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|
SUBROUTINE RKBESL(X,ALPHA,NB,IZE,BK,NCALC)
C-------------------------------------------------------------------
C
C This FORTRAN 77 routine calculates modified Bessel functions
C of the second kind, K SUB(N+ALPHA) (X), for non-negative
C argument X, and non-negative order N+ALPHA, with or without
C exponential scaling.
C
C Explanation of variables in the calling sequence
C
C Description of output values ..
C
C X - Working precision non-negative real argument for which
C K's or exponentially scaled K's (K*EXP(X))
C are to be calculated. If K's are to be calculated,
C X must not be greater than XMAX (see below).
C ALPHA - Working precision fractional part of order for which
C K's or exponentially scaled K's (K*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 K's are to be calculated,
C and 2 if exponentially scaled K's are to be calculated.
C BK - Working precision output vector of length NB. If the
C routine terminates normally (NCALC=NB), the vector BK
C contains the functions K(ALPHA,X), ... , K(NB-1+ALPHA,X),
C or the corresponding exponentially scaled functions.
C If (0 .LT. NCALC .LT. NB), BK(I) contains correct function
C values for I .LE. NCALC, and contains the ratios
C K(ALPHA+I-1,X)/K(ALPHA+I-2,X) for the rest of the array.
C NCALC - Integer output variable indicating possible errors.
C Before using the vector BK, 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 EPS = The smallest positive floating-point number such that
C 1.0+EPS .GT. 1.0
C XMAX = Upper limit on the magnitude of X when IZE=1; Solution
C to equation:
C W(X) * (1-1/8X+9/128X**2) = beta**minexp
C where W(X) = EXP(-X)*SQRT(PI/2X)
C SQXMIN = Square root of beta**minexp
C XINF = Largest positive machine number; approximately
C beta**maxexp
C XMIN = Smallest positive machine number; approximately
C beta**minexp
C
C
C Approximate values for some important machines are:
C
C beta minexp maxexp EPS
C
C CRAY-1 (S.P.) 2 -8193 8191 7.11E-15
C Cyber 180/185
C under NOS (S.P.) 2 -975 1070 3.55E-15
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 2 -126 128 1.19E-7
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 2 -1022 1024 2.22D-16
C IBM 3033 (D.P.) 16 -65 63 2.22D-16
C VAX (S.P.) 2 -128 127 5.96E-8
C VAX D-Format (D.P.) 2 -128 127 1.39D-17
C VAX G-Format (D.P.) 2 -1024 1023 1.11D-16
C
C
C SQXMIN XINF XMIN XMAX
C
C CRAY-1 (S.P.) 6.77E-1234 5.45E+2465 4.59E-2467 5674.858
C Cyber 180/855
C under NOS (S.P.) 1.77E-147 1.26E+322 3.14E-294 672.788
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 1.08E-19 3.40E+38 1.18E-38 85.337
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 1.49D-154 1.79D+308 2.23D-308 705.342
C IBM 3033 (D.P.) 7.35D-40 7.23D+75 5.40D-79 177.852
C VAX (S.P.) 5.42E-20 1.70E+38 2.94E-39 86.715
C VAX D-Format (D.P.) 5.42D-20 1.70D+38 2.94D-39 86.715
C VAX G-Format (D.P.) 7.46D-155 8.98D+307 5.57D-309 706.728
C
C*******************************************************************
C*******************************************************************
C
C Error returns
C
C In case of an error, NCALC .NE. NB, and not all K'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, the B-vector is not calculated,
C and NCALC is set to MIN0(NB,0)-2 so that NCALC .NE. NB.
C NCALC = -1: Either K(ALPHA,X) .GE. XINF or
C K(ALPHA+NB-1,X)/K(ALPHA+NB-2,X) .GE. XINF. In this case,
C the B-vector is not calculated. Note that again
C NCALC .NE. NB.
C
C 0 .LT. NCALC .LT. NB: Not all requested function values could
C be calculated accurately. BK(I) contains correct function
C values for I .LE. NCALC, and contains the ratios
C K(ALPHA+I-1,X)/K(ALPHA+I-2,X) for the rest of the array.
