sgelsd.f

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*> \brief <b> SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SGELSD( 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
*       REAL               RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       REAL               A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SGELSD 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 REAL 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 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 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 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)).
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*>          RCOND is 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.
*> \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 REAL 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 array WORK and the
*>          minimum size of the array IWORK, and returns these values as
*>          the first entries of the WORK and IWORK arrays, 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 realGEsolve
*
*> \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 sgelsd( M, N, NRHS, A, LDA, B, LDB, S, RCOND,
     $                   rank, work, lwork, iwork, info )
*
*  -- LAPACK driver routine (version 3.4.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
      REAL               RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      REAL               A( lda, * ), B( ldb, * ), S( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE, TWO
      parameter                ( zero = 0.0e0, one = 1.0e0, two = 2.0e0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
     $                   ldwork, liwork, maxmn, maxwrk, minmn, minwrk,
     $                   mm, mnthr, nlvl, nwork, smlsiz, wlalsd
      REAL               ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgebrd, sgelqf, sgeqrf, slabad, slacpy, slalsd,
     $                   slascl, slaset, sormbr, sormlq, sormqr, xerbla
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      REAL               SLAMCH, SLANGE
      EXTERNAL           slamch, slange, ilaenv
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          int, log, max, min, real
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
      info = 0
      minmn = min( m, n )
      maxmn = max( m, n )
      lquery = ( lwork.EQ.-1 )
      IF( m.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( nrhs.LT.0 ) THEN
         info = -3
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -5
      ELSE IF( ldb.LT.max( 1, maxmn ) ) THEN
         info = -7
      END IF
*
*     Compute workspace.
*     (Note: Comments in the code beginning "Workspace:" describe the
*     minimal amount of workspace needed at that point in the code,
*     as well as the preferred amount for good performance.
*     NB refers to the optimal block size for the immediately
*     following subroutine, as returned by ILAENV.)
*
      IF( info.EQ.0 ) THEN
         minwrk = 1
         maxwrk = 1
         liwork = 1
         IF( minmn.GT.0 ) THEN
            smlsiz = ilaenv( 9, 'SGELSD', ' ', 0, 0, 0, 0 )
            mnthr = ilaenv( 6, 'SGELSD', ' ', m, n, nrhs, -1 )
            nlvl = max( int( log( REAL( MINMN ) / REAL( SMLSIZ + 1 ) ) /
     $                  log( two ) ) + 1, 0 )
            liwork = 3*minmn*nlvl + 11*minmn
            mm = m
            IF( m.GE.n .AND. m.GE.mnthr ) THEN
*
*              Path 1a - overdetermined, with many more rows than
*                        columns.
*
               mm = n
               maxwrk = max( maxwrk, n + n*ilaenv( 1, 'SGEQRF', ' ', m,
     $                       n, -1, -1 ) )
               maxwrk = max( maxwrk, n + nrhs*ilaenv( 1, 'SORMQR', 'LT',
     $                       m, nrhs, n, -1 ) )
            END IF
            IF( m.GE.n ) THEN
*
*              Path 1 - overdetermined or exactly determined.
*
               maxwrk = max( maxwrk, 3*n + ( mm + n )*ilaenv( 1,
     $                       'SGEBRD', ' ', mm, n, -1, -1 ) )
               maxwrk = max( maxwrk, 3*n + nrhs*ilaenv( 1, 'SORMBR',
     $                       'QLT', mm, nrhs, n, -1 ) )
               maxwrk = max( maxwrk, 3*n + ( n - 1 )*ilaenv( 1,
     $                       'SORMBR', 'PLN', n, nrhs, n, -1 ) )
               wlalsd = 9*n + 2*n*smlsiz + 8*n*nlvl + n*nrhs +
     $                  ( smlsiz + 1 )**2
               maxwrk = max( maxwrk, 3*n + wlalsd )
               minwrk = max( 3*n + mm, 3*n + nrhs, 3*n + wlalsd )
            END IF
            IF( n.GT.m ) THEN
               wlalsd = 9*m + 2*m*smlsiz + 8*m*nlvl + m*nrhs +
     $                  ( smlsiz + 1 )**2
               IF( n.GE.mnthr ) THEN
*
*                 Path 2a - underdetermined, with many more columns
*                           than rows.
