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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
|