C
C
C Intrinsic functions required are:
C
C ABS, AINT, EXP, INT, LOG, MAX, MIN, SINH, SQRT
C
C
C Acknowledgement
C
C This program is based on a program written by J. B. Campbell
C (2) that computes values of the Bessel functions K of real
C argument and real order. Modifications include the addition
C of non-scaled functions, parameterization of machine
C dependencies, and the use of more accurate approximations
C for SINH and SIN.
C
C References: "On Temme's Algorithm for the Modified Bessel
C Functions of the Third Kind," Campbell, J. B.,
C TOMS 6(4), Dec. 1980, pp. 581-586.
C
C "A FORTRAN IV Subroutine for the Modified Bessel
C Functions of the Third Kind of Real Order and Real
C Argument," Campbell, J. B., Report NRC/ERB-925,
C National Research Council, Canada.
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 I,IEND,ITEMP,IZE,J,K,M,MPLUS1,NB,NCALC
CS REAL
DOUBLE PRECISION
1 A,ALPHA,BLPHA,BK,BK1,BK2,C,D,DM,D1,D2,D3,ENU,EPS,ESTF,ESTM,
2 EX,FOUR,F0,F1,F2,HALF,ONE,P,P0,Q,Q0,R,RATIO,S,SQXMIN,T,TINYX,
3 TWO,TWONU,TWOX,T1,T2,WMINF,X,XINF,XMAX,XMIN,X2BY4,ZERO
DIMENSION BK(1),P(8),Q(7),R(5),S(4),T(6),ESTM(6),ESTF(7)
C---------------------------------------------------------------------
C Mathematical constants
C A = LOG(2.D0) - Euler's constant
C D = SQRT(2.D0/PI)
C---------------------------------------------------------------------
CS DATA HALF,ONE,TWO,ZERO/0.5E0,1.0E0,2.0E0,0.0E0/
CS DATA FOUR,TINYX/4.0E0,1.0E-10/
CS DATA A/ 0.11593151565841244881E0/,D/0.797884560802865364E0/
DATA HALF,ONE,TWO,ZERO/0.5D0,1.0D0,2.0D0,0.0D0/
DATA FOUR,TINYX/4.0D0,1.0D-10/
DATA A/ 0.11593151565841244881D0/,D/0.797884560802865364D0/
C---------------------------------------------------------------------
C Machine dependent parameters
C---------------------------------------------------------------------
CS DATA EPS/1.19E-7/,SQXMIN/1.08E-19/,XINF/3.40E+38/
CS DATA XMIN/1.18E-38/,XMAX/85.337E0/
DATA EPS/2.22D-16/,SQXMIN/1.49D-154/,XINF/1.79D+308/
DATA XMIN/2.23D-308/,XMAX/705.342D0/
C---------------------------------------------------------------------
C P, Q - Approximation for LOG(GAMMA(1+ALPHA))/ALPHA
C + Euler's constant
C Coefficients converted from hex to decimal and modified
C by W. J. Cody, 2/26/82
C R, S - Approximation for (1-ALPHA*PI/SIN(ALPHA*PI))/(2.D0*ALPHA)
C T - Approximation for SINH(Y)/Y
C---------------------------------------------------------------------
CS DATA P/ 0.805629875690432845E00, 0.204045500205365151E02,
CS 1 0.157705605106676174E03, 0.536671116469207504E03,
CS 2 0.900382759291288778E03, 0.730923886650660393E03,
CS 3 0.229299301509425145E03, 0.822467033424113231E00/
CS DATA Q/ 0.294601986247850434E02, 0.277577868510221208E03,
CS 1 0.120670325591027438E04, 0.276291444159791519E04,
CS 2 0.344374050506564618E04, 0.221063190113378647E04,
