1, 2
X = XI(I)/TWO56
Y = BESY1(X)
T = (Y-YX(I))/YX(I)
IF (T .NE. ZERO) THEN
W = LOG(ABS(T))/ALBETA
ELSE
W = ALL9
END IF
W = MAX(AIT+W,ZERO)
WRITE (IOUT,1032) X,Y,W
330 CONTINUE
C--------------------------------------------------------------------
C Test of error returns
C--------------------------------------------------------------------
WRITE (IOUT,1033)
X = XMIN
IF (XMIN*XMAX .LT. ONE) THEN
WRITE (IOUT,1034) X
ELSE
WRITE (IOUT,1035) X
END IF
Y = BESY1(X)
WRITE (IOUT,1036) Y
X = -ONE
WRITE (IOUT,1034) X
Y = BESY1(X)
WRITE (IOUT,1036) Y
X = XMAX
WRITE (IOUT,1034) X
Y = BESY1(X)
WRITE (IOUT,1036) Y
WRITE (IOUT,1100)
STOP
C--------------------------------------------------------------------
1000 FORMAT('1Test of Y1(X) VS Multiplication Theorem' //)
1010 FORMAT(I7,' random arguments were tested from the interval ',
1 1H(,F5.1,1H,,F5.1,1H)//)
1011 FORMAT(' ABS(Y1(X)) was larger',I6,' times', /
1 15X,' agreed',I6,' times, and'/
1 11X,'was smaller',I6,' times.'//)
1020 FORMAT(' There are',I4,' base',I4,
1 ' significant digits in a floating-point number.' //)
1021 FORMAT(' The maximum relative error of',E15.4,3H = ,I4,3H **,
1 F7.2/4X,'occurred for X =',E13.6)
1022 FORMAT(' The estimated loss of base',I4,
1 ' significant digits is',F7.2//)
1023 FORMAT(' The root mean square relative error was',E15.4,
1 3H = ,I4,3H **,F7.2)
1024 FORMAT(' The maximum absolute error of',E15.4,3H = ,I4,3H **,
1 F7.2/4X,'occurred for X =',E13.6)
1025 FORMAT(' The root mean square absolute error was',E15.4,
1 3H = ,I4,3H **,F7.2)
1030 FORMAT('1Special Tests'//)
1031 FORMAT(' Accuracy near zeros'//10X,'X',15X,'BESY1(X)',
1 13X,'Loss of base',I3,' digits'/)
1032 FORMAT(E20.10,E25.15,8X,F7.2/)
1033 FORMAT(//' Test with extreme arguments'///)
1034 FORMAT(' Y1 will be called with the argument ',E17.10/
1 ' This may stop execution.'//)
1035 FORMAT(' Y1 will be called with the argument ',E17.10/
1 ' This should not stop execution.'//)
1036 FORMAT(' Y1 returned the value',E25.17/)
1100 FORMAT(' This concludes the tests.')
C---------- Last card of BESY1 test program ----------
END
SUBROUTINE CALJY0(ARG,RESULT,JINT)
C---------------------------------------------------------------------
C
C This packet computes zero-order Bessel functions of the first and
C second kind (J0 and Y0), for real arguments X, where 0 < X <= XMAX
C for Y0, and |X| <= XMAX for J0. It contains two function-type
C subprograms, BESJ0 and BESY0, and one subroutine-type
C subprogram, CALJY0. The calling statements for the primary
C entries are:
C
C Y = BESJ0(X)
C and
C Y = BESY0(X),
C
C where the entry points correspond to the functions J0(X) and Y0(X),
C respectively. The routine CALJY0 is intended for internal packet
C use only, all computations within the packet being concentrated in
C this one routine. The function subprograms invoke CALJY0 with
C the statement
C CALL CALJY0(ARG,RESULT,JINT),
C where the parameter usage is as follows:
C
C Function Parameters for CALJY0
C call ARG RESULT JINT
C
C BESJ0(ARG) |ARG| .LE. XMAX J0(ARG) 0
C BESY0(ARG) 0 .LT. ARG .LE. XMAX Y0(ARG) 1
C
C The main computation uses unpublished minimax rational
C approximations for X .LE. 8.0, and an approximation from the
C book Computer Approximations by Hart, et. al., Wiley and Sons,
C New York, 1968, for arguments larger than 8.0 Part of this
C transportable packet is patterned after the machine-dependent
C FUNPACK program BESJ0(X), but cannot match that version for
C efficiency or accuracy. This version uses rational functions
C that are theoretically accurate to at least 18 significant decimal
C digits for X <= 8, and at least 18 decimal places for X > 8. The
C accuracy achieved depends on the arithmetic system, the compiler,
C the intrinsic functions, and proper selection of the machine-
C dependent constants.
C
C*******************************************************************
C
C Explanation of machine-dependent constants
C
C XINF = largest positive machine number
C XMAX = largest acceptable argument. The functions AINT, SIN
C and COS must perform properly for ABS(X) .LE. XMAX.
