# drd.f

          DOUBLE PRECISION FUNCTION DRD (X, Y, Z, IER)
    C***BEGIN PROLOGUE  DRD
    C***PURPOSE  Compute the incomplete or complete elliptic integral of
    C            the 2nd kind. For X and Y nonnegative, X+Y and Z positive,
    C            DRD(X,Y,Z) = Integral from zero to infinity of
    C                                -1/2     -1/2     -3/2
    C                      (3/2)(t+X)    (t+Y)    (t+Z)    dt.
    C            If X or Y is zero, the integral is complete.
    C***LIBRARY   SLATEC
    C***CATEGORY  C14
    C***TYPE      DOUBLE PRECISION (RD-S, DRD-D)
    C***KEYWORDS  COMPLETE ELLIPTIC INTEGRAL, DUPLICATION THEOREM,
    C             INCOMPLETE ELLIPTIC INTEGRAL, INTEGRAL OF THE SECOND 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.     DRD
    C          Evaluate an INCOMPLETE (or COMPLETE) ELLIPTIC INTEGRAL
    C          of the second kind
    C          Standard FORTRAN function routine
    C          Double precision version
    C          The routine calculates an approximation result to
    C          DRD(X,Y,Z) = Integral from zero to infinity of
    C                              -1/2     -1/2     -3/2
    C                    (3/2)(t+X)    (t+Y)    (t+Z)    dt,
    C          where X and Y are nonnegative, X + Y is positive, and Z is
    C          positive.  If X or Y 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
    C          DRD( X, Y, Z, 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, nonnegative variable
    C
    C                   X + Y is positive
    C
    C          Z      - Double precision, positive variable
    C
    C
    C
    C          On Return    (values assigned by the DRD routine)
    C
    C          DRD     - Double precision approximation to the integral
    C
    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
    C          X, Y, Z are unaltered.
    C
    C   3.    Error Messages
    C
    C         Value of IER assigned by the DRD routine
    C
    C                  Value assigned         Error message printed
    C                  IER = 1                MIN(X,Y) .LT. 0.0D0
    C                      = 2                MIN(X + Y, Z ) .LT. LOLIM
    C                      = 3                MAX(X,Y,Z) .GT. UPLIM
    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 2 / (machine maximum) ** (2/3).
    C
    C          UPLIM  - Upper limit of valid arguments
    C
    C                 Not greater than (0.1D0 * ERRTOL / machine
    C                 minimum) ** (2/3), where ERRTOL is described below.
    C                 In the following table it is assumed that ERRTOL will
    C                 never be chosen smaller than 1.0D-5.
    C
    C
    C                    Acceptable values for:   LOLIM      UPLIM
    C                    IBM 360/370 SERIES   :   6.0D-51     1.0D+48
    C                    CDC 6000/7000 SERIES :   5.0D-215    2.0D+191
    C                    UNIVAC 1100 SERIES   :   1.0D-205    2.0D+201
    C                    CRAY                 :   3.0D-1644   1.69D+1640
    C                    VAX 11 SERIES        :   1.0D-25     4.5D+21
    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          ERRTOL    Relative error due to truncation is less than
    C                    3 * ERRTOL ** 6 / (1-ERRTOL) ** 3/2.
    C
    C
    C
    C        The accuracy of the computed approximation to the integral
    C        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 truncation
    C        error there will be round-off error, but in practice the
    C        total error from both sources is usually less than the
    C        amount given in the table.
    C
    C
    C
    C
    C          Sample choices:  ERRTOL   Relative truncation
    C                                    error less than
    C                           1.0D-3    4.0D-18
    C                           3.0D-3    3.0D-15
    C                           1.0D-2    4.0D-12
    C                           3.0D-2    3.0D-9
    C                           1.0D-1    4.0D-6
    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   DRD Special Comments
    C
    C
    C
    C          Check: DRD(X,Y,Z) + DRD(Y,Z,X) + DRD(Z,X,Y)
    C          = 3 / SQRT(X * Y * Z), where X, Y, and Z are positive.
