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SUBROUTINE APMM

PURPOSE
   APPROXIMATE A FUNCTION TABULATED IN N POINTS BY ANY LINEAR
   COMBINATION OF M GIVEN CONTINUOUS FUNCTIONS IN THE SENSE
   OF CHEBYSHEV.

USAGE
   CALL APMM(FCT,N,M,TOP,IHE,PIV,T,ITER,IER)
   PARAMETER FCT REQUIRES AN EXTERNAL STATEMENT IN THE
   CALLING PROGRAM.

DESCRIPTION OF PARAMETERS
   FCT	  - NAME OF SUBROUTINE TO BE SUPPLIED BY THE USER.
	    IT COMPUTES VALUES OF M GIVEN FUNCTIONS FOR
	    ARGUMENT VALUE X.
	    USAGE
	       CALL FCT(Y,X,K)
	    DESCRIPTION OF PARAMETERS
	       Y   - RESULT VECTOR OF DIMENSION M CONTAINING
		     THE VALUES OF GIVEN CONTINUOUS FUNCTIONS
		     FOR GIVEN ARGUMENT X
	       X   - ARGUMENT VALUE
	       K   - AN INTEGER VALUE WHICH IS EQUAL TO M-1
	    REMARKS
	       IF APPROXIMATION BY NORMAL CHEBYSHEV, SHIFTED
	       CHEBYSHEV, LEGENDRE, LAGUERRE, HERMITE POLYNO-
	       MIALS IS DESIRED SUBROUTINES CNP, CSP, LEP,
	       LAP, HEP, RESPECTIVELY FROM SSP COULD BE USED.
   N	  - NUMBER OF DATA POINTS DEFINING THE FUNCTION WHICH
	    IS TO BE APPROXIMATED
   M	  - NUMBER OF GIVEN CONTINUOUS FUNCTIONS FROM WHICH
	    THE APPROXIMATING FUNCTION IS CONSTRUCTED.
   TOP	  - VECTOR OF DIMENSION 3*N.
	    ON ENTRY IT MUST CONTAIN FROM TOP(1) UP TO TOP(N)
	    THE GIVEN N FUNCTION VALUES AND FROM TOP(N+1) UP
	    TO TOP(2*N) THE CORRESPONDING NODES
	    ON RETURN TOP CONTAINS FROM TOP(1) UP TO TOP(N)
	    THE ERRORS AT THOSE N NODES.
	    OTHER VALUES OF TOP ARE SCRATCH.
   IHE	  - INTEGER VECTOR OF DIMENSION 3*M+4*N+6
   PIV	  - VECTOR OF DIMENSION 3*M+6.
	    ON RETURN PIV CONTAINS AT PIV(1) UP TO PIV(M) THE
	    RESULTING COEFFICIENTS OF LINEAR APPROXIMATION.
   T	  - AUXILIARY VECTOR OF DIMENSION (M+2)*(M+2)
   ITER   - RESULTANT INTEGER WHICH SPECIFIES THE NUMBER OF
	    ITERATIONS NEEDED
   IER	  - RESULTANT ERROR PARAMETER CODED IN THE FOLLOWING
	    FORM
	     IER=0  - NO ERROR
	     IER=1  - THE NUMBER OF ITERATIONS HAS REACHED
		      THE INTERNAL MAXIMUM N+M
	     IER=-1 - NO RESULT BECAUSE OF WRONG INPUT PARA-
		      METER M OR N OR SINCE AT SOME ITERATION
		      NO SUITABLE PIVOT COULD BE FOUND

REMARKS
   NO ACTION BESIDES ERROR MESSAGE IN CASE M LESS THAN 1 OR
   N LESS THAN 2.

SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
   THE EXTERNAL SUBROUTINE FCT MUST BE FURNISHED BY THE USER.

METHOD
   THE PROBLEM OF APPROXIMATION A TABULATED FUNCTION BY ANY
   LINEAR COMBINATION OF GIVEN FUNCTIONS IN THE SENSE OF
   CHEBYSHEV (I.E. TO MINIMIZE THE MAXIMUM ERROR) IS TRANS-
   FORMED INTO A LINEAR PROGRAMMING PROBLEM. APMM USES A
   REVISED SIMPLEX METHOD TO SOLVE A CORRESPONDING DUAL
   PROBLEM. FOR REFERENCE, SEE
   I.BARRODALE/A.YOUNG, ALGORITHMS FOR BEST L-SUB-ONE AND
   L-SUB-INFINITY, LINEAR APPROXIMATIONS ON A DISCRETE SET,
   NUMERISCHE MATHEMATIK, VOL.8, ISS.3 (1966), PP.295-306.