C We recommend that XMAX be a small integer multiple of
C sqrt(1/eps), where eps is the smallest positive number
C such that 1+eps > 1.
C XSMALL = positive argument such that 1.0-(X/2)**2 = 1.0
C to machine precision for all ABS(X) .LE. XSMALL.
C We recommend that XSMALL < sqrt(eps)/beta, where beta
C is the floating-point radix (usually 2 or 16).
C
C Approximate values for some important machines are
C
C eps XMAX XSMALL XINF
C
C CDC 7600 (S.P.) 7.11E-15 1.34E+08 2.98E-08 1.26E+322
C CRAY-1 (S.P.) 7.11E-15 1.34E+08 2.98E-08 5.45E+2465
C IBM PC (8087) (S.P.) 5.96E-08 8.19E+03 1.22E-04 3.40E+38
C IBM PC (8087) (D.P.) 1.11D-16 2.68D+08 3.72D-09 1.79D+308
C IBM 195 (D.P.) 2.22D-16 6.87D+09 9.09D-13 7.23D+75
C UNIVAC 1108 (D.P.) 1.73D-18 4.30D+09 2.33D-10 8.98D+307
C VAX 11/780 (D.P.) 1.39D-17 1.07D+09 9.31D-10 1.70D+38
C
C*******************************************************************
C*******************************************************************
C
C Error Returns
C
C The program returns the value zero for X .GT. XMAX, and returns
C -XINF when BESLY0 is called with a negative or zero argument.
C
C
C Intrinsic functions required are:
C
C ABS, AINT, COS, LOG, SIN, SQRT
C
C
C Latest modification: June 2, 1989
C
C Author: W. J. Cody
C Mathematics and Computer Science Division
C Argonne National Laboratory
C Argonne, IL 60439
C
C--------------------------------------------------------------------
INTEGER I,JINT
CS REAL
CD DOUBLE PRECISION
1 ARG,AX,CONS,DOWN,EIGHT,FIVE5,FOUR,ONE,ONEOV8,PI2,PJ0,
2 PJ1,PLG,PROD,PY0,PY1,PY2,P0,P1,P17,QJ0,QJ1,QLG,QY0,QY1,
3 QY2,Q0,Q1,RESJ,RESULT,R0,R1,SIXTY4,THREE,TWOPI,TWOPI1,
4 TWOPI2,TWO56,UP,W,WSQ,XDEN,XINF,XMAX,XNUM,XSMALL,XJ0,
5 XJ1,XJ01,XJ02,XJ11,XJ12,XY,XY0,XY01,XY02,XY1,XY11,XY12,
6 XY2,XY21,XY22,Z,ZERO,ZSQ
DIMENSION PJ0(7),PJ1(8),PLG(4),PY0(6),PY1(7),PY2(8),P0(6),P1(6),
1 QJ0(5),QJ1(7),QLG(4),QY0(5),QY1(6),QY2(7),Q0(5),Q1(5)
C-------------------------------------------------------------------
C Mathematical constants
C CONS = ln(.5) + Euler's gamma
C-------------------------------------------------------------------
CS DATA ZERO,ONE,THREE,FOUR,EIGHT/0.0E0,1.0E0,3.0E0,4.0E0,8.0E0/,
CS 1 FIVE5,SIXTY4,ONEOV8,P17/5.5E0,64.0E0,0.125E0,1.716E-1/,
CS 2 TWO56,CONS/256.0E0,-1.1593151565841244881E-1/,
CS 3 PI2,TWOPI/6.3661977236758134308E-1,6.2831853071795864769E0/,
CS 4 TWOPI1,TWOPI2/6.28125E0,1.9353071795864769253E-3/
CD DATA ZERO,ONE,THREE,FOUR,EIGHT/0.0D0,1.0D0,3.0D0,4.0D0,8.0D0/,
CD 1 FIVE5,SIXTY4,ONEOV8,P17/5.5D0,64.0D0,0.125D0,1.716D-1/,
CD 2 TWO56,CONS/256.0D0,-1.1593151565841244881D-1/,
CD 3 PI2,TWOPI/6.3661977236758134308D-1,6.2831853071795864769D0/,
CD 4 TWOPI1,TWOPI2/6.28125D0,1.9353071795864769253D-3/
C-------------------------------------------------------------------
C Machine-dependent constants
C-------------------------------------------------------------------
CS DATA XMAX/8.19E+03/,XSMALL/1.22E-09/,XINF/1.7E+38/
CD DATA XMAX/1.07D+09/,XSMALL/9.31D-10/,XINF/1.7D+38/
C-------------------------------------------------------------------
C Zeroes of Bessel functions
C-------------------------------------------------------------------
CS DATA XJ0/2.4048255576957727686E+0/,XJ1/5.5200781102863106496E+0/,
CS 1 XY0/8.9357696627916752158E-1/,XY1/3.9576784193148578684E+0/,
CS 2 XY2/7.0860510603017726976E+0/,
CS 3 XJ01/ 616.0E+0/, XJ02/-1.4244423042272313784E-03/,
CS 4 XJ11/1413.0E+0/, XJ12/ 5.4686028631064959660E-04/,
CS 5 XY01/ 228.0E+0/, XY02/ 2.9519662791675215849E-03/,
CS 6 XY11/1013.0E+0/, XY12/ 6.4716931485786837568E-04/,
CS 7 XY21/1814.0E+0/, XY22/ 1.1356030177269762362E-04/
