SLATEC Routines --- DRC ---
*DECK DRC
DOUBLE PRECISION FUNCTION DRC (X, Y, IER)
C***BEGIN PROLOGUE DRC
C***PURPOSE Calculate a double precision approximation to
C DRC(X,Y) = Integral from zero to infinity of
C -1/2 -1
C (1/2)(t+X) (t+Y) dt,
C where X is nonnegative and Y is positive.
C***LIBRARY SLATEC
C***CATEGORY C14
C***TYPE DOUBLE PRECISION (RC-S, DRC-D)
C***KEYWORDS DUPLICATION THEOREM, ELEMENTARY FUNCTIONS,
C ELLIPTIC INTEGRAL, 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. DRC
C Standard FORTRAN function routine
C Double precision version
C The routine calculates an approximation result to
C DRC(X,Y) = integral from zero to infinity of
C
C -1/2 -1
C (1/2)(t+X) (t+Y) dt,
C
C where X is nonnegative and Y is positive. The duplication
C theorem is iterated until the variables are nearly equal,
C and the function is then expanded in Taylor series to fifth
C order. Logarithmic, inverse circular, and inverse hyper-
C bolic functions can be expressed in terms of DRC.
C
C 2. Calling Sequence
C DRC( X, Y, IER )
C
C Parameters On Entry
C Values assigned by the calling routine
C
C X - Double precision, nonnegative variable
C
C Y - Double precision, positive variable
C
C
C
C On Return (values assigned by the DRC routine)
C
C DRC - Double precision approximation to the integral
C
C IER - Integer to indicate normal or abnormal termination.
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 and Y are unaltered.
C
C 3. Error messages
C
C Value of IER assigned by the DRC routine
C
C Value assigned Error message printed
C IER = 1 X.LT.0.0D0.OR.Y.LE.0.0D0
C = 2 X+Y.LT.LOLIM
C = 3 MAX(X,Y) .GT. UPLIM
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 and Y
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 : 3.0D-78 1.0D+75
C CDC 6000/7000 SERIES : 1.0D-292 1.0D+321
C UNIVAC 1100 SERIES : 1.0D-307 1.0D+307
C CRAY : 2.3D-2466 1.0D+2465
C VAX 11 SERIES : 1.5D-38 3.0D+37
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 ERRTOL - relative error due to truncation is less than
C 16 * ERRTOL ** 6 / (1 - 2 * ERRTOL).
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 Sample choices: ERRTOL Relative truncation
C error less than
C 1.0D-3 2.0D-17
C 3.0D-3 2.0D-14
C 1.0D-2 2.0D-11
C 3.0D-2 2.0D-8
C 1.0D-1 2.0D-5
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 DRC special comments
C
C
C
C
C Check: DRC(X,X+Z) + DRC(Y,Y+Z) = DRC(0,Z)
C
C where X, Y, and Z are positive and X * Y = Z * Z
C
C
C On Input:
C
C X, and Y are the variables in the integral DRC(X,Y).
C
C On Output:
C
C X and Y are unaltered.
C
C
C
C DRC(0,1/4)=DRC(1/16,1/8)=PI=3.14159...
C
C DRC(9/4,2)=LN(2)
C
C
C
C ********************************************************
C
C WARNING: Changes in the program may improve speed at the
C expense of robustness.
C
C
C --------------------------------------------------------------------
C
C Special functions via DRC
C
C
C
C LN X X .GT. 0
C
C 2
C LN(X) = (X-1) DRC(((1+X)/2) , X )
C
C
C --------------------------------------------------------------------
C
C ARCSIN X -1 .LE. X .LE. 1
C
C 2
C ARCSIN X = X DRC (1-X ,1 )
C
C --------------------------------------------------------------------
C
C ARCCOS X 0 .LE. X .LE. 1
C
C
C 2 2
C ARCCOS X = SQRT(1-X ) DRC(X ,1 )
C
C --------------------------------------------------------------------
C
C ARCTAN X -INF .LT. X .LT. +INF
C
C 2
C ARCTAN X = X DRC(1,1+X )
C
C --------------------------------------------------------------------
C
C ARCCOT X 0 .LE. X .LT. INF
C
C 2 2
C ARCCOT X = DRC(X ,X +1 )
C
C --------------------------------------------------------------------
C
C ARCSINH X -INF .LT. X .LT. +INF
C
C 2
C ARCSINH X = X DRC(1+X ,1 )
C
C --------------------------------------------------------------------
C
C ARCCOSH X X .GE. 1
C
C 2 2
C ARCCOSH X = SQRT(X -1) DRC(X ,1 )
C
C --------------------------------------------------------------------
C
C ARCTANH X -1 .LT. X .LT. 1
C
C 2
C ARCTANH X = X DRC(1,1-X )
C
C --------------------------------------------------------------------
C
C ARCCOTH X X .GT. 1
C
C 2 2
C ARCCOTH X = DRC(X ,X -1 )
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 D1MACH, 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 DRC