      SUBROUTINE RYBESL(X,ALPHA,NB,BY,NCALC)
C----------------------------------------------------------------------
C
C  This routine calculates Bessel functions Y 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         Y's are to be calculated.
C ALPHA - Working precision fractional part of order for which
C         Y's are 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 BY    - Working precision output vector of length NB.  If the
C         routine terminates normally (NCALC=NB), the vector BY
C         contains the functions Y(ALPHA,X), ... , Y(NB-1+ALPHA,X),
C         If (0 .LT. NCALC .LT. NB), BY(I) contains correct function
C         values for I .LE. NCALC, and contains the ratios
C         Y(ALPHA+I-1,X)/Y(ALPHA+I-2,X) for the rest of the array.
C NCALC - Integer output variable indicating possible errors.
C         Before using the vector BY, 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   p      = Number of significant base-beta digits in the
C            significand of a floating-point number
C   minexp = Smallest representable power of beta
C   maxexp = Smallest power of beta that overflows
C   EPS    = beta ** (-p)
C   DEL    = Machine number below which sin(x)/x = 1; approximately
C            SQRT(EPS).
C   XMIN   = Smallest acceptable argument for RBESY; approximately
C            max(2*beta**minexp,2/XINF), rounded up
C   XINF   = Largest positive machine number; approximately
C            beta**maxexp
C   THRESH = Lower bound for use of the asymptotic form; approximately
C            AINT(-LOG10(EPS/2.0))+1.0
C   XLARGE = Upper bound on X; approximately 1/DEL, because the sine
C            and cosine functions have lost about half of their 
C            precision at that point.
C
C
C     Approximate values for some important machines are:
C
C                        beta    p     minexp      maxexp      EPS
C
C  CRAY-1        (S.P.)    2    48     -8193        8191    3.55E-15
C  Cyber 180/185 
C    under NOS   (S.P.)    2    48      -975        1070    3.55E-15
C  IEEE (IBM/XT,
C    SUN, etc.)  (S.P.)    2    24      -126         128    5.96E-8
C  IEEE (IBM/XT,
C    SUN, etc.)  (D.P.)    2    53     -1022        1024    1.11D-16
C  IBM 3033      (D.P.)   16    14       -65          63    1.39D-17
C  VAX           (S.P.)    2    24      -128         127    5.96E-8
C  VAX D-Format  (D.P.)    2    56      -128         127    1.39D-17
C  VAX G-Format  (D.P.)    2    53     -1024        1023    1.11D-16
C
C
C                         DEL      XMIN      XINF     THRESH  XLARGE
C
C CRAY-1        (S.P.)  5.0E-8  3.67E-2466 5.45E+2465  15.0E0  2.0E7
C Cyber 180/855
C   under NOS   (S.P.)  5.0E-8  6.28E-294  1.26E+322   15.0E0  2.0E7
C IEEE (IBM/XT,
C   SUN, etc.)  (S.P.)  1.0E-4  2.36E-38   3.40E+38     8.0E0  1.0E4
C IEEE (IBM/XT,
C   SUN, etc.)  (D.P.)  1.0D-8  4.46D-308  1.79D+308   16.0D0  1.0D8
C IBM 3033      (D.P.)  1.0D-8  2.77D-76   7.23D+75    17.0D0  1.0D8
C VAX           (S.P.)  1.0E-4  1.18E-38   1.70E+38     8.0E0  1.0E4
C VAX D-Format  (D.P.)  1.0D-9  1.18D-38   1.70D+38    17.0D0  1.0D9
C VAX G-Format  (D.P.)  1.0D-8  2.23D-308  8.98D+307   16.0D0  1.0D8
C
C*******************************************************************
C*******************************************************************
C
C Error returns
C
C  In case of an error, NCALC .NE. NB, and not all Y's are
C  calculated to the desired accuracy.
C
C  NCALC .LT. -1:  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.
