*DECK RC REAL FUNCTION RC (X, Y, IER) C***BEGIN PROLOGUE RC C***PURPOSE Calculate an approximation to C RC(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 SINGLE 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. RC C Standard FORTRAN function routine C Single precision version C The routine calculates an approximation to C RC(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 RC. C C C 2. Calling Sequence C RC( X, Y, 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, positive variable C C C C On Return (values assigned by the RC routine) C C RC - Single 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 C 3. Error Messages C C Value of IER assigned by the RC routine C C Value Assigned Error Message Printed C IER = 1 X.LT.0.0E0.OR.Y.LE.0.0E0 C = 2 X+Y.LT.LOLIM C = 3 MAX(X,Y) .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 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 : 3.0E-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 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 1.0E-3 2.0E-17 C 3.0E-3 2.0E-14 C 1.0E-2 2.0E-11 C 3.0E-2 2.0E-8 C 1.0E-1 2.0E-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 RC Special Comments C C C C C Check: RC(X,X+Z) + RC(Y,Y+Z) = RC(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 RC(X,Y). C C On Output: C C X and Y are unaltered. C C C C RC(0,1/4)=RC(1/16,1/8)=PI=3.14159... C C RC(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 RC C C C C LN X X .GT. 0 C C 2 C LN(X) = (X-1) RC(((1+X)/2) , X ) C C C -------------------------------------------------------------------- C C ARCSIN X -1 .LE. X .LE. 1 C C 2 C ARCSIN X = X RC (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 ) RC(X ,1 ) C C -------------------------------------------------------------------- C C ARCTAN X -INF .LT. X .LT. +INF C C 2 C ARCTAN X = X RC(1,1+X ) C C -------------------------------------------------------------------- C C ARCCOT X 0 .LE. X .LT. INF C C 2 2 C ARCCOT X = RC(X ,X +1 ) C C -------------------------------------------------------------------- C C ARCSINH X -INF .LT. X .LT. +INF C C 2 C ARCSINH X = X RC(1+X ,1 ) C C -------------------------------------------------------------------- C C ARCCOSH X X .GE. 1 C C 2 2 C ARCCOSH X = SQRT(X -1) RC(X ,1 ) C C -------------------------------------------------------------------- C C ARCTANH X -1 .LT. X .LT. 1 C C 2 C ARCTANH X = X RC(1,1-X ) C C -------------------------------------------------------------------- C C ARCCOTH X X .GT. 1 C C 2 2 C ARCCOTH X = RC(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 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 RC CHARACTER*16 XERN3, XERN4, XERN5 INTEGER IER REAL C1, C2, ERRTOL, LAMDA, LOLIM REAL MU, S, SN, UPLIM, X, XN, Y, YN LOGICAL FIRST SAVE ERRTOL,LOLIM,UPLIM,C1,C2,FIRST DATA FIRST /.TRUE./ C C***FIRST EXECUTABLE STATEMENT RC IF (FIRST) THEN ERRTOL = (R1MACH(3)/16.0E0)**(1.0E0/6.0E0) LOLIM = 5.0E0 * R1MACH(1) UPLIM = R1MACH(2) / 5.0E0 C C1 = 1.0E0/7.0E0 C2 = 9.0E0/22.0E0 ENDIF FIRST = .FALSE. C C CALL ERROR HANDLER IF NECESSARY. C RC = 0.0E0 IF (X.LT.0.0E0.OR.Y.LE.0.0E0) THEN IER = 1 WRITE (XERN3, '(1PE15.6)') X WRITE (XERN4, '(1PE15.6)') Y CALL XERMSG ('SLATEC', 'RC', * 'X.LT.0 .OR. Y.LE.0 WHERE X = ' // XERN3 // ' AND Y = ' // * XERN4, 1, 1) RETURN ENDIF C IF (MAX(X,Y).GT.UPLIM) THEN IER = 3 WRITE (XERN3, '(1PE15.6)') X WRITE (XERN4, '(1PE15.6)') Y WRITE (XERN5, '(1PE15.6)') UPLIM CALL XERMSG ('SLATEC', 'RC', * 'MAX(X,Y).GT.UPLIM WHERE X = ' // XERN3 // ' Y = ' // * XERN4 // ' AND UPLIM = ' // XERN5, 3, 1) RETURN ENDIF C IF (X+Y.LT.LOLIM) THEN IER = 2 WRITE (XERN3, '(1PE15.6)') X WRITE (XERN4, '(1PE15.6)') Y WRITE (XERN5, '(1PE15.6)') LOLIM CALL XERMSG ('SLATEC', 'RC', * 'X+Y.LT.LOLIM WHERE X = ' // XERN3 // ' Y = ' // XERN4 // * ' AND LOLIM = ' // XERN5, 2, 1) RETURN ENDIF C IER = 0 XN = X YN = Y C 30 MU = (XN+YN+YN)/3.0E0 SN = (YN+MU)/MU - 2.0E0 IF (ABS(SN).LT.ERRTOL) GO TO 40 LAMDA = 2.0E0*SQRT(XN)*SQRT(YN) + YN XN = (XN+LAMDA)*0.250E0 YN = (YN+LAMDA)*0.250E0 GO TO 30 C 40 S = SN*SN*(0.30E0+SN*(C1+SN*(0.3750E0+SN*C2))) RC = (1.0E0+S)/SQRT(MU) RETURN END