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decuslib20-02
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decus/20-0026/apfs.doc
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SUBROUTINE APFS
PURPOSE
PERFORM SYMMETRIC FACTORIZATION OF THE MATRIX OF THE NORMAL
EQUATIONS FOLLOWED BY CALCULATION OF THE LEAST SQUARES FIT
OPTIONALLY
USAGE
CALL APFS(WORK,IP,IRES,IOP,EPS,ETA,IER)
DESCRIPTION OF PARAMETERS
WORK - GIVEN SYMMETRIC COEFFICIENT MATRIX, STORED
COMPRESSED, I.E UPPER TRIANGULAR PART COLUMNWISE.
THE GIVEN RIGHT HAND SIDE OCCUPIES THE NEXT IP
LOCATIONS IN WORK. THE VERY LAST COMPONENT OF WORK
CONTAINS THE SQUARE SUM OF FUNCTION VALUES E0
THIS SCHEME OF STORAGE ALLOCATION IS PRODUCED E.G.
BY SUBROUTINE APLL.
THE GIVEN MATRIX IS FACTORED IN THE FORM
TRANSPOSE(T)*T AND THE GIVEN RIGHT HAND SIDE IS
DIVIDED BY TRANSPOSE(T).
THE UPPER TRIANGULAR FACTOR T IS RETURNED IN WORK IF
IOP EQUALS ZERO.
IN CASE OF NONZERO IOP THE CALCULATED SOLUTIONS ARE
STORED IN THE COLUMNS OF TRIANGULAR ARRAY WORK OF
CORRESPONDING DIMENSION AND E0 IS REPLACED BY THE
SQUARE SUM OF THE ERRORS FOR FIT OF DIMENSION IRES.
THE TOTAL DIMENSION OF WORK IS (IP+1)*(IP+2)/2
IP - NUMBER OF FUNDAMENTAL FUNCTIONS USED FOR LEAST
SQUARES FIT
IRES - DIMENSION OF CALCULATED LEAST SQUARES FIT.
LET N1, N2, DENOTE THE FOLLOWING NUMBERS
N1 = MAXIMAL DIMENSION FOR WHICH NO LOSS OF
SIGNIFICANCE WAS INDICATED DURING FACTORIZATION
N2 = SMALLEST DIMENSION FOR WHICH THE SQUARE SUM OF
THE ERRORS DOES NOT EXCEED TEST=ABS(ETA*FSQ)
THEN IRES=MINO(IP,N1) IF IOP IS NONNEGATIVE
AND IRES=MINO(IP,N1,N2) IF IOP IS NEGATIVE
IOP - INPUT PARAMETER FOR SELECTION OF OPERATION
IOP = 0 MEANS TRIANGULAR FACTORIZATION, DIVISION OF
THE RIGHT HAND SIDE BY TRANSPOSE(T) AND
CALCULATION OF THE SQUARE SUM OF ERRORS IS
PERFORMED ONLY
IOP = +1 OR -1 MEANS THE SOLUTION OF DIMENSION IRES
IS CALCULATED ADDITIONALLY
IOP = +2 OR -2 MEANS ALL SOLUTIONS FOR DIMENSION ONE
UP TO IRES ARE CALCULATED ADDITIONALLY
EPS - RELATIVE TOLERANCE FOR TEST ON LOSS OF SIGNIFICANCE.
A SENSIBLE VALUE IS BETWEEN 1.E-3 AND 1.E-6
ETA - RELATIVE TOLERANCE FOR TOLERATED SQUARE SUM OF
ERRORS. A REALISTIC VALUE IS BETWEEN 1.E0 AND 1.E-6
IER - RESULTANT ERROR PARAMETER
IER =-1 MEANS NONPOSITIVE IP
IER = 0 MEANS NO LOSS OF SIGNIFICANCE DETECTED
AND SPECIFIED TOLERANCE OF ERRORS REACHED
IER = 1 MEANS LOSS OF SIGNIFICANCE DETECTED OR
SPECIFIED TOLERANCE OF ERRORS NOT REACHED
REMARKS
THE ABSOLUTE TOLERANCE USED INTERNALLY FOR TEST ON LOSS OF
SIGNIFICANCE IS TOL=ABS(EPS*WORK(1)).
THE ABSOLUTE TOLERANCE USED INTERNALLY FOR THE SQUARE SUM OF
ERRORS IS ABS(ETA*FSQ).
IOP GREATER THAN 2 HAS THE SAME EFFECT AS IOP = 2.
IOP LESS THAN -2 HAS THE SAME EFFECT AS IOP =-2.
IRES = 0 MEANS THE ABSOLUTE VALUE OF EPS IS NOT LESS THAN
ONE AND/OR WORK(1) IS NOT POSITIVE AND/OR IP IS NOT POSITIVE
SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
NONE
METHOD
CALCULATION OF THE LEAST SQUARES FITS IS DONE USING
CHOLESKYS SQUARE ROOT METHOD FOR SYMMETRIC FACTORIZATION.
THE INCORPORATED TEST ON LOSS OF SIGNIFICANCE MEANS EACH
RADICAND MUST BE GREATER THAN THE INTERNAL ABSOLUTE
TOLERANCE TOL=ABS(EPS*WORK(1)).
IN CASE OF LOSS OF SIGNIFICANCE IN THE ABOVE SENSE ONLY A
SUBSYSTEM OF THE NORMAL EQUATIONS IS SOLVED.
IN CASE OF NEGATIVE IOP THE TRIANGULAR FACTORIZATION IS
TERMINATED PREMATURELY EITHER IF THE SQUARE SUM OF THE
ERRORS DOES NOT EXCEED ETA*FSQ OR IF THERE IS INDICATION
FOR LOSS OF SIGNIFICANCE