C       XMAX.  In this case, BY(1) = 0.0, the remainder of the
C       BY-vector is not calculated, and NCALC is set to
C       MIN0(NB,0)-2  so that NCALC .NE. NB.
C  NCALC = -1:  Y(ALPHA,X) .GE. XINF.  The requested function
C       values are set to 0.0.
C  1 .LT. NCALC .LT. NB: Not all requested function values could
C       be calculated accurately.  BY(I) contains correct function
C       values for I .LE. NCALC, and and the remaining NB-NCALC
C       array elements contain 0.0.
C
C
C Intrinsic functions required are:
C
C     DBLE, EXP, INT, MAX, MIN, REAL, SQRT
C
C
C Acknowledgement
C
C  This program draws heavily on Temme's Algol program for Y(a,x)
C  and Y(a+1,x) and on Campbell's programs for Y_nu(x).  Temme's
C  scheme is used for  x < THRESH, and Campbell's scheme is used
C  in the asymptotic region.  Segments of code from both sources
C  have been translated into Fortran 77, merged, and heavily modified.
C  Modifications include parameterization of machine dependencies,
C  use of a new approximation for ln(gamma(x)), and built-in
C  protection against over/underflow.
C
C References: "Bessel functions J_nu(x) and Y_nu(x) of real
C              order and real argument," Campbell, J. B.,
C              Comp. Phy. Comm. 18, 1979, pp. 133-142.
C
C             "On the numerical evaluation of the ordinary
C              Bessel function of the second kind," Temme,
C              N. M., J. Comput. Phys. 21, 1976, pp. 343-350.
C
C  Latest modification: March 19, 1990
C
C  Modified by: W. J. Cody
C               Applied Mathematics Division
C               Argonne National Laboratory
C               Argonne, IL  60439
C
C----------------------------------------------------------------------
      INTEGER I,K,NA,NB,NCALC
CS    REAL
      DOUBLE PRECISION
     1  ALFA,ALPHA,AYE,B,BY,C,CH,COSMU,D,DEL,DEN,DDIV,DIV,DMU,D1,D2,
     2  E,EIGHT,EN,ENU,EN1,EPS,EVEN,EX,F,FIVPI,G,GAMMA,H,HALF,ODD,
     3  ONBPI,ONE,ONE5,P,PA,PA1,PI,PIBY2,PIM5,Q,QA,QA1,Q0,R,S,SINMU,
     4  SQ2BPI,TEN9,TERM,THREE,THRESH,TWO,TWOBYX,X,XINF,XLARGE,XMIN,
     5  XNA,X2,YA,YA1,ZERO
      DIMENSION BY(NB),CH(21)
C----------------------------------------------------------------------
C  Mathematical constants
C    FIVPI = 5*PI
C    PIM5 = 5*PI - 15
C    ONBPI = 1/PI
C    PIBY2 = PI/2
C    SQ2BPI = SQUARE ROOT OF 2/PI
C----------------------------------------------------------------------
CS    DATA ZERO,HALF,ONE,TWO,THREE/0.0E0,0.5E0,1.0E0,2.0E0,3.0E0/
CS    DATA EIGHT,ONE5,TEN9/8.0E0,15.0E0,1.9E1/
CS    DATA FIVPI,PIBY2/1.5707963267948966192E1,1.5707963267948966192E0/
CS    DATA PI,SQ2BPI/3.1415926535897932385E0,7.9788456080286535588E-1/
CS    DATA PIM5,ONBPI/7.0796326794896619231E-1,3.1830988618379067154E-1/
      DATA ZERO,HALF,ONE,TWO,THREE/0.0D0,0.5D0,1.0D0,2.0D0,3.0D0/
      DATA EIGHT,ONE5,TEN9/8.0D0,15.0D0,1.9D1/
      DATA FIVPI,PIBY2/1.5707963267948966192D1,1.5707963267948966192D0/
      DATA PI,SQ2BPI/3.1415926535897932385D0,7.9788456080286535588D-1/
      DATA PIM5,ONBPI/7.0796326794896619231D-1,3.1830988618379067154D-1/
C----------------------------------------------------------------------
C  Machine-dependent constants
C----------------------------------------------------------------------
CS    DATA DEL,XMIN,XINF,EPS/1.0E-4,2.36E-38,3.40E38,5.96E-8/
CS    DATA THRESH,XLARGE/8.0E0,1.0E4/
      DATA DEL,XMIN,XINF,EPS/1.0D-8,4.46D-308,1.79D308,1.11D-16/
      DATA THRESH,XLARGE/16.0D0,1.0D8/
C----------------------------------------------------------------------
C  Coefficients for Chebyshev polynomial expansion of 
C         1/gamma(1-x), abs(x) .le. .5
C----------------------------------------------------------------------
