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CGELSD(l)					)					CGELSD(l)

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

			  INFO )



	   INTEGER	  IWORK( * )

	   REAL 	  RWORK( * ), S( * )

	   COMPLEX	  A( LDA, * ), B( LDB, * ), WORK( * )

       CGELSD 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 tranformations, 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 the matrix A. M >= 0.

       N       (input) INTEGER
	       The number of columns of the matrix 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/output) COMPLEX 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) COMPLEX 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,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) COMPLEX 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
	       2 * N + N * NRHS if M is greater than or equal to N or 2 * M + M * NRHS	if  M  is
	       less  than N, the code will execute correctly.  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.

       RWORK   (workspace) REAL array, dimension at least
	       10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + (SMLSIZ+1)**2 if M is greater  than
	       or equal to N or 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*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 )

       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 				CGELSD(l)
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