CD DATA XJ0/2.4048255576957727686D+0/,XJ1/5.5200781102863106496D+0/,
CD 1 XY0/8.9357696627916752158D-1/,XY1/3.9576784193148578684D+0/,
CD 2 XY2/7.0860510603017726976D+0/,
CD 3 XJ01/ 616.0D+0/, XJ02/-1.4244423042272313784D-03/,
CD 4 XJ11/1413.0D+0/, XJ12/ 5.4686028631064959660D-04/,
CD 5 XY01/ 228.0D+0/, XY02/ 2.9519662791675215849D-03/,
CD 6 XY11/1013.0D+0/, XY12/ 6.4716931485786837568D-04/,
CD 7 XY21/1814.0D+0/, XY22/ 1.1356030177269762362D-04/
C-------------------------------------------------------------------
C Coefficients for rational approximation to ln(x/a)
C--------------------------------------------------------------------
CS DATA PLG/-2.4562334077563243311E+01,2.3642701335621505212E+02,
CS 1 -5.4989956895857911039E+02,3.5687548468071500413E+02/
CS DATA QLG/-3.5553900764052419184E+01,1.9400230218539473193E+02,
CS 1 -3.3442903192607538956E+02,1.7843774234035750207E+02/
CD DATA PLG/-2.4562334077563243311D+01,2.3642701335621505212D+02,
CD 1 -5.4989956895857911039D+02,3.5687548468071500413D+02/
CD DATA QLG/-3.5553900764052419184D+01,1.9400230218539473193D+02,
CD 1 -3.3442903192607538956D+02,1.7843774234035750207D+02/
C-------------------------------------------------------------------
C Coefficients for rational approximation of
C J0(X) / (X**2 - XJ0**2), XSMALL < |X| <= 4.0
C--------------------------------------------------------------------
CS DATA PJ0/6.6302997904833794242E+06,-6.2140700423540120665E+08,
CS 1 2.7282507878605942706E+10,-4.1298668500990866786E+11,
CS 2 -1.2117036164593528341E-01, 1.0344222815443188943E+02,
CS 3 -3.6629814655107086448E+04/
CS DATA QJ0/4.5612696224219938200E+05, 1.3985097372263433271E+08,
CS 1 2.6328198300859648632E+10, 2.3883787996332290397E+12,
CS 2 9.3614022392337710626E+02/
CD DATA PJ0/6.6302997904833794242D+06,-6.2140700423540120665D+08,
CD 1 2.7282507878605942706D+10,-4.1298668500990866786D+11,
CD 2 -1.2117036164593528341D-01, 1.0344222815443188943D+02,
CD 3 -3.6629814655107086448D+04/
CD DATA QJ0/4.5612696224219938200D+05, 1.3985097372263433271D+08,
CD 1 2.6328198300859648632D+10, 2.3883787996332290397D+12,
CD 2 9.3614022392337710626D+02/
C-------------------------------------------------------------------
C Coefficients for rational approximation of
C J0(X) / (X**2 - XJ1**2), 4.0 < |X| <= 8.0
C-------------------------------------------------------------------
CS DATA PJ1/4.4176707025325087628E+03, 1.1725046279757103576E+04,
CS 1 1.0341910641583726701E+04,-7.2879702464464618998E+03,
CS 2 -1.2254078161378989535E+04,-1.8319397969392084011E+03,
CS 3 4.8591703355916499363E+01, 7.4321196680624245801E+02/
CS DATA QJ1/3.3307310774649071172E+02,-2.9458766545509337327E+03,
CS 1 1.8680990008359188352E+04,-8.4055062591169562211E+04,
CS 2 2.4599102262586308984E+05,-3.5783478026152301072E+05,
CS 3 -2.5258076240801555057E+01/
CD DATA PJ1/4.4176707025325087628D+03, 1.1725046279757103576D+04,
CD 1 1.0341910641583726701D+04,-7.2879702464464618998D+03,
CD 2 -1.2254078161378989535D+04,-1.8319397969392084011D+03,
CD 3 4.8591703355916499363D+01, 7.4321196680624245801D+02/
CD DATA QJ1/3.3307310774649071172D+02,-2.9458766545509337327D+03,
CD 1 1.8680990008359188352D+04,-8.4055062591169562211D+04,
CD 2 2.4599102262586308984D+05,-3.5783478026152301072D+05,
CD 3 -2.5258076240801555057D+01/
C-------------------------------------------------------------------
C Coefficients for rational approximation of
C (Y0(X) - 2 LN(X/XY0) J0(X)) / (X**2 - XY0**2),
C XSMALL < |X| <= 3.0
C--------------------------------------------------------------------
CS DATA PY0/1.0102532948020907590E+04,-2.1287548474401797963E+06,
CS 1 2.0422274357376619816E+08,-8.3716255451260504098E+09,