CS    DATA CH/-0.67735241822398840964E-23,-0.61455180116049879894E-22,
CS   1         0.29017595056104745456E-20, 0.13639417919073099464E-18,
CS   2         0.23826220476859635824E-17,-0.90642907957550702534E-17,
CS   3        -0.14943667065169001769E-14,-0.33919078305362211264E-13,
CS   4        -0.17023776642512729175E-12, 0.91609750938768647911E-11,
CS   5         0.24230957900482704055E-09, 0.17451364971382984243E-08,
CS   6        -0.33126119768180852711E-07,-0.86592079961391259661E-06,
CS   7        -0.49717367041957398581E-05, 0.76309597585908126618E-04,
CS   8         0.12719271366545622927E-02, 0.17063050710955562222E-02,
CS   9        -0.76852840844786673690E-01,-0.28387654227602353814E+00,
CS   A         0.92187029365045265648E+00/
      DATA CH/-0.67735241822398840964D-23,-0.61455180116049879894D-22,
     1         0.29017595056104745456D-20, 0.13639417919073099464D-18,
     2         0.23826220476859635824D-17,-0.90642907957550702534D-17,
     3        -0.14943667065169001769D-14,-0.33919078305362211264D-13,
     4        -0.17023776642512729175D-12, 0.91609750938768647911D-11,
     5         0.24230957900482704055D-09, 0.17451364971382984243D-08,
     6        -0.33126119768180852711D-07,-0.86592079961391259661D-06,
     7        -0.49717367041957398581D-05, 0.76309597585908126618D-04,
     8         0.12719271366545622927D-02, 0.17063050710955562222D-02,
     9        -0.76852840844786673690D-01,-0.28387654227602353814D+00,
     A         0.92187029365045265648D+00/
C----------------------------------------------------------------------
      EX = X
      ENU = ALPHA
      IF ((NB .GT. 0) .AND. (X .GE. XMIN) .AND. (EX .LT. XLARGE)
     1       .AND. (ENU .GE. ZERO) .AND. (ENU .LT. ONE))  THEN
            XNA = AINT(ENU+HALF)
            NA = INT(XNA)
            IF (NA .EQ. 1) ENU = ENU - XNA
            IF (ENU .EQ. -HALF) THEN
                  P = SQ2BPI/SQRT(EX)
                  YA = P * SIN(EX)
                  YA1 = -P * COS(EX)
               ELSE IF (EX .LT. THREE) THEN
C----------------------------------------------------------------------
C  Use Temme's scheme for small X
C----------------------------------------------------------------------
                  B = EX * HALF
                  D = -LOG(B)
                  F = ENU * D
                  E = B**(-ENU)
                  IF (ABS(ENU) .LT. DEL) THEN
                        C = ONBPI
                     ELSE
                        C = ENU / SIN(ENU*PI)
                  END IF
C----------------------------------------------------------------------
C  Computation of sinh(f)/f
C----------------------------------------------------------------------
                  IF (ABS(F) .LT. ONE) THEN
                        X2 = F*F
                        EN = TEN9
                        S = ONE
                        DO 80 I = 1, 9
                           S = S*X2/EN/(EN-ONE)+ONE
                           EN = EN - TWO
   80                   CONTINUE
                     ELSE 
                        S = (E - ONE/E) * HALF / F
                  END IF
C----------------------------------------------------------------------
C  Computation of 1/gamma(1-a) using Chebyshev polynomials
C----------------------------------------------------------------------
                  X2 = ENU*ENU*EIGHT
                  AYE = CH(1)
                  EVEN = ZERO
                  ALFA = CH(2)
                  ODD = ZERO
                  DO 40 I = 3, 19, 2
                     EVEN = -(AYE+AYE+EVEN)
                     AYE = -EVEN*X2 - AYE + CH(I)
                     ODD = -(ALFA+ALFA+ODD)