CS 3 0.572267338359892221E03/
CS DATA R/-0.48672575865218401848E+0, 0.13079485869097804016E+2,
CS 1 -0.10196490580880537526E+3, 0.34765409106507813131E+3,
CS 2 0.34958981245219347820E-3/
CS DATA S/-0.25579105509976461286E+2, 0.21257260432226544008E+3,
CS 1 -0.61069018684944109624E+3, 0.42269668805777760407E+3/
CS DATA T/ 0.16125990452916363814E-9, 0.25051878502858255354E-7,
CS 1 0.27557319615147964774E-5, 0.19841269840928373686E-3,
CS 2 0.83333333333334751799E-2, 0.16666666666666666446E+0/
CS DATA ESTM/5.20583E1, 5.7607E0, 2.7782E0, 1.44303E1, 1.853004E2,
CS 1 9.3715E0/
CS DATA ESTF/4.18341E1, 7.1075E0, 6.4306E0, 4.25110E1, 1.35633E0,
CS 1 8.45096E1, 2.0E1/
DATA P/ 0.805629875690432845D00, 0.204045500205365151D02,
1 0.157705605106676174D03, 0.536671116469207504D03,
2 0.900382759291288778D03, 0.730923886650660393D03,
3 0.229299301509425145D03, 0.822467033424113231D00/
DATA Q/ 0.294601986247850434D02, 0.277577868510221208D03,
1 0.120670325591027438D04, 0.276291444159791519D04,
2 0.344374050506564618D04, 0.221063190113378647D04,
3 0.572267338359892221D03/
DATA R/-0.48672575865218401848D+0, 0.13079485869097804016D+2,
1 -0.10196490580880537526D+3, 0.34765409106507813131D+3,
2 0.34958981245219347820D-3/
DATA S/-0.25579105509976461286D+2, 0.21257260432226544008D+3,
1 -0.61069018684944109624D+3, 0.42269668805777760407D+3/
DATA T/ 0.16125990452916363814D-9, 0.25051878502858255354D-7,
1 0.27557319615147964774D-5, 0.19841269840928373686D-3,
2 0.83333333333334751799D-2, 0.16666666666666666446D+0/
DATA ESTM/5.20583D1, 5.7607D0, 2.7782D0, 1.44303D1, 1.853004D2,
1 9.3715D0/
DATA ESTF/4.18341D1, 7.1075D0, 6.4306D0, 4.25110D1, 1.35633D0,
1 8.45096D1, 2.0D1/
C---------------------------------------------------------------------
EX = X
ENU = ALPHA
NCALC = MIN(NB,0)-2
IF ((NB .GT. 0) .AND. ((ENU .GE. ZERO) .AND. (ENU .LT. ONE))
1 .AND. ((IZE .GE. 1) .AND. (IZE .LE. 2)) .AND.
2 ((IZE .NE. 1) .OR. (EX .LE. XMAX)) .AND.
3 (EX .GT. ZERO)) THEN
K = 0
IF (ENU .LT. SQXMIN) ENU = ZERO
IF (ENU .GT. HALF) THEN
K = 1
ENU = ENU - ONE
END IF
TWONU = ENU+ENU
IEND = NB+K-1
C = ENU*ENU
D3 = -C
IF (EX .LE. ONE) THEN
C---------------------------------------------------------------------
C Calculation of P0 = GAMMA(1+ALPHA) * (2/X)**ALPHA
C Q0 = GAMMA(1-ALPHA) * (X/2)**ALPHA
C---------------------------------------------------------------------
D1 = ZERO
D2 = P(1)
T1 = ONE
T2 = Q(1)
DO 10 I = 2,7,2
D1 = C*D1+P(I)
D2 = C*D2+P(I+1)
T1 = C*T1+Q(I)
T2 = C*T2+Q(I+1)
10 CONTINUE
D1 = ENU*D1
T1 = ENU*T1
F1 = LOG(EX)
F0 = A+ENU*(P(8)-ENU*(D1+D2)/(T1+T2))-F1
Q0 = EXP(-ENU*(A-ENU*(P(8)+ENU*(D1-D2)/(T1-T2))-F1))
F1 = ENU*F0
P0 = EXP(F1)
C---------------------------------------------------------------------
C Calculation of F0 =
C---------------------------------------------------------------------
D1 = R(5)
T1 = ONE
