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k1.f
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module k1
!! from https://netlib.org/specfun/k1
implicit none
private
public:: besk1
contains
SUBROUTINE CALCK1(ARG,RESULT,JINT)
C--------------------------------------------------------------------
C
C This packet computes modified Bessel functions of the second kind
C and order one, K1(X) and EXP(X)*K1(X), for real arguments X.
C It contains two function type subprograms, BESK1 and BESEK1,
C and one subroutine type subprogram, CALCK1. The calling
C statements for the primary entries are
C
C Y=BESK1(X)
C and
C Y=BESEK1(X)
C
C where the entry points correspond to the functions K1(X) and
C EXP(X)*K1(X), respectively. The routine CALCK1 is intended
C for internal packet use only, all computations within the
C packet being concentrated in this routine. The function
C subprograms invoke CALCK1 with the statement
C CALL CALCK1(ARG,RESULT,JINT)
C where the parameter usage is as follows
C
C Function Parameters for CALCK1
C Call ARG RESULT JINT
C
C BESK1(ARG) XLEAST .LT. ARG .LT. XMAX K1(ARG) 1
C BESEK1(ARG) XLEAST .LT. ARG EXP(ARG)*K1(ARG) 2
C
C The main computation evaluates slightly modified forms of near
C minimax rational approximations generated by Russon and Blair,
C Chalk River (Atomic Energy of Canada Limited) Report AECL-3461,
C 1969. This transportable program is patterned after the
C machine-dependent FUNPACK packet NATSK1, but cannot match that
C version for efficiency or accuracy. This version uses rational
C functions that theoretically approximate K-SUB-1(X) to at
C least 18 significant decimal digits. The accuracy achieved
C depends on the arithmetic system, the compiler, the intrinsic
C functions, and proper selection of the machine-dependent
C constants.
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 XLEAST = Smallest acceptable argument, i.e., smallest machine
C number X such that 1/X is machine representable.
C XSMALL = Argument below which BESK1(X) and BESEK1(X) may
C each be represented by 1/X. A safe value is the
C largest X such that 1.0 + X = 1.0 to machine
C precision.
C XINF = Largest positive machine number; approximately
C beta**maxexp
C XMAX = Largest argument acceptable to BESK1; Solution to
C equation:
C W(X) * (1+3/8X-15/128X**2) = beta**minexp
C where W(X) = EXP(-X)*SQRT(PI/2X)
C
C
C Approximate values for some important machines are:
C
C beta minexp maxexp
C
C CRAY-1 (S.P.) 2 -8193 8191
C Cyber 180/185
C under NOS (S.P.) 2 -975 1070
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 2 -126 128
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 2 -1022 1024
C IBM 3033 (D.P.) 16 -65 63
C VAX D-Format (D.P.) 2 -128 127
C VAX G-Format (D.P.) 2 -1024 1023
C
C
C XLEAST XSMALL XINF XMAX
C
C CRAY-1 1.84E-2466 3.55E-15 5.45E+2465 5674.858
C Cyber 180/855
C under NOS (S.P.) 3.14E-294 1.77E-15 1.26E+322 672.789
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 1.18E-38 5.95E-8 3.40E+38 85.343
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 2.23D-308 1.11D-16 1.79D+308 705.343
C IBM 3033 (D.P.) 1.39D-76 1.11D-16 7.23D+75 177.855
C VAX D-Format (D.P.) 5.88D-39 6.95D-18 1.70D+38 86.721
C VAX G-Format (D.P.) 1.12D-308 5.55D-17 8.98D+307 706.728
C
C*******************************************************************
C*******************************************************************
C
C Error returns
C
C The program returns the value XINF for ARG .LE. 0.0 and the
C BESK1 entry returns the value 0.0 for ARG .GT. XMAX.