CS 2 1.0723538782003176831E+11,-1.8402381979244993524E+01/
CS DATA QY0/6.6475986689240190091E+02, 2.3889393209447253406E+05,
CS 1 5.5662956624278251596E+07, 8.1617187777290363573E+09,
CS 2 5.8873865738997033405E+11/
CD DATA PY0/1.0102532948020907590D+04,-2.1287548474401797963D+06,
CD 1 2.0422274357376619816D+08,-8.3716255451260504098D+09,
CD 2 1.0723538782003176831D+11,-1.8402381979244993524D+01/
CD DATA QY0/6.6475986689240190091D+02, 2.3889393209447253406D+05,
CD 1 5.5662956624278251596D+07, 8.1617187777290363573D+09,
CD 2 5.8873865738997033405D+11/
C-------------------------------------------------------------------
C Coefficients for rational approximation of
C (Y0(X) - 2 LN(X/XY1) J0(X)) / (X**2 - XY1**2),
C 3.0 < |X| <= 5.5
C--------------------------------------------------------------------
CS DATA PY1/-1.4566865832663635920E+04, 4.6905288611678631510E+06,
CS 1 -6.9590439394619619534E+08, 4.3600098638603061642E+10,
CS 2 -5.5107435206722644429E+11,-2.2213976967566192242E+13,
CS 3 1.7427031242901594547E+01/
CS DATA QY1/ 8.3030857612070288823E+02, 4.0669982352539552018E+05,
CS 1 1.3960202770986831075E+08, 3.4015103849971240096E+10,
CS 2 5.4266824419412347550E+12, 4.3386146580707264428E+14/
CD DATA PY1/-1.4566865832663635920D+04, 4.6905288611678631510D+06,
CD 1 -6.9590439394619619534D+08, 4.3600098638603061642D+10,
CD 2 -5.5107435206722644429D+11,-2.2213976967566192242D+13,
CD 3 1.7427031242901594547D+01/
CD DATA QY1/ 8.3030857612070288823D+02, 4.0669982352539552018D+05,
CD 1 1.3960202770986831075D+08, 3.4015103849971240096D+10,
CD 2 5.4266824419412347550D+12, 4.3386146580707264428D+14/
C-------------------------------------------------------------------
C Coefficients for rational approximation of
C (Y0(X) - 2 LN(X/XY2) J0(X)) / (X**2 - XY2**2),
C 5.5 < |X| <= 8.0
C--------------------------------------------------------------------
CS DATA PY2/ 2.1363534169313901632E+04,-1.0085539923498211426E+07,
CS 1 2.1958827170518100757E+09,-1.9363051266772083678E+11,
CS 2 -1.2829912364088687306E+11, 6.7016641869173237784E+14,
CS 3 -8.0728726905150210443E+15,-1.7439661319197499338E+01/
CS DATA QY2/ 8.7903362168128450017E+02, 5.3924739209768057030E+05,
CS 1 2.4727219475672302327E+08, 8.6926121104209825246E+10,
CS 2 2.2598377924042897629E+13, 3.9272425569640309819E+15,
CS 3 3.4563724628846457519E+17/
CD DATA PY2/ 2.1363534169313901632D+04,-1.0085539923498211426D+07,
CD 1 2.1958827170518100757D+09,-1.9363051266772083678D+11,
CD 2 -1.2829912364088687306D+11, 6.7016641869173237784D+14,
CD 3 -8.0728726905150210443D+15,-1.7439661319197499338D+01/
CD DATA QY2/ 8.7903362168128450017D+02, 5.3924739209768057030D+05,
CD 1 2.4727219475672302327D+08, 8.6926121104209825246D+10,
CD 2 2.2598377924042897629D+13, 3.9272425569640309819D+15,
CD 3 3.4563724628846457519D+17/
C-------------------------------------------------------------------
C Coefficients for Hart,s approximation, |X| > 8.0
C-------------------------------------------------------------------
CS DATA P0/3.4806486443249270347E+03, 2.1170523380864944322E+04,
CS 1 4.1345386639580765797E+04, 2.2779090197304684302E+04,
CS 2 8.8961548424210455236E-01, 1.5376201909008354296E+02/
CS DATA Q0/3.5028735138235608207E+03, 2.1215350561880115730E+04,
CS 1 4.1370412495510416640E+04, 2.2779090197304684318E+04,
CS 2 1.5711159858080893649E+02/
CS DATA P1/-2.2300261666214198472E+01,-1.1183429920482737611E+02,
CS 1 -1.8591953644342993800E+02,-8.9226600200800094098E+01,
CS 2 -8.8033303048680751817E-03,-1.2441026745835638459E+00/
CS DATA Q1/1.4887231232283756582E+03, 7.2642780169211018836E+03,
CS 1 1.1951131543434613647E+04, 5.7105024128512061905E+03,
