SLATEC Routines --- DPSIFN ---
*DECK DPSIFN
SUBROUTINE DPSIFN (X, N, KODE, M, ANS, NZ, IERR)
C***BEGIN PROLOGUE DPSIFN
C***PURPOSE Compute derivatives of the Psi function.
C***LIBRARY SLATEC
C***CATEGORY C7C
C***TYPE DOUBLE PRECISION (PSIFN-S, DPSIFN-D)
C***KEYWORDS DERIVATIVES OF THE GAMMA FUNCTION, POLYGAMMA FUNCTION,
C PSI FUNCTION
C***AUTHOR Amos, D. E., (SNLA)
C***DESCRIPTION
C
C The following definitions are used in DPSIFN:
C
C Definition 1
C PSI(X) = d/dx (ln(GAMMA(X)), the first derivative of
C the log GAMMA function.
C Definition 2
C K K
C PSI(K,X) = d /dx (PSI(X)), the K-th derivative of PSI(X).
C ___________________________________________________________________
C DPSIFN computes a sequence of SCALED derivatives of
C the PSI function; i.e. for fixed X and M it computes
C the M-member sequence
C
C ((-1)**(K+1)/GAMMA(K+1))*PSI(K,X)
C for K = N,...,N+M-1
C
C where PSI(K,X) is as defined above. For KODE=1, DPSIFN returns
C the scaled derivatives as described. KODE=2 is operative only
C when K=0 and in that case DPSIFN returns -PSI(X) + LN(X). That
C is, the logarithmic behavior for large X is removed when KODE=2
C and K=0. When sums or differences of PSI functions are computed
C the logarithmic terms can be combined analytically and computed
C separately to help retain significant digits.
C
C Note that CALL DPSIFN(X,0,1,1,ANS) results in
C ANS = -PSI(X)
C
C Input X is DOUBLE PRECISION
C X - Argument, X .gt. 0.0D0
C N - First member of the sequence, 0 .le. N .le. 100
C N=0 gives ANS(1) = -PSI(X) for KODE=1
C -PSI(X)+LN(X) for KODE=2
C KODE - Selection parameter
C KODE=1 returns scaled derivatives of the PSI
C function.
C KODE=2 returns scaled derivatives of the PSI
C function EXCEPT when N=0. In this case,
C ANS(1) = -PSI(X) + LN(X) is returned.
C M - Number of members of the sequence, M.ge.1
C
C Output ANS is DOUBLE PRECISION
C ANS - A vector of length at least M whose first M
C components contain the sequence of derivatives
C scaled according to KODE.
C NZ - Underflow flag
C NZ.eq.0, A normal return
C NZ.ne.0, Underflow, last NZ components of ANS are
C set to zero, ANS(M-K+1)=0.0, K=1,...,NZ
C IERR - Error flag
C IERR=0, A normal return, computation completed
C IERR=1, Input error, no computation
C IERR=2, Overflow, X too small or N+M-1 too
C large or both
C IERR=3, Error, N too large. Dimensioned
C array TRMR(NMAX) is not large enough for N
C
C The nominal computational accuracy is the maximum of unit
C roundoff (=D1MACH(4)) and 1.0D-18 since critical constants
C are given to only 18 digits.
C
C PSIFN is the single precision version of DPSIFN.
C
C *Long Description:
C
C The basic method of evaluation is the asymptotic expansion
C for large X.ge.XMIN followed by backward recursion on a two
C term recursion relation
C
C W(X+1) + X**(-N-1) = W(X).
C
C This is supplemented by a series
C
C SUM( (X+K)**(-N-1) , K=0,1,2,... )
C
C which converges rapidly for large N. Both XMIN and the
C number of terms of the series are calculated from the unit
C roundoff of the machine environment.
C
C***REFERENCES Handbook of Mathematical Functions, National Bureau
C of Standards Applied Mathematics Series 55, edited
C by M. Abramowitz and I. A. Stegun, equations 6.3.5,
C 6.3.18, 6.4.6, 6.4.9 and 6.4.10, pp.258-260, 1964.
C D. E. Amos, A portable Fortran subroutine for
C derivatives of the Psi function, Algorithm 610, ACM
C Transactions on Mathematical Software 9, 4 (1983),
C pp. 494-502.
C***ROUTINES CALLED D1MACH, I1MACH
C***REVISION HISTORY (YYMMDD)
C 820601 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890911 Removed unnecessary intrinsics. (WRB)
C 891006 Cosmetic changes to prologue. (WRB)
C 891006 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE DPSIFN