                     ALFA = -ODD*X2 - ALFA + CH(I+1)
   40             CONTINUE
                  EVEN = (EVEN*HALF+AYE)*X2 - AYE + CH(21)
                  ODD = (ODD+ALFA)*TWO
                  GAMMA = ODD*ENU + EVEN
C----------------------------------------------------------------------
C  End of computation of 1/gamma(1-a)
C----------------------------------------------------------------------
                  G = E * GAMMA
                  E = (E + ONE/E) * HALF
                  F = TWO*C*(ODD*E+EVEN*S*D)
                  E = ENU*ENU
                  P = G*C
                  Q = ONBPI / G
                  C = ENU*PIBY2
                  IF (ABS(C) .LT. DEL) THEN
                        R = ONE
                     ELSE 
                        R = SIN(C)/C
                  END IF
                  R = PI*C*R*R
                  C = ONE
                  D = - B*B
                  H = ZERO
                  YA = F + R*Q
                  YA1 = P
                  EN = ZERO
  100             EN = EN + ONE
                  IF (ABS(G/(ONE+ABS(YA)))
     1                      + ABS(H/(ONE+ABS(YA1))) .GT. EPS) THEN
                        F = (F*EN+P+Q)/(EN*EN-E)
                        C = C * D/EN
                        P = P/(EN-ENU)
                        Q = Q/(EN+ENU)
                        G = C*(F+R*Q)
                        H = C*P - EN*G
                        YA = YA + G
                        YA1 = YA1+H
                        GO TO 100
                  END IF
                  YA = -YA
                  YA1 = -YA1/B
               ELSE IF (EX .LT. THRESH) THEN
C----------------------------------------------------------------------
C  Use Temme's scheme for moderate X
C----------------------------------------------------------------------
                  C = (HALF-ENU)*(HALF+ENU)
                  B = EX + EX
                  E = (EX*ONBPI*COS(ENU*PI)/EPS)
                  E = E*E
                  P = ONE
                  Q = -EX
                  R = ONE + EX*EX
                  S = R
                  EN = TWO
  200             IF (R*EN*EN .LT. E) THEN
                        EN1 = EN+ONE
                        D = (EN-ONE+C/EN)/S
                        P = (EN+EN-P*D)/EN1
                        Q = (-B+Q*D)/EN1
                        S = P*P + Q*Q
                        R = R*S
                        EN = EN1
                        GO TO 200
                  END IF
                  F = P/S
                  P = F
                  G = -Q/S
                  Q = G
  220             EN = EN - ONE  
                  IF (EN .GT. ZERO) THEN
                        R = EN1*(TWO-P)-TWO
                        S = B + EN1*Q
                        D = (EN-ONE+C/EN)/(R*R+S*S)
                        P = D*R
                        Q = D*S
                        E = F + ONE
                        F = P*E - G*Q
                        G = Q*E + P*G
                        EN1 = EN
                        GO TO 220
                  END IF
                  F = ONE + F
                  D = F*F + G*G
                  PA = F/D
                  QA = -G/D
                  D = ENU + HALF -P
                  Q = Q + EX
                  PA1 = (PA*Q-QA*D)/EX
                  QA1 = (QA*Q+PA*D)/EX
                  B = EX - PIBY2*(ENU+HALF)
                  C = COS(B)
                  S = SIN(B)
                  D = SQ2BPI/SQRT(EX)
                  YA = D*(PA*S+QA*C)
                  YA1 = D*(QA1*S-PA1*C)
               ELSE
C----------------------------------------------------------------------
C  Use Campbell's asymptotic scheme.