CS 2 9.0593769594993125859E+01/
CD DATA P0/3.4806486443249270347D+03, 2.1170523380864944322D+04,
CD 1 4.1345386639580765797D+04, 2.2779090197304684302D+04,
CD 2 8.8961548424210455236D-01, 1.5376201909008354296D+02/
CD DATA Q0/3.5028735138235608207D+03, 2.1215350561880115730D+04,
CD 1 4.1370412495510416640D+04, 2.2779090197304684318D+04,
CD 2 1.5711159858080893649D+02/
CD DATA P1/-2.2300261666214198472D+01,-1.1183429920482737611D+02,
CD 1 -1.8591953644342993800D+02,-8.9226600200800094098D+01,
CD 2 -8.8033303048680751817D-03,-1.2441026745835638459D+00/
CD DATA Q1/1.4887231232283756582D+03, 7.2642780169211018836D+03,
CD 1 1.1951131543434613647D+04, 5.7105024128512061905D+03,
CD 2 9.0593769594993125859D+01/
C-------------------------------------------------------------------
C Check for error conditions
C-------------------------------------------------------------------
AX = ABS(ARG)
IF ((JINT .EQ. 1) .AND. (ARG .LE. ZERO)) THEN
RESULT = -XINF
GO TO 2000
ELSE IF (AX .GT. XMAX) THEN
RESULT = ZERO
GO TO 2000
END IF
IF (AX .GT. EIGHT) GO TO 800
IF (AX .LE. XSMALL) THEN
IF (JINT .EQ. 0) THEN
RESULT = ONE
ELSE
RESULT = PI2 * (LOG(AX) + CONS)
END IF
GO TO 2000
END IF
C-------------------------------------------------------------------
C Calculate J0 for appropriate interval, preserving
C accuracy near the zero of J0
C-------------------------------------------------------------------
ZSQ = AX * AX
IF (AX .LE. FOUR) THEN
XNUM = (PJ0(5) * ZSQ + PJ0(6)) * ZSQ + PJ0(7)
XDEN = ZSQ + QJ0(5)
DO 50 I = 1, 4
XNUM = XNUM * ZSQ + PJ0(I)
XDEN = XDEN * ZSQ + QJ0(I)
50 CONTINUE
PROD = ((AX - XJ01/TWO56) - XJ02) * (AX + XJ0)
ELSE
WSQ = ONE - ZSQ / SIXTY4
XNUM = PJ1(7) * WSQ + PJ1(8)
XDEN = WSQ + QJ1(7)
DO 220 I = 1, 6
XNUM = XNUM * WSQ + PJ1(I)
XDEN = XDEN * WSQ + QJ1(I)
220 CONTINUE
PROD = (AX + XJ1) * ((AX - XJ11/TWO56) - XJ12)
END IF
RESULT = PROD * XNUM / XDEN
IF (JINT .EQ. 0) GO TO 2000
C-------------------------------------------------------------------
C Calculate Y0. First find RESJ = pi/2 ln(x/xn) J0(x),
C where xn is a zero of Y0
C-------------------------------------------------------------------
IF (AX .LE. THREE) THEN
UP = (AX-XY01/TWO56)-XY02
XY = XY0
ELSE IF (AX .LE. FIVE5) THEN
UP = (AX-XY11/TWO56)-XY12
XY = XY1
ELSE
UP = (AX-XY21/TWO56)-XY22
XY = XY2
END IF
DOWN = AX + XY
IF (ABS(UP) .LT. P17*DOWN) THEN
W = UP/DOWN
WSQ = W*W
XNUM = PLG(1)
XDEN = WSQ + QLG(1)
DO 320 I = 2, 4
XNUM = XNUM*WSQ + PLG(I)
XDEN = XDEN*WSQ + QLG(I)
320 CONTINUE
RESJ = PI2 * RESULT * W * XNUM/XDEN
ELSE
RESJ = PI2 * RESULT * LOG(AX/XY)
END IF
C-------------------------------------------------------------------
C Now calculate Y0 for appropriate interval, preserving
C accuracy near the zero of Y0
C-------------------------------------------------------------------
IF (AX .LE. THREE) THEN
XNUM = PY0(6) * ZSQ + PY0(1)
XDEN = ZSQ + QY0(1)
DO 340 I = 2, 5
XNUM = XNUM * ZSQ + PY0(I)
XDEN = XDEN * ZSQ + QY0(I)
340 CONTINUE
ELSE IF (AX .LE. FIVE5) THEN
XNUM = PY1(7) * ZSQ + PY1(1)
XDEN = ZSQ + QY1(1)
DO 360 I = 2, 6
XNUM = XNUM * ZSQ + PY1(I)
XDEN = XDEN * ZSQ + QY1(I)
360 CONTINUE
ELSE
XNUM = PY2(8) * ZSQ + PY2(1)
XDEN = ZSQ + QY2(1)
DO 380 I = 2, 7
XNUM = XNUM * ZSQ + PY2(I)
XDEN = XDEN * ZSQ + QY2(I)
380 CONTINUE
END IF
RESULT = RESJ + UP * DOWN * XNUM / XDEN
GO TO 2000
C-------------------------------------------------------------------
C Calculate J0 or Y0 for |ARG| > 8.0
C-------------------------------------------------------------------
800 Z = EIGHT / AX
W = AX / TWOPI
W = AINT(W) + ONEOV8
W = (AX - W * TWOPI1) - W * TWOPI2
ZSQ = Z * Z
XNUM = P0(5) * ZSQ + P0(6)
