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SUBROUTINE LLSQ

PURPOSE
   TO SOLVE LINEAR LEAST SQUARES PROBLEMS, I.E. TO MINIMIZE
   THE EUCLIDEAN NORM OF B-A*X, WHERE A IS A M BY N MATRIX
   WITH M NOT LESS THAN N. IN THE SPECIAL CASE M=N SYSTEMS OF
   LINEAR EQUATIONS MAY BE SOLVED.

USAGE
   CALL LLSQ (A,B,M,N,L,X,IPIV,EPS,IER,AUX)

DESCRIPTION OF PARAMETERS
   A	  - M BY N COEFFICIENT MATRIX (DESTROYED).
   B	  - M BY L RIGHT HAND SIDE MATRIX (DESTROYED).
   M	  - ROW NUMBER OF MATRICES A AND B.
   N	  - COLUMN NUMBER OF MATRIX A, ROW NUMBER OF MATRIX X.
   L	  - COLUMN NUMBER OF MATRICES B AND X.
   X	  - N BY L SOLUTION MATRIX.
   IPIV   - INTEGER OUTPUT VECTOR OF DIMENSION N WHICH
	    CONTAINS INFORMATIONS ON COLUMN INTERCHANGES
	    IN MATRIX A. (SEE REMARK NO.3).
   EPS	  - INPUT PARAMETER WHICH SPECIFIES A RELATIVE
	    TOLERANCE FOR DETERMINATION OF RANK OF MATRIX A.
   IER	  - A RESULTING ERROR PARAMETER.
   AUX	  - AUXILIARY STORAGE ARRAY OF DIMENSION MAX(2*N,L).
	    ON RETURN FIRST L LOCATIONS OF AUX CONTAIN THE
	    RESULTING LEAST SQUARES.

REMARKS
   (1) NO ACTION BESIDES ERROR MESSAGE IER=-2 IN CASE
       M LESS THAN N.
   (2) NO ACTION BESIDES ERROR MESSAGE IER=-1 IN CASE
       OF A ZERO-MATRIX A.
   (3) IF RANK K OF MATRIX A IS FOUND TO BE LESS THAN N BUT
       GREATER THAN 0, THE PROCEDURE RETURNS WITH ERROR CODE
       IER=K INTO CALLING PROGRAM. THE LAST N-K ELEMENTS OF
       VECTOR IPIV DENOTE THE USELESS COLUMNS IN MATRIX A.
       THE REMAINING USEFUL COLUMNS FORM A BASE OF MATRIX A.
   (4) IF THE PROCEDURE WAS SUCCESSFUL, ERROR PARAMETER IER
       IS SET TO 0.

SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
   NONE

METHOD
   HOUSEHOLDER TRANSFORMATIONS ARE USED TO TRANSFORM MATRIX A
   TO UPPER TRIANGULAR FORM. AFTER HAVING APPLIED THE SAME
   TRANSFORMATION TO THE RIGHT HAND SIDE MATRIX B, AN
   APPROXIMATE SOLUTION OF THE PROBLEM IS COMPUTED BY
   BACK SUBSTITUTION. FOR REFERENCE, SEE
   G. GOLUB, NUMERICAL METHODS FOR SOLVING LINEAR LEAST
   SQUARES PROBLEMS, NUMERISCHE MATHEMATIK, VOL.7,
   ISS.3 (1965), PP.206-216.