DO 20 I = 1,4
D1 = C*D1+R(I)
T1 = C*T1+S(I)
20 CONTINUE
IF (ABS(F1) .LE. HALF) THEN
F1 = F1*F1
D2 = ZERO
DO 30 I = 1,6
D2 = F1*D2+T(I)
30 CONTINUE
D2 = F0+F0*F1*D2
ELSE
D2 = SINH(F1)/ENU
END IF
F0 = D2-ENU*D1/(T1*P0)
IF (EX .LE. TINYX) THEN
C--------------------------------------------------------------------
C X.LE.1.0E-10
C Calculation of K(ALPHA,X) and X*K(ALPHA+1,X)/K(ALPHA,X)
C--------------------------------------------------------------------
BK(1) = F0+EX*F0
IF (IZE .EQ. 1) BK(1) = BK(1)-EX*BK(1)
RATIO = P0/F0
C = EX*XINF
IF (K .NE. 0) THEN
C--------------------------------------------------------------------
C Calculation of K(ALPHA,X) and X*K(ALPHA+1,X)/K(ALPHA,X),
C ALPHA .GE. 1/2
C--------------------------------------------------------------------
NCALC = -1
IF (BK(1) .GE. C/RATIO) GO TO 500
BK(1) = RATIO*BK(1)/EX
TWONU = TWONU+TWO
RATIO = TWONU
END IF
NCALC = 1
IF (NB .EQ. 1) GO TO 500
C--------------------------------------------------------------------
C Calculate K(ALPHA+L,X)/K(ALPHA+L-1,X), L = 1, 2, ... , NB-1
C--------------------------------------------------------------------
NCALC = -1
DO 80 I = 2,NB
IF (RATIO .GE. C) GO TO 500
BK(I) = RATIO/EX
TWONU = TWONU+TWO
RATIO = TWONU
80 CONTINUE
NCALC = 1
GO TO 420
ELSE
C--------------------------------------------------------------------
C 1.0E-10 .LT. X .LE. 1.0
C--------------------------------------------------------------------
C = ONE
X2BY4 = EX*EX/FOUR
P0 = HALF*P0
Q0 = HALF*Q0
D1 = -ONE
D2 = ZERO
BK1 = ZERO
BK2 = ZERO
F1 = F0
F2 = P0
100 D1 = D1+TWO
D2 = D2+ONE
D3 = D1+D3
C = X2BY4*C/D2
F0 = (D2*F0+P0+Q0)/D3
P0 = P0/(D2-ENU)
Q0 = Q0/(D2+ENU)
T1 = C*F0
T2 = C*(P0-D2*F0)
BK1 = BK1+T1
BK2 = BK2+T2
IF ((ABS(T1/(F1+BK1)) .GT. EPS) .OR.
1 (ABS(T2/(F2+BK2)) .GT. EPS)) GO TO 100
BK1 = F1+BK1
BK2 = TWO*(F2+BK2)/EX
IF (IZE .EQ. 2) THEN
D1 = EXP(EX)
BK1 = BK1*D1
BK2 = BK2*D1
END IF
WMINF = ESTF(1)*EX+ESTF(2)
END IF
ELSE IF (EPS*EX .GT. ONE) THEN
C--------------------------------------------------------------------
C X .GT. ONE/EPS
C--------------------------------------------------------------------
NCALC = NB
BK1 = ONE / (D*SQRT(EX))
DO 110 I = 1, NB
BK(I) = BK1
110 CONTINUE
GO TO 500
ELSE
C--------------------------------------------------------------------
C X .GT. 1.0
C--------------------------------------------------------------------
TWOX = EX+EX
BLPHA = ZERO
RATIO = ZERO
IF (EX .LE. FOUR) THEN
C--------------------------------------------------------------------
C Calculation of K(ALPHA+1,X)/K(ALPHA,X), 1.0 .LE. X .LE. 4.0
C--------------------------------------------------------------------
D2 = AINT(ESTM(1)/EX+ESTM(2))
M = INT(D2)
D1 = D2+D2
D2 = D2-HALF
D2 = D2*D2
DO 120 I = 2,M
D1 = D1-TWO
D2 = D2-D1
RATIO = (D3+D2)/(TWOX+D1-RATIO)
120 CONTINUE
C--------------------------------------------------------------------