C
C
C Intrinsic functions required are:
C
C LOG, SQRT, EXP
C
C
C Authors: W. J. Cody and Laura Stoltz
C Mathematics and Computer Science Division
C Argonne National Laboratory
C Argonne, IL 60439
C
C Latest modification: January 28, 1988
C
C--------------------------------------------------------------------
INTEGER I,JINT
CS REAL
DOUBLE PRECISION
1 ARG,F,G,ONE,P,PP,Q,QQ,RESULT,SUMF,SUMG,
2 SUMP,SUMQ,X,XINF,XMAX,XLEAST,XSMALL,XX,ZERO
DIMENSION P(5),Q(3),PP(11),QQ(9),F(5),G(3)
C--------------------------------------------------------------------
C Mathematical constants
C--------------------------------------------------------------------
CS DATA ONE/1.0E0/,ZERO/0.0E0/
DATA ONE/1.0D0/,ZERO/0.0D0/
C--------------------------------------------------------------------
C Machine-dependent constants
C--------------------------------------------------------------------
CS DATA XLEAST/1.18E-38/,XSMALL/5.95E-8/,XINF/3.40E+38/,
CS 1 XMAX/85.343E+0/
DATA XLEAST/2.23D-308/,XSMALL/1.11D-16/,XINF/1.79D+308/,
1 XMAX/705.343D+0/
C--------------------------------------------------------------------
C Coefficients for XLEAST .LE. ARG .LE. 1.0
C--------------------------------------------------------------------
CS DATA P/ 4.8127070456878442310E-1, 9.9991373567429309922E+1,
CS 1 7.1885382604084798576E+3, 1.7733324035147015630E+5,
CS 2 7.1938920065420586101E+5/
CS DATA Q/-2.8143915754538725829E+2, 3.7264298672067697862E+4,
CS 1 -2.2149374878243304548E+6/
CS DATA F/-2.2795590826955002390E-1,-5.3103913335180275253E+1,
CS 1 -4.5051623763436087023E+3,-1.4758069205414222471E+5,
CS 2 -1.3531161492785421328E+6/
CS DATA G/-3.0507151578787595807E+2, 4.3117653211351080007E+4,
CS 2 -2.7062322985570842656E+6/
DATA P/ 4.8127070456878442310D-1, 9.9991373567429309922D+1,
1 7.1885382604084798576D+3, 1.7733324035147015630D+5,
2 7.1938920065420586101D+5/
DATA Q/-2.8143915754538725829D+2, 3.7264298672067697862D+4,
1 -2.2149374878243304548D+6/
DATA F/-2.2795590826955002390D-1,-5.3103913335180275253D+1,
1 -4.5051623763436087023D+3,-1.4758069205414222471D+5,
2 -1.3531161492785421328D+6/
DATA G/-3.0507151578787595807D+2, 4.3117653211351080007D+4,
2 -2.7062322985570842656D+6/
C--------------------------------------------------------------------
C Coefficients for 1.0 .LT. ARG
C--------------------------------------------------------------------
CS DATA PP/ 6.4257745859173138767E-2, 7.5584584631176030810E+0,
CS 1 1.3182609918569941308E+2, 8.1094256146537402173E+2,
CS 2 2.3123742209168871550E+3, 3.4540675585544584407E+3,
CS 3 2.8590657697910288226E+3, 1.3319486433183221990E+3,
CS 4 3.4122953486801312910E+2, 4.4137176114230414036E+1,
CS 5 2.2196792496874548962E+0/
CS DATA QQ/ 3.6001069306861518855E+1, 3.3031020088765390854E+2,
CS 1 1.2082692316002348638E+3, 2.1181000487171943810E+3,
CS 2 1.9448440788918006154E+3, 9.6929165726802648634E+2,
CS 3 2.5951223655579051357E+2, 3.4552228452758912848E+1,
CS 4 1.7710478032601086579E+0/
DATA PP/ 6.4257745859173138767D-2, 7.5584584631176030810D+0,
1 1.3182609918569941308D+2, 8.1094256146537402173D+2,
2 2.3123742209168871550D+3, 3.4540675585544584407D+3,
3 2.8590657697910288226D+3, 1.3319486433183221990D+3,
4 3.4122953486801312910D+2, 4.4137176114230414036D+1,
