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SUBROUTINE DCAR

POSE
TO COMPUTE, AT A GIVEN POINT X, AN APPROXIMATION Z TO THE
DERIVATIVE OF AN ANALYTICALLY GIVEN FUNCTION FCT THAT IS 11-
TIMES DIFFERENTIABLE IN A DOMAIN CONTAINING A CLOSED, 2-SIDED
SYMMETRIC INTERVAL OF RADIUS ABSOLUTE H ABOUT X, USING FUNCTION
VALUES ONLY ON THAT CLOSED INTERVAL.

GE
   CALL DCAR (X,H,IH,FCT,Z)
PARAMETER FCT REQUIRES AN EXTERNAL STATEMENT

CRIPTION OF PARAMETERS
X   - THE POINT AT WHICH THE DERIVATIVE IS TO BE COMPUTED
H   - THE NUMBER WHOSE ABSOLUTE VALUE DEFINES THE CLOSED,
      SYMMETRIC 2-SIDED INTERVAL ABOUT X (SEE PURPOSE)
IH  - INPUT PARAMETER (SEE REMARKS AND METHOD)
      IH NON-ZERO - THE SUBROUTINE GENERATES THE INTERNAL
		    VALUE HH
      IH    =	0 - THE INTERNAL VALUE HH IS SET TO ABSOLUTE H
FCT - THE NAME OF THE EXTERNAL FUNCTION SUBPROGRAM THAT WILL
      GENERATE THE NECESSARY FUNCTION VALUES
Z   - RESULTING DERIVATIVE VALUE

ARKS
(1)  IF H = 0, THEN THERE IS NO COMPUTATION.
(2)  THE INTERNAL VALUE HH, WHICH IS DETERMINED ACCORDING TO
     IH, IS THE MAXIMUM STEP-SIZE USED IN THE COMPUTATION OF
     THE CENTRAL DIVIDED DIFFERENCES (SEE METHOD.)  IF IH IS
     NON-ZERO, THEN THE SUBROUTINE GENERATES HH ACCORDING TO
     CRITERIA THAT BALANCE ROUND-OFF AND TRUNCATION ERROR.  HH
     IS ALWAYS LESS THAN OR EQUAL TO ABSOLUTE H IN ABSOLUTE
     VALUE, SO THAT ALL COMPUTATION OCCURS WITHIN A RADIUS
     ABSOLUTE H OF X.

ROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
THE EXTERNAL FUNCTION SUBPROGRAM FCT(T) MUST BE FURNISHED BY
THE USER.

HOD
THE COMPUTATION OF Z IS BASED ON RICHARDSON'S AND ROMBERG'S
EXTRAPOLATION METHOD AS APPLIED TO THE SEQUENCE OF CENTRAL
DIVIDED DIFFERENCES ASSOCIATED WITH THE POINT PAIRS
(X-(K*HH)/5,X+(K*HH)/5) K=1,...,5.  (SEE FILLIPI, S. AND
ENGELS, H., ALTES UND NEUES ZUR NUMERISCHEN DIFFERENTIATION,
ELECTRONISCHE DATENVERARBEITUNG, ISS. 2 (1966), PP. 57-65.)