poch1.f
FUNCTION POCH1 (A, X)
C***BEGIN PROLOGUE POCH1
C***PURPOSE Calculate a generalization of Pochhammer's symbol starting
C from first order.
C***LIBRARY SLATEC (FNLIB)
C***CATEGORY C1, C7A
C***TYPE SINGLE PRECISION (POCH1-S, DPOCH1-D)
C***KEYWORDS FIRST ORDER, FNLIB, POCHHAMMER, SPECIAL FUNCTIONS
C***AUTHOR Fullerton, W., (LANL)
C***DESCRIPTION
C
C Evaluate a generalization of Pochhammer's symbol for special
C situations that require especially accurate values when X is small in
C POCH1(A,X) = (POCH(A,X)-1)/X
C = (GAMMA(A+X)/GAMMA(A) - 1.0)/X .
C This specification is particularly suited for stably computing
C expressions such as
C (GAMMA(A+X)/GAMMA(A) - GAMMA(B+X)/GAMMA(B))/X
C = POCH1(A,X) - POCH1(B,X)
C Note that POCH1(A,0.0) = PSI(A)
C
C When ABS(X) is so small that substantial cancellation will occur if
C the straightforward formula is used, we use an expansion due
C to Fields and discussed by Y. L. Luke, The Special Functions and Their
C Approximations, Vol. 1, Academic Press, 1969, page 34.
C
C The ratio POCH(A,X) = GAMMA(A+X)/GAMMA(A) is written by Luke as
C (A+(X-1)/2)**X * polynomial in (A+(X-1)/2)**(-2) .
C In order to maintain significance in POCH1, we write for positive A
C (A+(X-1)/2)**X = EXP(X*LOG(A+(X-1)/2)) = EXP(Q)
C = 1.0 + Q*EXPREL(Q) .
C Likewise the polynomial is written
C POLY = 1.0 + X*POLY1(A,X) .
C Thus,
C POCH1(A,X) = (POCH(A,X) - 1) / X
C = EXPREL(Q)*(Q/X + Q*POLY1(A,X)) + POLY1(A,X)
C
C***REFERENCES (NONE)
C***ROUTINES CALLED COT, EXPREL, POCH, PSI, R1MACH, XERMSG
C***REVISION HISTORY (YYMMDD)
C 770801 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 900727 Added EXTERNAL statement. (WRB)
C***END PROLOGUE POCH1