# rf.f

          REAL FUNCTION RF (X, Y, Z, IER)
    C***BEGIN PROLOGUE  RF
    C***PURPOSE  Compute the incomplete or complete elliptic integral of the
    C            1st kind.  For X, Y, and Z non-negative and at most one of
    C            them zero, RF(X,Y,Z) = Integral from zero to infinity of
    C                                -1/2     -1/2     -1/2
    C                      (1/2)(t+X)    (t+Y)    (t+Z)    dt.
    C            If X, Y or Z is zero, the integral is complete.
    C***LIBRARY   SLATEC
    C***CATEGORY  C14
    C***TYPE      SINGLE PRECISION (RF-S, DRF-D)
    C***KEYWORDS  COMPLETE ELLIPTIC INTEGRAL, DUPLICATION THEOREM,
    C             INCOMPLETE ELLIPTIC INTEGRAL, INTEGRAL OF THE FIRST KIND,
    C             TAYLOR SERIES
    C***AUTHOR  Carlson, B. C.
    C             Ames Laboratory-DOE
    C             Iowa State University
    C             Ames, IA  50011
    C           Notis, E. M.
    C             Ames Laboratory-DOE
    C             Iowa State University
    C             Ames, IA  50011
    C           Pexton, R. L.
    C             Lawrence Livermore National Laboratory
    C             Livermore, CA  94550
    C***DESCRIPTION
    C
    C   1.     RF
    C          Evaluate an INCOMPLETE (or COMPLETE) ELLIPTIC INTEGRAL
    C          of the first kind
    C          Standard FORTRAN function routine
    C          Single precision version
    C          The routine calculates an approximation result to
    C          RF(X,Y,Z) = Integral from zero to infinity of
    C
    C                               -1/2     -1/2     -1/2
    C                     (1/2)(t+X)    (t+Y)    (t+Z)    dt,
    C
    C          where X, Y, and Z are nonnegative and at most one of them
    C          is zero.  If one of them is zero, the integral is COMPLETE.
    C          The duplication theorem is iterated until the variables are
    C          nearly equal, and the function is then expanded in Taylor
    C          series to fifth order.
    C
    C   2.     Calling Sequence
    C          RF( X, Y, Z, IER )
    C
    C          Parameters on Entry
    C          Values assigned by the calling routine
    C
    C          X      - Single precision, nonnegative variable
    C
    C          Y      - Single precision, nonnegative variable
    C
    C          Z      - Single precision, nonnegative variable
    C
    C
    C
    C          On Return     (values assigned by the RF routine)
    C
    C          RF     - Single precision approximation to the integral
    C
    C          IER    - Integer
    C
    C                   IER = 0 Normal and reliable termination of the
    C                           routine.  It is assumed that the requested
    C                           accuracy has been achieved.
    C
    C                   IER >  0 Abnormal termination of the routine
    C
    C          X, Y, Z are unaltered.
    C
    C
    C   3.    Error Messages
    C
    C         Value of IER assigned by the RF routine
    C
    C                  Value assigned         Error Message Printed
    C                  IER = 1                MIN(X,Y,Z) .LT. 0.0E0
    C                      = 2                MIN(X+Y,X+Z,Y+Z) .LT. LOLIM
    C                      = 3                MAX(X,Y,Z) .GT. UPLIM
    C
    C
    C
    C   4.     Control Parameters
    C
    C                  Values of LOLIM, UPLIM, and ERRTOL are set by the
    C                  routine.
    C
    C          LOLIM and UPLIM determine the valid range of X, Y and Z
    C
    C          LOLIM  - Lower limit of valid arguments
    C
    C                   Not less than 5 * (machine minimum).
    C
    C          UPLIM  - Upper limit of valid arguments
    C
    C                   Not greater than (machine maximum) / 5.
    C
    C
    C                     Acceptable Values For:   LOLIM      UPLIM
    C                     IBM 360/370 SERIES   :   3.0E-78     1.0E+75
    C                     CDC 6000/7000 SERIES :   1.0E-292    1.0E+321
    C                     UNIVAC 1100 SERIES   :   1.0E-37     1.0E+37
    C                     CRAY                 :   2.3E-2466   1.09E+2465
    C                     VAX 11 SERIES        :   1.5E-38     3.0E+37
    C
    C
    C
    C          ERRTOL determines the accuracy of the answer
    C
    C                 The value assigned by the routine will result
    C                 in solution precision within 1-2 decimals of
    C                 "machine precision".
