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decus/20-0026/dapch.doc
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SUBROUTINE DAPCH
PURPOSE
SET UP NORMAL EQUATIONS OF LEAST SQUARES FIT IN TERMS OF
CHEBYSHEV POLYNOMIALS FOR A GIVEN DISCRETE FUNCTION
USAGE
CALL DAPCH(DATI,N,IP,XD,X0,WORK,IER)
DESCRIPTION OF PARAMETERS
DATI - VECTOR OF DIMENSION 3*N (OR DIMENSION 2*N+1)
CONTAINING THE GIVEN ARGUMENTS, FOLLOWED BY THE
FUNCTION VALUES AND N (RESPECTIVELY 1) WEIGHT
VALUES. THE CONTENT OF VECTOR DATI REMAINS
UNCHANGED.
DATI MUST BE OF DOUBLE PRECISION
N - NUMBER OF GIVEN POINTS
IP - DIMENSION OF LEAST SQUARES FIT, I.E. NUMBER OF
CHEBYSHEV POLYNOMIALS USED AS FUNDAMENTAL FUNCTIONS
IP SHOULD NOT EXCEED N
XD - RESULTANT MULTIPLICATIVE CONSTANT FOR LINEAR
TRANSFORMATION OF ARGUMENT RANGE
XD MUST BE DOUBLE PRECISION
X0 - RESULTANT ADDITIVE CONSTANT FOR LINEAR
TRANSFORMATION OF ARGUMENT RANGE
X0 MUST BE DOUBLE PRECISION
WORK - WORKING STORAGE OF DIMENSION (IP+1)*(IP+2)/2
ON RETURN WORK CONTAINS THE SYMMETRIC COEFFICIENT
MATRIX OF THE NORMAL EQUATIONS IN COMPRESSED FORM
FOLLOWED IMMEDIATELY BY RIGHT HAND SIDE
AND SQUARE SUM OF FUNCTION VALUES
WORK MUST BE OF DOUBLE PRECISION
IER - RESULTING ERROR PARAMETER
IER =-1 MEANS FORMAL ERRORS IN DIMENSION
IER = 0 MEANS NO ERRORS
IER = 1 MEANS COINCIDING ARGUMENTS
REMARKS
NO WEIGHTS ARE USED IF THE VALUE OF DATI(2*N+1) IS
NOT POSITIVE.
EXECUTION OF SUBROUTINE DAPCH IS A PREPARATORY STEP FOR
CALCULATION OF LEAST SQUARES FITS IN CHEBYSHEV POLYNOMIALS
IT SHOULD BE FOLLOWED BY EXECUTION OF SUBROUTINE DAPFS
SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
NONE
METHOD
THE LEAST SQUARE FIT IS DETERMINED USING CHEBYSHEV
POLYNOMIALS AS FUNDAMENTAL FUNCTION SYSTEM.
THE METHOD IS DISCUSSED IN THE ARTICLE
A.T.BERZTISS, LEAST SQUARES FITTING TO IRREGULARLY SPACED
DATA, SIAM REVIEW, VOL.6, ISS.3, 1964, PP. 203-227.