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SUBROUTINE DMLSS

PURPOSE
   SUBROUTINE DMLSS IS THE SECOND STEP IN THE PROCEDURE FOR
   CALCULATING THE LEAST SQUARES SOLUTION OF MINIMAL LENGTH
   OF A SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS WITH SYMMETRIC
   POSITIVE SEMI-DEFINITE COEFFICIENT MATRIX.

USAGE
   CALL DMLSS(A,N,IRANK,TRAC,INC,RHS,IER)

DESCRIPTION OF PARAMETERS
   A	 - COEFFICIENT MATRIX IN FACTORED FORM AS GENERATED
	   BY SUBROUTINE MFSS FROM INITIALLY GIVEN SYMMETRIC
	   COEFFICIENT MATRIX A STORED IN N*(N+1)/2 LOCATIONS
	   A REMAINS UNCHANGED
	   A MUST BE OF DOUBLE PRECISION
   N	 - DIMENSION OF COEFFICIENT MATRIX
   IRANK - RANK OF COEFFICIENT MATRIX, CALCULATED BY MEANS OF
	   SUBROUTINE DMFSS
   TRAC  - VECTOR OF DIMENSION N CONTAINING THE
	   SUBSCRIPTS OF PIVOT ROWS AND COLUMNS, I.E. THE
	   PRODUCT REPRESENTATION IN TRANSPOSITIONS OF THE
	   PERMUTATION WHICH WAS APPLIED TO ROWS AND COLUMNS
	   OF A IN THE FACTORIZATION PROCESS
	   TRAC IS A RESULTANT ARRAY OF SUBROUTINE MFSS
	   TRAC MUST BE OF DOUBLE PRECISION
   INC	 - INPUT VARIABLE WHICH SHOULD CONTAIN THE VALUE ZERO
	   IF THE SYSTEM OF SIMULTANEOUS EQUATIONS IS KNOWN
	   TO BE COMPATIBLE AND A NONZERO VALUE OTHERWISE
   RHS	 - VECTOR OF DIMENSION N CONTAINING THE RIGHT HAND SIDE
	   ON RETURN RHS CONTAINS THE MINIMAL LENGTH SOLUTION
	   RHS MUST BE OF DOUBLE PRECISION
   IER	 - RESULTANT ERROR PARAMETER
	   IER = 0 MEANS NO ERRORS
	   IER =-1 MEANS N AND/OR IRANK IS NOT POSITIVE AND/OR
		   IRANK IS GREATER THAN N
	   IER = 1 MEANS THE FACTORIZATION CONTAINED IN A HAS
		   ZERO DIVISORS AND/OR TRAC CONTAINS
		   VALUES OUTSIDE THE FEASIBLE RANGE 1 UP TO N

REMARKS
   THE MINIMAL LENGTH SOLUTION IS PRODUCED IN THE STORAGE
   LOCATIONS OCCUPIED BY THE RIGHT HAND SIDE.
   SUBROUTINE DMLSS DOES TAKE CARE OF THE PERMUTATION
   WHICH WAS APPLIED TO ROWS AND COLUMNS OF A.
   OPERATION IS BYPASSED IN CASE OF A NON POSITIVE VALUE
   OF IRANK

SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
   NONE

METHOD
   LET T, U, TU BE THE COMPONENTS OF THE FACTORIZATION OF A,
   AND LET THE RIGHT HAND SIDE BE PARTITIONED INTO A FIRST
   PART X1 OF DIMENSION IRANK AND A SECOND PART X2 OF DIMENSION
   N-IRANK. THEN THE FOLLOWING OPERATIONS ARE APPLIED IN
   SEQUENCE
   (1) INTERCHANGE RIGHT HAND SIDE
   (2) X1 = X1 + U * X2
   (3) X2 =-TRANSPOSE(U) * X1
   (4) X2 = INVERSE(TU) * INVERSE(TRANSPOSE(TU)) * X2
   (5) X1 = X1 + U * X2
   (6) X1 = INVERSE(T) * INVERSE(TRANSPOSE(T)) * X1
   (7) X2 =-TRANSPOSE(U) * X1
   (8) X2 = INVERSE(TU) * INVERSE(TRANSPOSE(TU)) * X2
   (9) X1 = X1 + U * X2
   (10)X2 = TRANSPOSE(U) * X1
   (11) REINTERCHANGE CALCULATED SOLUTION
   IF THE SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS IS SPECIFIED
   TO BE COMPATIBLE THEN STEPS (2), (3), (4) AND (5) ARE
   CANCELLED.
   IF THE COEFFICIENT MATRIX HAS RANK N, THEN THE ONLY STEPS
   PERFORMED ARE (1), (6) AND (11).