*
                  maxwrk = m + m*ilaenv( 1, 'SGELQF', ' ', m, n, -1,
     $                                  -1 )
                  maxwrk = max( maxwrk, m*m + 4*m + 2*m*ilaenv( 1,
     $                          'SGEBRD', ' ', m, m, -1, -1 ) )
                  maxwrk = max( maxwrk, m*m + 4*m + nrhs*ilaenv( 1,
     $                          'SORMBR', 'QLT', m, nrhs, m, -1 ) )
                  maxwrk = max( maxwrk, m*m + 4*m + ( m - 1 )*ilaenv( 1,
     $                          'SORMBR', 'PLN', m, nrhs, m, -1 ) )
                  IF( nrhs.GT.1 ) THEN
                     maxwrk = max( maxwrk, m*m + m + m*nrhs )
                  ELSE
                     maxwrk = max( maxwrk, m*m + 2*m )
                  END IF
                  maxwrk = max( maxwrk, m + nrhs*ilaenv( 1, 'SORMLQ',
     $                          'LT', n, nrhs, m, -1 ) )
                  maxwrk = max( maxwrk, m*m + 4*m + wlalsd )
!     XXX: Ensure the Path 2a case below is triggered.  The workspace
!     calculation should use queries for all routines eventually.
                  maxwrk = max( maxwrk,
     $                 4*m+m*m+max( m, 2*m-4, nrhs, n-3*m ) )
               ELSE
*
*                 Path 2 - remaining underdetermined cases.
*
                  maxwrk = 3*m + ( n + m )*ilaenv( 1, 'SGEBRD', ' ', m,
     $                     n, -1, -1 )
                  maxwrk = max( maxwrk, 3*m + nrhs*ilaenv( 1, 'SORMBR',
     $                          'QLT', m, nrhs, n, -1 ) )
                  maxwrk = max( maxwrk, 3*m + m*ilaenv( 1, 'SORMBR',
     $                          'PLN', n, nrhs, m, -1 ) )
                  maxwrk = max( maxwrk, 3*m + wlalsd )
               END IF
               minwrk = max( 3*m + nrhs, 3*m + m, 3*m + wlalsd )
            END IF
         END IF
         minwrk = min( minwrk, maxwrk )
         work( 1 ) = maxwrk
         iwork( 1 ) = liwork
*
         IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
            info = -12
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SGELSD', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( m.EQ.0 .OR. n.EQ.0 ) THEN
         rank = 0
         RETURN
      END IF
*
*     Get machine parameters.
*
      eps = slamch( 'P' )
      sfmin = slamch( 'S' )
      smlnum = sfmin / eps
      bignum = one / smlnum
      CALL slabad( smlnum, bignum )
*
*     Scale A if max entry outside range [SMLNUM,BIGNUM].
*
      anrm = slange( 'M', m, n, a, lda, work )
      iascl = 0
      IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
*
*        Scale matrix norm up to SMLNUM.
*
         CALL slascl( 'G', 0, 0, anrm, smlnum, m, n, a, lda, info )
         iascl = 1
      ELSE IF( anrm.GT.bignum ) THEN
*
*        Scale matrix norm down to BIGNUM.
*
         CALL slascl( 'G', 0, 0, anrm, bignum, m, n, a, lda, info )
         iascl = 2
      ELSE IF( anrm.EQ.zero ) THEN
*
*        Matrix all zero. Return zero solution.
*
         CALL slaset( 'F', max( m, n ), nrhs, zero, zero, b, ldb )
         CALL slaset( 'F', minmn, 1, zero, zero, s, 1 )
         rank = 0
         GO TO 10
      END IF
*
*     Scale B if max entry outside range [SMLNUM,BIGNUM].
