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SUBROUTINE AHI

PURPOSE
   TO INTERPOLATE FUNCTION VALUE Y FOR A GIVEN ARGUMENT VALUE
   X USING A GIVEN TABLE (ARG,VAL) OF ARGUMENT, FUNCTION, AND
   DERIVATIVE VALUES.

USAGE
   CALL AHI (X,ARG,VAL,Y,NDIM,EPS,IER)

DESCRIPTION OF PARAMETERS
   X	  - THE ARGUMENT VALUE SPECIFIED BY INPUT.
   ARG	  - THE INPUT VECTOR (DIMENSION NDIM) OF ARGUMENT
	    VALUES OF THE TABLE (NOT DESTROYED).
   VAL	  - THE INPUT VECTOR (DIMENSION 2*NDIM) OF FUNCTION
	    AND DERIVATIVE VALUES OF THE TABLE (DESTROYED).
	    FUNCTION AND DERIVATIVE VALUES MUST BE STORED IN
	    PAIRS, THAT MEANS BEGINNING WITH FUNCTION VALUE AT
	    POINT ARG(1) EVERY FUNCTION VALUE MUST BE FOLLOWED
	    BY THE VALUE OF DERIVATIVE AT THE SAME POINT.
   Y	  - THE RESULTING INTERPOLATED FUNCTION VALUE.
   NDIM   - AN INPUT VALUE WHICH SPECIFIES THE NUMBER OF
	    POINTS IN TABLE (ARG,VAL).
   EPS	  - AN INPUT CONSTANT WHICH IS USED AS UPPER BOUND
	    FOR THE ABSOLUTE ERROR.
   IER	  - A RESULTING ERROR PARAMETER.

REMARKS
   (1) TABLE (ARG,VAL) SHOULD REPRESENT A SINGLE-VALUED
       FUNCTION AND SHOULD BE STORED IN SUCH A WAY, THAT THE
       DISTANCES ABS(ARG(I)-X) INCREASE WITH INCREASING
       SUBSCRIPT I. TO GENERATE THIS ORDER IN TABLE (ARG,VAL),
       SUBROUTINES ATSG, ATSM OR ATSE COULD BE USED IN A
       PREVIOUS STAGE.
   (2) NO ACTION BESIDES ERROR MESSAGE IN CASE NDIM LESS
       THAN 1.
   (3) INTERPOLATION IS TERMINATED EITHER IF THE DIFFERENCE
       BETWEEN TWO SUCCESSIVE INTERPOLATED VALUES IS
       ABSOLUTELY LESS THAN TOLERANCE EPS, OR IF THE ABSOLUTE
       VALUE OF THIS DIFFERENCE STOPS DIMINISHING, OR AFTER
       (2*NDIM-2) STEPS. FURTHER IT IS TERMINATED IF THE
       PROCEDURE DISCOVERS TWO ARGUMENT VALUES IN VECTOR ARG
       WHICH ARE IDENTICAL. DEPENDENT ON THESE FOUR CASES,
       ERROR PARAMETER IER IS CODED IN THE FOLLOWING FORM
	IER=0 - IT WAS POSSIBLE TO REACH THE REQUIRED
		ACCURACY (NO ERROR).
	IER=1 - IT WAS IMPOSSIBLE TO REACH THE REQUIRED
		ACCURACY BECAUSE OF ROUNDING ERRORS.
	IER=2 - IT WAS IMPOSSIBLE TO CHECK ACCURACY BECAUSE
		NDIM IS LESS THAN 2, OR THE REQUIRED ACCURACY
		COULD NOT BE REACHED BY MEANS OF THE GIVEN
		TABLE. NDIM SHOULD BE INCREASED.
	IER=3 - THE PROCEDURE DISCOVERED TWO ARGUMENT VALUES
		IN VECTOR ARG WHICH ARE IDENTICAL.

SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
   NONE

METHOD
   INTERPOLATION IS DONE BY MEANS OF AITKENS SCHEME OF
   HERMITE INTERPOLATION. ON RETURN Y CONTAINS AN INTERPOLATED
   FUNCTION VALUE AT POINT X, WHICH IS IN THE SENSE OF REMARK
   (3) OPTIMAL WITH RESPECT TO GIVEN TABLE. FOR REFERENCE, SEE
   F.B.HILDEBRAND, INTRODUCTION TO NUMERICAL ANALYSIS,
   MCGRAW-HILL, NEW YORK/TORONTO/LONDON, 1956, PP.314-317, AND
   GERSHINSKY/LEVINE, AITKEN-HERMITE INTERPOLATION,
   JACM, VOL.11, ISS.3 (1964), PP.352-356.