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SUBROUTINE DMFSS

PURPOSE
   GIVEN A SYMMETRIC POSITIVE SEMI DEFINITE MATRIX ,DMFSS WILL
   (1) DETERMINE THE RANK AND LINEARLY INDEPENDENT ROWS AND
       COLUMNS
   (2) FACTOR A SYMMETRIC SUBMATRIX OF MAXIMAL RANK
   (3) EXPRESS NONBASIC ROWS IN TERMS OF BASIC ONES,
       EXPRESS NONBASIC COLUMNS IN TERMS OF BASIC ONES
       EXPRESS BASIC VARIABLES IN TERMS OF FREE ONES
   SUBROUTINE DMFSS MAY BE USED AS A PREPARATORY STEP FOR THE
   CALCULATION OF THE LEAST SQUARES SOLUTION OF MINIMAL
   LENGTH OF A SYSTEM OF LINEAR EQUATIONS WITH SYMMETRIC
   POSITIVE SEMI-DEFINITE COEFFICIENT MATRIX

USAGE
   CALL DMFSS(A,N,EPS,IRANK,TRAC)

DESCRIPTION OF PARAMETERS
   A	 - UPPER TRIANGULAR PART OF GIVEN SYMMETRIC SEMI-
	   DEFINITE MATRIX STORED COLUMNWISE IN COMPRESSED FORM
	   ON RETURN A CONTAINS THE MATRIX T AND, IF IRANK IS
	   LESS THAN N, THE MATRICES U AND TU
	   A MUST BE OF DOUBLE PRECISION
   N	 - DIMENSION OF GIVEN MATRIX A
   EPS	 - TESTVALUE FOR ZERO AFFECTED BY ROUND-OFF NOISE
   IRANK - RESULTANT VARIABLE, CONTAINING THE RANK OF GIVEN
	   MATRIX A IF A IS SEMI-DEFINITE
	   IRANK = 0 MEANS A HAS NO POSITIVE DIAGONAL ELEMENT
		     AND/OR EPS IS NOT ABSOLUTELY LESS THAN ONE
	   IRANK =-1 MEANS DIMENSION N IS NOT POSITIVE
	   IRANK =-2 MEANS COMPLETE FAILURE, POSSIBLY DUE TO
		     INADEQUATE RELATIVE TOLERANCE EPS
   TRAC  - VECTOR OF DIMENSION N CONTAINING THE
	   SOURCE INDEX OF THE I-TH PIVOT ROW IN ITS I-TH
	   LOCATION, THIS MEANS THAT TRAC CONTAINS THE
	   PRODUCT REPRESENTATION OF THE PERMUTATION WHICH
	   IS APPLIED TO ROWS AND COLUMNS OF A IN TERMS OF
	   TRANSPOSITIONS
	   TRAC MUST BE OF DOUBLE PRECISION

REMARKS
   EPS MUST BE ABSOLUTELY LESS THAN ONE. A SENSIBLE VALUE IS
   SOMEWHERE IN BETWEEN 10**(-4) AND 10**(-6)
   THE ABSOLUTE VALUE OF INPUT PARAMETER EPS IS USED AS
   RELATIVE TOLERANCE.
   IN ORDER TO PRESERVE SYMMETRY ONLY PIVOTING ALONG THE
   DIAGONAL IS BUILT IN.
   ALL PIVOTELEMENTS MUST BE GREATER THAN THE ABSOLUTE VALUE
   OF EPS TIMES ORIGINAL DIAGONAL ELEMENT
   OTHERWISE THEY ARE TREATED AS IF THEY WERE ZERO
   MATRIX A REMAINS UNCHANGED IF THE RESULTANT VALUE IRANK
   EQUALS ZERO

SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
   NONE

METHOD
   THE SQUARE ROOT METHOD WITH DIAGONAL PIVOTING IS USED FOR
   CALCULATION OF THE RIGHT HAND TRIANGULAR FACTOR.
   IN CASE OF AN ONLY SEMI-DEFINITE MATRIX THE SUBROUTINE
   RETURNS THE IRANK X IRANK UPPER TRIANGULAR FACTOR T OF A
   SUBMATRIX OF MAXIMAL RANK, THE IRANK X (N-IRANK) MATRIX U
   AND THE (N-IRANK) X (N-IRANK) UPPER TRIANGULAR TU SUCH
   THAT TRANSPOSE(TU)*TU=I+TRANSPOSE(U)*U