    C
    C
    C
    C          ERRTOL - Relative error due to truncation is less than
    C                   ERRTOL ** 6 / (4 * (1-ERRTOL)  .
    C
    C
    C
    C              The accuracy of the computed approximation to the inte-
    C              gral can be controlled by choosing the value of ERRTOL.
    C              Truncation of a Taylor series after terms of fifth order
    C              introduces an error less than the amount shown in the
    C              second column of the following table for each value of
    C              ERRTOL in the first column.  In addition to the trunca-
    C              tion error there will be round-off error, but in prac-
    C              tice the total error from both sources is usually less
    C              than the amount given in the table.
    C
    C
    C
    C
    C
    C          Sample Choices:  ERRTOL   Relative Truncation
    C                                    error less than
    C                           1.0E-3    3.0E-19
    C                           3.0E-3    2.0E-16
    C                           1.0E-2    3.0E-13
    C                           3.0E-2    2.0E-10
    C                           1.0E-1    3.0E-7
    C
    C
    C                    Decreasing ERRTOL by a factor of 10 yields six more
    C                    decimal digits of accuracy at the expense of one or
    C                    two more iterations of the duplication theorem.
    C
    C *Long Description:
    C
    C   RF Special Comments
    C
    C
    C
    C          Check by addition theorem: RF(X,X+Z,X+W) + RF(Y,Y+Z,Y+W)
    C          = RF(0,Z,W), where X,Y,Z,W are positive and X * Y = Z * W.
    C
    C
    C          On Input:
    C
    C          X, Y, and Z are the variables in the integral RF(X,Y,Z).
    C
    C
    C          On Output:
    C
    C
    C          X, Y, and Z are unaltered.
    C
    C
    C
    C          ********************************************************
    C
    C          Warning: Changes in the program may improve speed at the
    C                   expense of robustness.
    C
    C
    C
    C   Special Functions via RF
    C
    C
    C                  Legendre form of ELLIPTIC INTEGRAL of 1st kind
    C                  ----------------------------------------------
    C
    C
    C                                            2         2   2
    C                  F(PHI,K) = SIN(PHI) RF(COS (PHI),1-K SIN (PHI),1)
    C
    C
    C                                 2
    C                  K(K) = RF(0,1-K ,1)
    C
    C                         PI/2     2   2      -1/2
    C                       = INT  (1-K SIN (PHI) )   D PHI
    C                          0
    C
    C
    C
    C
    C
    C                  Bulirsch form of ELLIPTIC INTEGRAL of 1st kind
    C                  ----------------------------------------------
    C
    C
    C                                         2 2    2
    C                  EL1(X,KC) = X RF(1,1+KC X ,1+X )
    C
    C
    C
    C
    C                  Lemniscate constant A
    C                  ---------------------
    C
    C
    C                       1      4 -1/2
    C                  A = INT (1-S )    DS = RF(0,1,2) = RF(0,2,1)
    C                       0
    C
    C
    C    -------------------------------------------------------------------
    C
    C***REFERENCES  B. C. Carlson and E. M. Notis, Algorithms for incomplete
    C                 elliptic integrals, ACM Transactions on Mathematical
    C                 Software 7, 3 (September 1981), pp. 398-403.
    C               B. C. Carlson, Computing elliptic integrals by
    C                 duplication, Numerische Mathematik 33, (1979),
    C                 pp. 1-16.
    C               B. C. Carlson, Elliptic integrals of the first kind,
    C                 SIAM Journal of Mathematical Analysis 8, (1977),
    C                 pp. 231-242.
    C***ROUTINES CALLED  R1MACH, XERMSG
    C***REVISION HISTORY  (YYMMDD)
    C   790801  DATE WRITTEN
    C   890531  Changed all specific intrinsics to generic.  (WRB)
    C   891009  Removed unreferenced statement labels.  (WRB)
    C   891009  REVISION DATE from Version 3.2
    C   891214  Prologue converted to Version 4.0 format.  (BAB)
    C   900315  CALLs to XERROR changed to CALLs to XERMSG.  (THJ)
    C   900326  Removed duplicate information from DESCRIPTION section.
    C           (WRB)
    C   900510  Changed calls to XERMSG to standard form, and some
    C           editorial changes.  (RWC))
    C   920501  Reformatted the REFERENCES section.  (WRB)
    C***END PROLOGUE  RF