 
      DOUBLE PRECISION FUNCTION DPCHDF(K,X,S,IERR)
C***BEGIN PROLOGUE  DPCHDF
C***REFER TO  DPCHCE,DPCHSP
C***ROUTINES CALLED  XERROR
C***REVISION DATE  870707   (YYMMDD)
C***DESCRIPTION
C
C          DPCHDF:   DPCHIP Finite Difference Formula
C
C     Uses a divided difference formulation to compute a K-point approx-
C     imation to the derivative at X(K) based on the data in X and S.
C
C     Called by  DPCHCE  and  DPCHSP  to compute 3- and 4-point boundary
C     derivative approximations.
C
C ----------------------------------------------------------------------
C
C     On input:
C        K      is the order of the desired derivative approximation.
C               K must be at least 3 (error return if not).
C        X      contains the K values of the independent variable.
C               X need not be ordered, but the values **MUST** be
C               distinct.  (Not checked here.)
C        S      contains the associated slope values:
C                  S(I) = (F(I+1)-F(I))/(X(I+1)-X(I)), I=1(1)K-1.
C               (Note that S need only be of length K-1.)
C
C     On return:
C        S      will be destroyed.
C        IERR   will be set to -1 if K.LT.2 .
C        DPCHDF  will be set to the desired derivative approximation if
C               IERR=0 or to zero if IERR=-1.
C
C ----------------------------------------------------------------------
C
C  Reference:  Carl de Boor, A Practical Guide to Splines, Springer-
C              Verlag (New York, 1978), pp. 10-16.
C
C***END PROLOGUE  DPCHDF
 
 
