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version 1.8, 2010/12/21 13:53:25 version 1.9, 2011/11/21 20:42:51
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   *> \brief <b> DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DGELSD + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelsd.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelsd.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelsd.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
   *                          WORK, LWORK, IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
   *       DOUBLE PRECISION   RCOND
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> 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
   *> singular 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 digits, but we know of none.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of rows of A. M >= 0.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of columns of A. N >= 0.
   *> \endverbatim
   *>
   *> \param[in] NRHS
   *> \verbatim
   *>          NRHS is INTEGER
   *>          The number of right hand sides, i.e., the number of columns
   *>          of the matrices B and X. NRHS >= 0.
   *> \endverbatim
   *>
   *> \param[in] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   *>          On entry, the M-by-N matrix A.
   *>          On exit, A has been destroyed.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,M).
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is 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.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B. LDB >= max(1,max(M,N)).
   *> \endverbatim
   *>
   *> \param[out] S
   *> \verbatim
   *>          S is 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)).
   *> \endverbatim
   *>
   *> \param[in] RCOND
   *> \verbatim
   *>          RCOND is DOUBLE PRECISION
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] RANK
   *> \verbatim
   *>          RANK is INTEGER
   *>          The effective rank of A, i.e., the number of singular values
   *>          which are greater than RCOND*S(1).
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
   *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is 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.
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
   *>          LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),
   *>          where MINMN = MIN( M,N ).
   *>          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is 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.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleGEsolve
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Ming Gu and Ren-Cang Li, Computer Science Division, University of
   *>       California at Berkeley, USA \n
   *>     Osni Marques, LBNL/NERSC, USA \n
   *
   *  =====================================================================
       SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,        SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
      $                   WORK, LWORK, IWORK, INFO )       $                   WORK, LWORK, IWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2.2) --  *  -- LAPACK driver routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     June 2010  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK        INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
Line 15 Line 223
       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )        DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  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  
 *  singular 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 digits, but we know of none.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  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   (input) DOUBLE PRECISION  
 *          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 (MAX(1,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 (MAX(1,LIWORK))  
 *          LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),  
 *          where MINMN = MIN( M,N ).  
 *          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.  
 *  
 *  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.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Based on contributions by  
 *     Ming Gu and Ren-Cang Li, Computer Science Division, University of  
 *       California at Berkeley, USA  
 *     Osni Marques, LBNL/NERSC, USA  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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