SLATEC Routines --- RF ---
*DECK RF
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