XDEN = ZSQ + Q0(5)
UP = P1(5) * ZSQ + P1(6)
DOWN = ZSQ + Q1(5)
DO 850 I = 1, 4
XNUM = XNUM * ZSQ + P0(I)
XDEN = XDEN * ZSQ + Q0(I)
UP = UP * ZSQ + P1(I)
DOWN = DOWN * ZSQ + Q1(I)
850 CONTINUE
R0 = XNUM / XDEN
R1 = UP / DOWN
IF (JINT .EQ. 0) THEN
RESULT = SQRT(PI2/AX) * (R0*COS(W) - Z*R1*SIN(W))
ELSE
RESULT = SQRT(PI2/AX) * (R0*SIN(W) + Z*R1*COS(W))
END IF
2000 RETURN
C---------- Last line of CALJY0 ----------
END
CD DOUBLE PRECISION FUNCTION BESJ0(X)
CS REAL FUNCTION BESJ0(X)
C--------------------------------------------------------------------
C
C This subprogram computes approximate values for Bessel functions
C of the first kind of order zero for arguments |X| <= XMAX
C (see comments heading CALJY0).
C
C--------------------------------------------------------------------
INTEGER JINT
CS REAL X, RESULT
CD DOUBLE PRECISION X, RESULT
C--------------------------------------------------------------------
JINT=0
CALL CALJY0(X,RESULT,JINT)
BESJ0 = RESULT
RETURN
C---------- Last line of BESJ0 ----------
END
CD DOUBLE PRECISION FUNCTION BESY0(X)
CS REAL FUNCTION BESY0(X)
C--------------------------------------------------------------------
C
C This subprogram computes approximate values for Bessel functions
C of the second kind of order zero for arguments 0 < X <= XMAX
C (see comments heading CALJY0).
C
C--------------------------------------------------------------------
INTEGER JINT
CS REAL X, RESULT
CD DOUBLE PRECISION X, RESULT
C--------------------------------------------------------------------
JINT=1
CALL CALJY0(X,RESULT,JINT)
BESY0 = RESULT
RETURN
C---------- Last line of BESY0 ----------
END
SUBROUTINE MACHAR(IBETA,IT,IRND,NGRD,MACHEP,NEGEP,IEXP,MINEXP,
1 MAXEXP,EPS,EPSNEG,XMIN,XMAX)
C----------------------------------------------------------------------
C This Fortran 77 subroutine is intended to determine the parameters
C of the floating-point arithmetic system specified below. The
C determination of the first three uses an extension of an algorithm
C due to M. Malcolm, CACM 15 (1972), pp. 949-951, incorporating some,
C but not all, of the improvements suggested by M. Gentleman and S.
C Marovich, CACM 17 (1974), pp. 276-277. An earlier version of this
C program was published in the book Software Manual for the
C Elementary Functions by W. J. Cody and W. Waite, Prentice-Hall,
C Englewood Cliffs, NJ, 1980.
C
C The program as given here must be modified before compiling. If
C a single (double) precision version is desired, change all
C occurrences of CS (CD) in columns 1 and 2 to blanks.
C
C Parameter values reported are as follows:
C
C IBETA - the radix for the floating-point representation
C IT - the number of base IBETA digits in the floating-point
C significand
C IRND - 0 if floating-point addition chops
C 1 if floating-point addition rounds, but not in the
C IEEE style
C 2 if floating-point addition rounds in the IEEE style
C 3 if floating-point addition chops, and there is
C partial underflow
C 4 if floating-point addition rounds, but not in the
C IEEE style, and there is partial underflow
C 5 if floating-point addition rounds in the IEEE style,
C and there is partial underflow
C NGRD - the number of guard digits for multiplication with
C truncating arithmetic. It is
C 0 if floating-point arithmetic rounds, or if it
C truncates and only IT base IBETA digits
C participate in the post-normalization shift of the
C floating-point significand in multiplication;
C 1 if floating-point arithmetic truncates and more
C than IT base IBETA digits participate in the
C post-normalization shift of the floating-point
C significand in multiplication.