*
      bnrm = slange( 'M', m, nrhs, b, ldb, work )
      ibscl = 0
      IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
*
*        Scale matrix norm up to SMLNUM.
*
         CALL slascl( 'G', 0, 0, bnrm, smlnum, m, nrhs, b, ldb, info )
         ibscl = 1
      ELSE IF( bnrm.GT.bignum ) THEN
*
*        Scale matrix norm down to BIGNUM.
*
         CALL slascl( 'G', 0, 0, bnrm, bignum, m, nrhs, b, ldb, info )
         ibscl = 2
      END IF
*
*     If M < N make sure certain entries of B are zero.
*
      IF( m.LT.n )
     $   CALL slaset( 'F', n-m, nrhs, zero, zero, b( m+1, 1 ), ldb )
*
*     Overdetermined case.
*
      IF( m.GE.n ) THEN
*
*        Path 1 - overdetermined or exactly determined.
*
         mm = m
         IF( m.GE.mnthr ) THEN
*
*           Path 1a - overdetermined, with many more rows than columns.
*
            mm = n
            itau = 1
            nwork = itau + n
*
*           Compute A=Q*R.
*           (Workspace: need 2*N, prefer N+N*NB)
*
            CALL sgeqrf( m, n, a, lda, work( itau ), work( nwork ),
     $                   lwork-nwork+1, info )
*
*           Multiply B by transpose(Q).
*           (Workspace: need N+NRHS, prefer N+NRHS*NB)
*
            CALL sormqr( 'L', 'T', m, nrhs, n, a, lda, work( itau ), b,
     $                   ldb, work( nwork ), lwork-nwork+1, info )
*
*           Zero out below R.
*
            IF( n.GT.1 ) THEN
               CALL slaset( 'L', n-1, n-1, zero, zero, a( 2, 1 ), lda )
            END IF
         END IF
*
         ie = 1
         itauq = ie + n
         itaup = itauq + n
         nwork = itaup + n
*
*        Bidiagonalize R in A.
*        (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
*
         CALL sgebrd( mm, n, a, lda, s, work( ie ), work( itauq ),
     $                work( itaup ), work( nwork ), lwork-nwork+1,
     $                info )
*
*        Multiply B by transpose of left bidiagonalizing vectors of R.
*        (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
*
         CALL sormbr( 'Q', 'L', 'T', mm, nrhs, n, a, lda, work( itauq ),
     $                b, ldb, work( nwork ), lwork-nwork+1, info )
*
*        Solve the bidiagonal least squares problem.
*
         CALL slalsd( 'U', smlsiz, n, nrhs, s, work( ie ), b, ldb,
     $                rcond, rank, work( nwork ), iwork, info )
         IF( info.NE.0 ) THEN
            GO TO 10
         END IF
*
*        Multiply B by right bidiagonalizing vectors of R.
*
         CALL sormbr( 'P', 'L', 'N', n, nrhs, n, a, lda, work( itaup ),
     $                b, ldb, work( nwork ), lwork-nwork+1, info )
*
      ELSE IF( n.GE.mnthr .AND. lwork.GE.4*m+m*m+
     $         max( m, 2*m-4, nrhs, n-3*m, wlalsd ) ) THEN
*
*        Path 2a - underdetermined, with many more columns than rows
*        and sufficient workspace for an efficient algorithm.
*
         ldwork = m
         IF( lwork.GE.max( 4*m+m*lda+max( m, 2*m-4, nrhs, n-3*m ),
     $       m*lda+m+m*nrhs, 4*m+m*lda+wlalsd ) )ldwork = lda
         itau = 1
         nwork = m + 1
*
*        Compute A=L*Q.
*        (Workspace: need 2*M, prefer M+M*NB)
*
         CALL sgelqf( m, n, a, lda, work( itau ), work( nwork ),
     $                lwork-nwork+1, info )
         il = nwork
*
*        Copy L to WORK(IL), zeroing out above its diagonal.
