SLATEC Routines --- RD ---


*DECK RD
      REAL FUNCTION RD (X, Y, Z, IER)
C***BEGIN PROLOGUE  RD
C***PURPOSE  Compute the incomplete or complete elliptic integral of the
C            2nd kind.  For X and Y nonnegative, X+Y and Z positive,
C             RD(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      SINGLE 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.     RD
C          Evaluate an INCOMPLETE (or COMPLETE) ELLIPTIC INTEGRAL
C          of the second kind
C          Standard FORTRAN function routine
C          Single precision version
C          The routine calculates an approximation result to
C          RD(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          RD( 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                   X + Y is positive
C
C          Z      - Real, positive variable
C
C
C
C          On Return     (values assigned by the RD routine)
C
C          RD     - Real 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 RD routine
C
C                  Value Assigned         Error Message Printed
C                  IER = 1                MIN(X,Y) .LT. 0.0E0
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 (0.1E0 * ERRTOL / machine
C                    minimum) ** (2/3), where ERRTOL is described below.
C                    In the following table it is assumed that ERRTOL
C                    will never be chosen smaller than 1.0E-5.
C
C
C                    Acceptable Values For:   LOLIM      UPLIM
C                    IBM 360/370 SERIES   :   6.0E-51     1.0E+48
C                    CDC 6000/7000 SERIES :   5.0E-215    2.0E+191
C                    UNIVAC 1100 SERIES   :   1.0E-25     2.0E+21
C                    CRAY                 :   3.0E-1644   1.69E+1640
C                    VAX 11 SERIES        :   1.0E-25     4.5E+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 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          Sample Choices:  ERRTOL   Relative Truncation
C                                    error less than
C                           1.0E-3    4.0E-18
C                           3.0E-3    3.0E-15
C                           1.0E-2    4.0E-12
C                           3.0E-2    3.0E-9
C                           1.0E-1    4.0E-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   RD Special Comments
C
C
C
C          Check: RD(X,Y,Z) + RD(Y,Z,X) + RD(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 RD(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    -------------------------------------------------------------------
C
C
C   Special Functions via RD and RF
C
C
C                  Legendre form of ELLIPTIC INTEGRAL of 2nd kind
C                  ----------------------------------------------
C
C
C                                            2         2   2
C                  E(PHI,K) = SIN(PHI) RF(COS (PHI),1-K SIN (PHI),1) -
C
C                     2      3            2         2   2
C                  -(K/3) SIN (PHI) RD(COS (PHI),1-K SIN (PHI),1)
C
C
C                                 2        2           2
C                  E(K) = RF(0,1-K ,1) - (K/3) RD(3,1-K ,1)
C
C
C                         PI/2     2   2      1/2
C                       = INT  (1-K SIN (PHI) )  D PHI
C                          0
C
C
C
C                  Bulirsch form of ELLIPTIC INTEGRAL of 2nd kind
C                  ----------------------------------------------
C
C                                              2 2    2
C                  EL2(X,KC,A,B) = AX RF(1,1+KC X ,1+X ) +
C
C                                              3         2 2    2
C                                 +(1/3)(B-A) X RD(1,1+KC X ,1+X )
C
C
C
C                  Legendre form of alternative ELLIPTIC INTEGRAL of 2nd
C                  -----------------------------------------------------
C                        kind
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)  RD(COS Q,1-K SIN Q,1)
C
C
C
C
C
C                  Lemniscate constant B
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)RD (0,2,1)
C
C
C
C
C                  Heuman's LAMBDA function
C                  ------------------------
C
C
C
C                  (PI/2) LAMBDA0(A,B) =
C
C                                       2                2
C                     = SIN(B) (RF(0,COS (A),1)-(1/3) SIN (A) *
C
C                               2              2         2       2
C                      *RD(0,COS (A),1)) RF(COS (B),1-COS (A) SIN (B),1)
C
C                               2       3            2
C                     -(1/3) COS (A) SIN (B) RF(0,COS (A),1) *
C
C                             2         2       2
C                      *RD(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) RF(COS (B),1-K SIN (B),1)
C
C
C                                      2            2
C                             *RD(0,1-K ,1)/RF(0,1-K ,1)
C
C                               2       3          2       2   2
C                            -(K /3) SIN (B) RD(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  R1MACH, 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  RD