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DGELSD(1)	      LAPACK driver routine (version 3.2)	     DGELSD(1)

       DGELSD  -  computes  the	 minimum-norm  solution to a real linear least
       squares problem




	   INTEGER	  IWORK( * )

	   DOUBLE	  PRECISION  A( LDA, * ), B( LDB, * ), S( * ), WORK( *

       DGELSD computes the  minimum-norm  solution  to	a  real	 linear	 least
       squares problem:
	   minimize 2-norm(| b - A*x |)
       using  the  singular  value  decomposition  (SVD)  of A. A is an M-by-N
       matrix which may be rank-deficient.
       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	singu‐
       lar values which are less than RCOND times the largest singular value.
       The  divide  and	 conquer  algorithm  makes very mild assumptions about
       floating point arithmetic. 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 dig‐
       its, 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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  elements
	       n+1:m in that column.

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

       S       (output) DOUBLE PRECISION 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 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)   DOUBLE	PRECISION   array,   dimension
	       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 sub‐
	       problems	 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  mes‐
	       sage related to LWORK is issued by XERBLA.

       IWORK   (workspace) INTEGER array, dimension (MAX(1,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 bidiag‐
	       onal 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 driver routine (version 3November 2008			     DGELSD(1)

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