C Calculation of I(|ALPHA|,X) and I(|ALPHA|+1,X) by backward
C recurrence and K(ALPHA,X) from the wronskian
C--------------------------------------------------------------------
D2 = AINT(ESTM(3)*EX+ESTM(4))
M = INT(D2)
C = ABS(ENU)
D3 = C+C
D1 = D3-ONE
F1 = XMIN
F0 = (TWO*(C+D2)/EX+HALF*EX/(C+D2+ONE))*XMIN
DO 130 I = 3,M
D2 = D2-ONE
F2 = (D3+D2+D2)*F0
BLPHA = (ONE+D1/D2)*(F2+BLPHA)
F2 = F2/EX+F1
F1 = F0
F0 = F2
130 CONTINUE
F1 = (D3+TWO)*F0/EX+F1
D1 = ZERO
T1 = ONE
DO 140 I = 1,7
D1 = C*D1+P(I)
T1 = C*T1+Q(I)
140 CONTINUE
P0 = EXP(C*(A+C*(P(8)-C*D1/T1)-LOG(EX)))/EX
F2 = (C+HALF-RATIO)*F1/EX
BK1 = P0+(D3*F0-F2+F0+BLPHA)/(F2+F1+F0)*P0
IF (IZE .EQ. 1) BK1 = BK1*EXP(-EX)
WMINF = ESTF(3)*EX+ESTF(4)
ELSE
C--------------------------------------------------------------------
C Calculation of K(ALPHA,X) and K(ALPHA+1,X)/K(ALPHA,X), by backward
C recurrence, for X .GT. 4.0
C--------------------------------------------------------------------
DM = AINT(ESTM(5)/EX+ESTM(6))
M = INT(DM)
D2 = DM-HALF
D2 = D2*D2
D1 = DM+DM
DO 160 I = 2,M
DM = DM-ONE
D1 = D1-TWO
D2 = D2-D1
RATIO = (D3+D2)/(TWOX+D1-RATIO)
BLPHA = (RATIO+RATIO*BLPHA)/DM
160 CONTINUE
BK1 = ONE/((D+D*BLPHA)*SQRT(EX))
IF (IZE .EQ. 1) BK1 = BK1*EXP(-EX)
WMINF = ESTF(5)*(EX-ABS(EX-ESTF(7)))+ESTF(6)
END IF
C--------------------------------------------------------------------
C Calculation of K(ALPHA+1,X) from K(ALPHA,X) and
C K(ALPHA+1,X)/K(ALPHA,X)
C--------------------------------------------------------------------
BK2 = BK1+BK1*(ENU+HALF-RATIO)/EX
END IF
C--------------------------------------------------------------------
C Calculation of 'NCALC', K(ALPHA+I,X), I = 0, 1, ... , NCALC-1,
C K(ALPHA+I,X)/K(ALPHA+I-1,X), I = NCALC, NCALC+1, ... , NB-1
C--------------------------------------------------------------------
NCALC = NB
BK(1) = BK1
IF (IEND .EQ. 0) GO TO 500
J = 2-K
IF (J .GT. 0) BK(J) = BK2
IF (IEND .EQ. 1) GO TO 500
M = MIN(INT(WMINF-ENU),IEND)
DO 190 I = 2,M
T1 = BK1
BK1 = BK2
TWONU = TWONU+TWO
IF (EX .LT. ONE) THEN
IF (BK1 .GE. (XINF/TWONU)*EX) GO TO 195
GO TO 187
ELSE
IF (BK1/EX .GE. XINF/TWONU) GO TO 195
END IF
187 CONTINUE
BK2 = TWONU/EX*BK1+T1
ITEMP = I
J = J+1
IF (J .GT. 0) BK(J) = BK2
190 CONTINUE
195 M = ITEMP
IF (M .EQ. IEND) GO TO 500
RATIO = BK2/BK1
MPLUS1 = M+1
NCALC = -1
DO 410 I = MPLUS1,IEND
TWONU = TWONU+TWO
RATIO = TWONU/EX+ONE/RATIO
J = J+1
IF (J .GT. 1) THEN
BK(J) = RATIO
ELSE
IF (BK2 .GE. XINF/RATIO) GO TO 500
BK2 = RATIO*BK2
END IF
410 CONTINUE
NCALC = MAX(MPLUS1-K,1)
IF (NCALC .EQ. 1) BK(1) = BK2
IF (NB .EQ. 1) GO TO 500
420 J = NCALC+1
DO 430 I = J,NB
IF (BK(NCALC) .GE. XINF/BK(I)) GO TO 500
BK(I) = BK(NCALC)*BK(I)
NCALC = I
430 CONTINUE
END IF
500 RETURN
C---------- Last line of RKBESL ----------
END
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