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SGELSD(l)					)					SGELSD(l)

       SGELSD - compute the minimum-norm solution to a real linear least squares problem




	   INTEGER	  IWORK( * )

	   REAL 	  A( LDA, * ), B( LDB, * ), S( * ), WORK( * )

       SGELSD computes the minimum-norm solution to a real linear least squares problem:     min-
       imize 2-norm(| b - A*x |)
       using the singular value decomposition (SVD) of A. A is an  M-by-N  matrix  which  may  be

       Several	right hand side vectors b and solution vectors x can be handled in a single call;
       they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS
       solution matrix X.

       The problem is solved in three steps:
       (1) Reduce the coefficient matrix A to bidiagonal form with
	   Householder transformations, reducing the original problem
	   into a "bidiagonal least squares problem" (BLS)
       (2) Solve the BLS using a divide and conquer approach.
       (3) Apply back all the Householder tranformations to solve
	   the original least squares problem.

       The  effective rank of A is determined by treating as zero those singular values which are
       less than RCOND times the largest singular value.

       The divide and conquer algorithm makes very mild assumptions about floating  point  arith-
       metic.  It  will  work  on machines with a guard digit in add/subtract, or on those binary
       machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90,  or
       Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits,
       but we know of none.

       M       (input) INTEGER
	       The number of rows of A. M >= 0.

       N       (input) INTEGER
	       The number of columns of A. N >= 0.

       NRHS    (input) INTEGER
	       The number of right hand sides, i.e., the number of columns of the matrices B  and
	       X. NRHS >= 0.

       A       (input) REAL array, dimension (LDA,N)
	       On entry, the M-by-N matrix A.  On exit, A has been destroyed.

       LDA     (input) INTEGER
	       The leading dimension of the array A.  LDA >= max(1,M).

       B       (input/output) REAL array, dimension (LDB,NRHS)
	       On  entry,  the	M-by-NRHS right hand side matrix B.  On exit, B is overwritten by
	       the N-by-NRHS solution matrix X.  If m >= n and RANK =  n,  the	residual  sum-of-
	       squares for the solution in the i-th column is given by the sum of squares of ele-
	       ments n+1:m in that column.

       LDB     (input) INTEGER
	       The leading dimension of the array B. LDB >= max(1,max(M,N)).

       S       (output) REAL array, dimension (min(M,N))
	       The singular values of A in decreasing order.  The condition number of  A  in  the
	       2-norm = S(1)/S(min(m,n)).

       RCOND   (input) REAL
	       RCOND  is  used	to  determine  the  effective rank of A.  Singular values S(i) <=
	       RCOND*S(1) are treated as zero.	If RCOND < 0, machine precision is used instead.

       RANK    (output) INTEGER
	       The effective rank of A, i.e., the number of singular  values  which  are  greater
	       than RCOND*S(1).

       WORK    (workspace/output) REAL array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The  dimension  of  the	array  WORK. LWORK must be at least 1.	The exact minimum
	       amount of workspace needed depends on M, N and NRHS. As long as LWORK is at  least
	       12*N  +	2*N*SMLSIZ  +  8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, if M is greater than or
	       equal to N or 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is  less
	       than  N,  the  code  will  execute correctly.  SMLSIZ is returned by ILAENV and is
	       equal to the maximum size of the subproblems at the bottom of the computation tree
	       (usually about 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
	       For good performance, LWORK should generally be larger.

	       If LWORK = -1, then a workspace query is assumed; the routine only calculates  the
	       optimal	size of the WORK array, returns this value as the first entry of the WORK
	       array, and no error message related to LWORK is issued by XERBLA.

       IWORK   (workspace) INTEGER array, dimension (LIWORK)
	       LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, where MINMN = MIN( M,N ).

       INFO    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value.
	       > 0:  the algorithm for computing the SVD failed to converge; if INFO = i, i  off-
	       diagonal elements of an intermediate bidiagonal form did not converge to zero.

       Based on contributions by
	  Ming Gu and Ren-Cang Li, Computer Science Division, University of
	    California at Berkeley, USA
	  Osni Marques, LBNL/NERSC, USA

LAPACK version 3.0			   15 June 2000 				SGELSD(l)
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