C MACHEP - the largest negative integer such that
C 1.0+FLOAT(IBETA)**MACHEP .NE. 1.0, except that
C MACHEP is bounded below by -(IT+3)
C NEGEPS - the largest negative integer such that
C 1.0-FLOAT(IBETA)**NEGEPS .NE. 1.0, except that
C NEGEPS is bounded below by -(IT+3)
C IEXP - the number of bits (decimal places if IBETA = 10)
C reserved for the representation of the exponent
C (including the bias or sign) of a floating-point
C number
C MINEXP - the largest in magnitude negative integer such that
C FLOAT(IBETA)**MINEXP is positive and normalized
C MAXEXP - the smallest positive power of BETA that overflows
C EPS - FLOAT(IBETA)**MACHEP.
C EPSNEG - FLOAT(IBETA)**NEGEPS.
C XMIN - the smallest non-vanishing normalized floating-point
C power of the radix, i.e., XMIN = FLOAT(IBETA)**MINEXP
C XMAX - the largest finite floating-point number. In
C particular XMAX = (1.0-EPSNEG)*FLOAT(IBETA)**MAXEXP
C Note - on some machines XMAX will be only the
C second, or perhaps third, largest number, being
C too small by 1 or 2 units in the last digit of
C the significand.
C
C Latest modification: May 30, 1989
C
C Author: W. J. Cody
C Mathematics and Computer Science Division
C Argonne National Laboratory
C Argonne, IL 60439
C
C----------------------------------------------------------------------
INTEGER I,IBETA,IEXP,IRND,IT,ITEMP,IZ,J,K,MACHEP,MAXEXP,
1 MINEXP,MX,NEGEP,NGRD,NXRES
CS REAL
CD DOUBLE PRECISION
1 A,B,BETA,BETAIN,BETAH,CONV,EPS,EPSNEG,ONE,T,TEMP,TEMPA,
2 TEMP1,TWO,XMAX,XMIN,Y,Z,ZERO
C----------------------------------------------------------------------
CS CONV(I) = REAL(I)
CD CONV(I) = DBLE(I)
ONE = CONV(1)
TWO = ONE + ONE
ZERO = ONE - ONE
C----------------------------------------------------------------------
C Determine IBETA, BETA ala Malcolm.
C----------------------------------------------------------------------
A = ONE
10 A = A + A
TEMP = A+ONE
TEMP1 = TEMP-A
IF (TEMP1-ONE .EQ. ZERO) GO TO 10
B = ONE
20 B = B + B
TEMP = A+B
ITEMP = INT(TEMP-A)
IF (ITEMP .EQ. 0) GO TO 20
IBETA = ITEMP
BETA = CONV(IBETA)
C----------------------------------------------------------------------
C Determine IT, IRND.
C----------------------------------------------------------------------
IT = 0
B = ONE
100 IT = IT + 1
B = B * BETA
TEMP = B+ONE
TEMP1 = TEMP-B
IF (TEMP1-ONE .EQ. ZERO) GO TO 100
IRND = 0
BETAH = BETA / TWO
TEMP = A+BETAH
IF (TEMP-A .NE. ZERO) IRND = 1
TEMPA = A + BETA
TEMP = TEMPA+BETAH
IF ((IRND .EQ. 0) .AND. (TEMP-TEMPA .NE. ZERO)) IRND = 2
C----------------------------------------------------------------------
C Determine NEGEP, EPSNEG.
C----------------------------------------------------------------------
NEGEP = IT + 3
BETAIN = ONE / BETA
A = ONE
DO 200 I = 1, NEGEP
A = A * BETAIN
200 CONTINUE
B = A
210 TEMP = ONE-A
IF (TEMP-ONE .NE. ZERO) GO TO 220
A = A * BETA
NEGEP = NEGEP - 1
GO TO 210
220 NEGEP = -NEGEP
EPSNEG = A
C----------------------------------------------------------------------
C Determine MACHEP, EPS.
C----------------------------------------------------------------------
MACHEP = -IT - 3
A = B
300 TEMP = ONE+A
IF (TEMP-ONE .NE. ZERO) GO TO 320
A = A * BETA
MACHEP = MACHEP + 1
GO TO 300
320 EPS = A
C----------------------------------------------------------------------
C Determine NGRD.
C----------------------------------------------------------------------
NGRD = 0
TEMP = ONE+EPS
IF ((IRND .EQ. 0) .AND. (TEMP*ONE-ONE .NE. ZERO)) NGRD = 1
C----------------------------------------------------------------------
C Determine IEXP, MINEXP, XMIN.
C
C Loop to determine largest I and K = 2**I such that
C (1/BETA) ** (2**(I))
C does not underflow.
C Exit from loop is signaled by an underflow.
