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SUBROUTINE QATR

PURPOSE
   TO COMPUTE AN APPROXIMATION FOR INTEGRAL(FCT(X), SUMMED
   OVER X FROM XL TO XU).

USAGE
   CALL QATR (XL,XU,EPS,NDIM,FCT,Y,IER,AUX)
   PARAMETER FCT REQUIRES AN EXTERNAL STATEMENT.

DESCRIPTION OF PARAMETERS
   XL	  - THE LOWER BOUND OF THE INTERVAL.
   XU	  - THE UPPER BOUND OF THE INTERVAL.
   EPS	  - THE UPPER BOUND OF THE ABSOLUTE ERROR.
   NDIM   - THE DIMENSION OF THE AUXILIARY STORAGE ARRAY AUX.
	    NDIM-1 IS THE MAXIMAL NUMBER OF BISECTIONS OF
	    THE INTERVAL (XL,XU).
   FCT	  - THE NAME OF THE EXTERNAL FUNCTION SUBPROGRAM USED.
   Y	  - THE RESULTING APPROXIMATION FOR THE INTEGRAL VALUE.
   IER	  - A RESULTING ERROR PARAMETER.
   AUX	  - AN AUXILIARY STORAGE ARRAY WITH DIMENSION NDIM.

REMARKS
   ERROR PARAMETER IER IS CODED IN THE FOLLOWING FORM
   IER=0  - IT WAS POSSIBLE TO REACH THE REQUIRED ACCURACY.
	    NO ERROR.
   IER=1  - IT IS IMPOSSIBLE TO REACH THE REQUIRED ACCURACY
	    BECAUSE OF ROUNDING ERRORS.
   IER=2  - IT WAS IMPOSSIBLE TO CHECK ACCURACY BECAUSE NDIM
	    IS LESS THAN 5, OR THE REQUIRED ACCURACY COULD NOT
	    BE REACHED WITHIN NDIM-1 STEPS. NDIM SHOULD BE
	    INCREASED.

SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
   THE EXTERNAL FUNCTION SUBPROGRAM FCT(X) MUST BE CODED BY
   THE USER. ITS ARGUMENT X SHOULD NOT BE DESTROYED.

METHOD
   EVALUATION OF Y IS DONE BY MEANS OF TRAPEZOIDAL RULE IN
   CONNECTION WITH ROMBERGS PRINCIPLE. ON RETURN Y CONTAINS
   THE BEST POSSIBLE APPROXIMATION OF THE INTEGRAL VALUE AND
   VECTOR AUX THE UPWARD DIAGONAL OF ROMBERG SCHEME.
   COMPONENTS AUX(I) (I=1,2,...,IEND, WITH IEND LESS THAN OR
   EQUAL TO NDIM) BECOME APPROXIMATIONS TO INTEGRAL VALUE WITH
   DECREASING ACCURACY BY MULTIPLICATION WITH (XU-XL).
   FOR REFERENCE, SEE
   (1) FILIPPI, DAS VERFAHREN VON ROMBERG-STIEFEL-BAUER ALS
       SPEZIALFALL DES ALLGEMEINEN PRINZIPS VON RICHARDSON,
       MATHEMATIK-TECHNIK-WIRTSCHAFT, VOL.11, ISS.2 (1964),
       PP.49-54.
   (2) BAUER, ALGORITHM 60, CACM, VOL.4, ISS.6 (1961), PP.255.