    C
    C
    C          On Input:
    C
    C          X, Y, and Z are the variables in the integral DRD(X,Y,Z).
    C
    C
    C          On Output:
    C
    C
    C          X, Y, 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    -------------------------------------------------------------------
    C
    C
    C   Special double precision functions via DRD and DRF
    C
    C
    C                  Legendre form of ELLIPTIC INTEGRAL of 2nd kind
    C
    C                  -----------------------------------------
    C
    C
    C                                             2         2   2
    C                  E(PHI,K) = SIN(PHI) DRF(COS (PHI),1-K SIN (PHI),1) -
    C
    C                     2      3             2         2   2
    C                  -(K/3) SIN (PHI) DRD(COS (PHI),1-K SIN (PHI),1)
    C
    C
    C                                  2        2            2
    C                  E(K) = DRF(0,1-K ,1) - (K/3) DRD(0,1-K ,1)
    C
    C                         PI/2     2   2      1/2
    C                       = INT  (1-K SIN (PHI) )  D PHI
    C                          0
    C
    C                  Bulirsch form of ELLIPTIC INTEGRAL of 2nd kind
    C
    C                  -----------------------------------------
    C
    C                                               2 2    2
    C                  EL2(X,KC,A,B) = AX DRF(1,1+KC X ,1+X ) +
    C
    C                                              3          2 2    2
    C                                 +(1/3)(B-A) X DRD(1,1+KC X ,1+X )
    C
    C
    C
    C
    C                  Legendre form of alternative ELLIPTIC INTEGRAL
    C                  of 2nd kind
    C
    C                  -----------------------------------------
    C
    C
    C
    C                            Q     2       2   2  -1/2
    C                  D(Q,K) = INT SIN P  (1-K SIN P)     DP
    C                            0
    C
    C
    C
    C                                     3          2     2   2
    C                  D(Q,K) = (1/3) (SIN Q) DRD(COS Q,1-K SIN Q,1)
    C
    C
    C
    C
    C                  Lemniscate constant  B
    C
    C                  -----------------------------------------
    C
    C
    C
    C
    C                       1    2    4 -1/2
    C                  B = INT  S (1-S )    DS
    C                       0
    C
    C
    C                  B = (1/3) DRD (0,2,1)
    C
    C
    C                  Heuman's LAMBDA function
    C
    C                  -----------------------------------------
    C
    C
    C
    C                  (PI/2) LAMBDA0(A,B) =
    C
    C                                    2                2
    C                 = SIN(B) (DRF(0,COS (A),1)-(1/3) SIN (A) *
    C
    C                            2               2         2       2
    C                  *DRD(0,COS (A),1)) DRF(COS (B),1-COS (A) SIN (B),1)
    C
    C                            2       3             2
    C                  -(1/3) COS (A) SIN (B) DRF(0,COS (A),1) *
    C
    C                           2         2       2
    C                   *DRD(COS (B),1-COS (A) SIN (B),1)
    C
    C
    C
    C                  Jacobi ZETA function
    C
    C                  -----------------------------------------
    C
    C                             2                 2       2   2
    C                  Z(B,K) = (K/3) SIN(B) DRF(COS (B),1-K SIN (B),1)
    C
    C
    C                                       2             2
    C                             *DRD(0,1-K ,1)/DRF(0,1-K ,1)
    C
    C                               2       3           2       2   2
    C                            -(K /3) SIN (B) DRD(COS (B),1-K SIN (B),1)
    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  D1MACH, XERMSG
    C***REVISION HISTORY  (YYMMDD)
    C   790801  DATE WRITTEN
    C   890531  Changed all specific intrinsics to generic.  (WRB)
    C   890531  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  Modify calls to XERMSG to put in standard form.  (RWC)
    C   920501  Reformatted the REFERENCES section.  (WRB)
    C***END PROLOGUE  DRD