C----------------------------------------------------------------------
I = 0
K = 1
Z = BETAIN
T = ONE + EPS
NXRES = 0
400 Y = Z
Z = Y * Y
C----------------------------------------------------------------------
C Check for underflow here.
C----------------------------------------------------------------------
A = Z * ONE
TEMP = Z * T
IF ((A+A .EQ. ZERO) .OR. (ABS(Z) .GE. Y)) GO TO 410
TEMP1 = TEMP * BETAIN
IF (TEMP1*BETA .EQ. Z) GO TO 410
I = I + 1
K = K + K
GO TO 400
410 IF (IBETA .EQ. 10) GO TO 420
IEXP = I + 1
MX = K + K
GO TO 450
C----------------------------------------------------------------------
C This segment is for decimal machines only.
C----------------------------------------------------------------------
420 IEXP = 2
IZ = IBETA
430 IF (K .LT. IZ) GO TO 440
IZ = IZ * IBETA
IEXP = IEXP + 1
GO TO 430
440 MX = IZ + IZ - 1
C----------------------------------------------------------------------
C Loop to determine MINEXP, XMIN.
C Exit from loop is signaled by an underflow.
C----------------------------------------------------------------------
450 XMIN = Y
Y = Y * BETAIN
C----------------------------------------------------------------------
C Check for underflow here.
C----------------------------------------------------------------------
A = Y * ONE
TEMP = Y * T
IF (((A+A) .EQ. ZERO) .OR. (ABS(Y) .GE. XMIN)) GO TO 460
K = K + 1
TEMP1 = TEMP * BETAIN
IF ((TEMP1*BETA .NE. Y) .OR. (TEMP .EQ. Y)) THEN
GO TO 450
ELSE
NXRES = 3
XMIN = Y
END IF
460 MINEXP = -K
C----------------------------------------------------------------------
C Determine MAXEXP, XMAX.
C----------------------------------------------------------------------
IF ((MX .GT. K+K-3) .OR. (IBETA .EQ. 10)) GO TO 500
MX = MX + MX
IEXP = IEXP + 1
500 MAXEXP = MX + MINEXP
C----------------------------------------------------------------------
C Adjust IRND to reflect partial underflow.
C----------------------------------------------------------------------
IRND = IRND + NXRES
C----------------------------------------------------------------------
C Adjust for IEEE-style machines.
C----------------------------------------------------------------------
IF (IRND .GE. 2) MAXEXP = MAXEXP - 2
C----------------------------------------------------------------------
C Adjust for machines with implicit leading bit in binary
C significand, and machines with radix point at extreme
C right of significand.
C----------------------------------------------------------------------
I = MAXEXP + MINEXP
IF ((IBETA .EQ. 2) .AND. (I .EQ. 0)) MAXEXP = MAXEXP - 1
IF (I .GT. 20) MAXEXP = MAXEXP - 1
IF (A .NE. Y) MAXEXP = MAXEXP - 2
XMAX = ONE - EPSNEG
IF (XMAX*ONE .NE. XMAX) XMAX = ONE - BETA * EPSNEG
XMAX = XMAX / (BETA * BETA * BETA * XMIN)
I = MAXEXP + MINEXP + 3
IF (I .LE. 0) GO TO 520
DO 510 J = 1, I
IF (IBETA .EQ. 2) XMAX = XMAX + XMAX
IF (IBETA .NE. 2) XMAX = XMAX * BETA
510 CONTINUE
520 RETURN
C---------- Last line of MACHAR ----------
END
FUNCTION REN(K)
C---------------------------------------------------------------------
C Random number generator - based on Algorithm 266 by Pike and
C Hill (modified by Hansson), Communications of the ACM,
C Vol. 8, No. 10, October 1965.
C
C This subprogram is intended for use on computers with
C fixed point wordlength of at least 29 bits. It is
C best if the floating-point significand has at most
C 29 bits.
C
C Latest modification: May 30, 1989
C
C Author: W. J. Cody
C Mathematics and Computer Science Division
C Argonne National Laboratory
C Argonne, IL 60439
C
C---------------------------------------------------------------------
INTEGER IY,J,K
CS REAL CONV,C1,C2,C3,ONE,REN
CD DOUBLE PRECISION CONV,C1,C2,C3,ONE,REN
DATA IY/100001/
CS DATA ONE,C1,C2,C3/1.0E0,2796203.0E0,1.0E-6,1.0E-12/
CD DATA ONE,C1,C2,C3/1.0D0,2796203.0D0,1.0D-6,1.0D-12/
C---------------------------------------------------------------------
C Statement functions for conversion between integer and float
C---------------------------------------------------------------------
CS CONV(J) = REAL(J)
CD CONV(J) = DBLE(J)
C---------------------------------------------------------------------
J = K
IY = IY * 125
IY = IY - (IY/2796203) * 2796203
REN = CONV(IY) / C1 * (ONE + C2 + C3)
RETURN
C---------- Last card of REN ----------
END