C----------------------------------------------------------------------
                  NA = 0
                  D1 = AINT(EX/FIVPI)
                  I = INT(D1)
                  DMU = ((EX-ONE5*D1)-D1*PIM5)-(ALPHA+HALF)*PIBY2
                  IF (I-2*(I/2) .EQ. 0) THEN
                        COSMU = COS(DMU)
                        SINMU = SIN(DMU)
                     ELSE
                        COSMU = -COS(DMU)
                        SINMU = -SIN(DMU)
                  END IF
                  DDIV = EIGHT * EX
                  DMU = ALPHA
                  DEN = SQRT(EX)
                  DO 350 K = 1, 2
                     P = COSMU
                     COSMU = SINMU
                     SINMU = -P
                     D1 = (TWO*DMU-ONE)*(TWO*DMU+ONE)
                     D2 = ZERO
                     DIV = DDIV
                     P = ZERO
                     Q = ZERO
                     Q0 = D1/DIV
                     TERM = Q0
                     DO 310 I = 2, 20
                        D2 = D2 + EIGHT
                        D1 = D1 - D2
                        DIV = DIV + DDIV
                        TERM = -TERM*D1/DIV
                        P = P + TERM
                        D2 = D2 + EIGHT
                        D1 = D1 - D2
                        DIV = DIV + DDIV
                        TERM = TERM*D1/DIV
                        Q = Q + TERM
                        IF (ABS(TERM) .LE. EPS) GO TO 320
  310                CONTINUE
  320                P = P + ONE
                     Q = Q + Q0
                     IF (K .EQ. 1) THEN
                           YA = SQ2BPI * (P*COSMU-Q*SINMU) / DEN
                        ELSE
                           YA1 = SQ2BPI * (P*COSMU-Q*SINMU) / DEN
                     END IF
                     DMU = DMU + ONE
  350             CONTINUE
            END IF
            IF (NA .EQ. 1) THEN
               H = TWO*(ENU+ONE)/EX
               IF (H .GT. ONE) THEN
                  IF (ABS(YA1) .GT. XINF/H) THEN
                     H = ZERO
                     YA = ZERO
                  END IF
               END IF
               H = H*YA1 - YA
               YA = YA1
               YA1 = H
            END IF
C----------------------------------------------------------------------
C  Now have first one or two Y's
C----------------------------------------------------------------------
            BY(1) = YA
            BY(2) = YA1
            IF (YA1 .EQ. ZERO) THEN
                  NCALC = 1
               ELSE
                  AYE = ONE + ALPHA
                  TWOBYX = TWO/EX
                  NCALC = 2
                  DO 400 I = 3, NB
                     IF (TWOBYX .LT. ONE) THEN
                           IF (ABS(BY(I-1))*TWOBYX .GE. XINF/AYE)
     1                                                     GO TO 450
                        ELSE
                           IF (ABS(BY(I-1)) .GE. XINF/AYE/TWOBYX )
     1                                                     GO TO 450
                     END IF
                     BY(I) = TWOBYX*AYE*BY(I-1) - BY(I-2) 
                     AYE = AYE + ONE
                     NCALC = NCALC + 1
  400             CONTINUE
            END IF
  450       DO 460 I = NCALC+1, NB
               BY(I) = ZERO
  460       CONTINUE
         ELSE
            BY(1) = ZERO
            NCALC = MIN(NB,0) - 1
      END IF
  900 RETURN
C---------- Last line of RYBESL ----------
      END