5 2.2196792496874548962D+0/
DATA QQ/ 3.6001069306861518855D+1, 3.3031020088765390854D+2,
1 1.2082692316002348638D+3, 2.1181000487171943810D+3,
2 1.9448440788918006154D+3, 9.6929165726802648634D+2,
3 2.5951223655579051357D+2, 3.4552228452758912848D+1,
4 1.7710478032601086579D+0/
C--------------------------------------------------------------------
X = ARG
IF (X .LT. XLEAST) THEN
C--------------------------------------------------------------------
C Error return for ARG .LT. XLEAST
C--------------------------------------------------------------------
RESULT = XINF
ELSE IF (X .LE. 1) THEN
C--------------------------------------------------------------------
C XLEAST .LE. ARG .LE. 1.0
C--------------------------------------------------------------------
IF (X .LT. XSMALL) THEN
C--------------------------------------------------------------------
C Return for small ARG
C--------------------------------------------------------------------
RESULT = 1 / X
ELSE
XX = X * X
SUMP = ((((P(1)*XX + P(2))*XX + P(3))*XX + P(4))*XX
1 + P(5))*XX + Q(3)
SUMQ = ((XX + Q(1))*XX + Q(2))*XX + Q(3)
SUMF = (((F(1)*XX + F(2))*XX + F(3))*XX + F(4))*XX
1 + F(5)
SUMG = ((XX + G(1))*XX + G(2))*XX + G(3)
RESULT = (XX * LOG(X) * SUMF/SUMG + SUMP/SUMQ) / X
IF (JINT .EQ. 2) RESULT = RESULT * EXP(X)
END IF
ELSE IF ((JINT .EQ. 1) .AND. (X .GT. XMAX)) THEN
C--------------------------------------------------------------------
C Error return for ARG .GT. XMAX
C--------------------------------------------------------------------
RESULT = 0
ELSE
C--------------------------------------------------------------------
C 1.0 .LT. ARG
C--------------------------------------------------------------------
XX = 1 / X
SUMP = PP(1)
DO 120 I = 2, 11
SUMP = SUMP * XX + PP(I)
120 CONTINUE
SUMQ = XX
DO 140 I = 1, 8
SUMQ = (SUMQ + QQ(I)) * XX
140 CONTINUE
SUMQ = SUMQ + QQ(9)
RESULT = SUMP / SUMQ / SQRT(X)
IF (JINT .EQ. 1) RESULT = RESULT * EXP(-X)
END IF
RETURN
C---------- Last line of CALCK1 ----------
END SUBROUTINE CALCK1
CS REAL
DOUBLE PRECISION FUNCTION BESK1(X)
C--------------------------------------------------------------------
C
C This function program computes approximate values for the
C modified Bessel function of the second kind of order one
C for arguments XLEAST .LE. ARG .LE. XMAX.
C
C--------------------------------------------------------------------
INTEGER JINT
CS REAL
DOUBLE PRECISION X, RESULT
C--------------------------------------------------------------------
JINT = 1
CALL CALCK1(X,RESULT,JINT)
BESK1 = RESULT
RETURN
C---------- Last line of BESK1 ----------
END FUNCTION BESK1
CS REAL
DOUBLE PRECISION FUNCTION BESEK1(X)
C--------------------------------------------------------------------
C
C This function program computes approximate values for the
C modified Bessel function of the second kind of order one
C multiplied by the exponential function, for arguments
C XLEAST .LE. ARG .LE. XMAX.
C
C--------------------------------------------------------------------
INTEGER JINT
CS REAL
DOUBLE PRECISION X, RESULT
C--------------------------------------------------------------------
JINT = 2
CALL CALCK1(X,RESULT,JINT)
BESEK1 = RESULT
RETURN
C---------- Last line of BESEK1 ----------
END FUNCTION BESEK1
end module k1