*
         CALL slacpy( 'L', m, m, a, lda, work( il ), ldwork )
         CALL slaset( 'U', m-1, m-1, zero, zero, work( il+ldwork ),
     $                ldwork )
         ie = il + ldwork*m
         itauq = ie + m
         itaup = itauq + m
         nwork = itaup + m
*
*        Bidiagonalize L in WORK(IL).
*        (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
*
         CALL sgebrd( m, m, work( il ), ldwork, s, work( ie ),
     $                work( itauq ), work( itaup ), work( nwork ),
     $                lwork-nwork+1, info )
*
*        Multiply B by transpose of left bidiagonalizing vectors of L.
*        (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
*
         CALL sormbr( 'Q', 'L', 'T', m, nrhs, m, work( il ), ldwork,
     $                work( itauq ), b, ldb, work( nwork ),
     $                lwork-nwork+1, info )
*
*        Solve the bidiagonal least squares problem.
*
         CALL slalsd( 'U', smlsiz, m, nrhs, s, work( ie ), b, ldb,
     $                rcond, rank, work( nwork ), iwork, info )
         IF( info.NE.0 ) THEN
            GO TO 10
         END IF
*
*        Multiply B by right bidiagonalizing vectors of L.
*
         CALL sormbr( 'P', 'L', 'N', m, nrhs, m, work( il ), ldwork,
     $                work( itaup ), b, ldb, work( nwork ),
     $                lwork-nwork+1, info )
*
*        Zero out below first M rows of B.
*
         CALL slaset( 'F', n-m, nrhs, zero, zero, b( m+1, 1 ), ldb )
         nwork = itau + m
*
*        Multiply transpose(Q) by B.
*        (Workspace: need M+NRHS, prefer M+NRHS*NB)
*
         CALL sormlq( 'L', 'T', n, nrhs, m, a, lda, work( itau ), b,
     $                ldb, work( nwork ), lwork-nwork+1, info )
*
      ELSE
*
*        Path 2 - remaining underdetermined cases.
*
         ie = 1
         itauq = ie + m
         itaup = itauq + m
         nwork = itaup + m
*
*        Bidiagonalize A.
*        (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
*
         CALL sgebrd( m, n, a, lda, s, work( ie ), work( itauq ),
     $                work( itaup ), work( nwork ), lwork-nwork+1,
     $                info )
*
*        Multiply B by transpose of left bidiagonalizing vectors.
*        (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
*
         CALL sormbr( 'Q', 'L', 'T', m, nrhs, n, a, lda, work( itauq ),
     $                b, ldb, work( nwork ), lwork-nwork+1, info )
*
*        Solve the bidiagonal least squares problem.
*
         CALL slalsd( 'L', smlsiz, m, nrhs, s, work( ie ), b, ldb,
     $                rcond, rank, work( nwork ), iwork, info )
         IF( info.NE.0 ) THEN
            GO TO 10
         END IF
*
*        Multiply B by right bidiagonalizing vectors of A.
*
         CALL sormbr( 'P', 'L', 'N', n, nrhs, m, a, lda, work( itaup ),
     $                b, ldb, work( nwork ), lwork-nwork+1, info )
*
      END IF
*
*     Undo scaling.
*
      IF( iascl.EQ.1 ) THEN
         CALL slascl( 'G', 0, 0, anrm, smlnum, n, nrhs, b, ldb, info )
         CALL slascl( 'G', 0, 0, smlnum, anrm, minmn, 1, s, minmn,
     $                info )
      ELSE IF( iascl.EQ.2 ) THEN
         CALL slascl( 'G', 0, 0, anrm, bignum, n, nrhs, b, ldb, info )
         CALL slascl( 'G', 0, 0, bignum, anrm, minmn, 1, s, minmn,
     $                info )
      END IF
      IF( ibscl.EQ.1 ) THEN
         CALL slascl( 'G', 0, 0, smlnum, bnrm, n, nrhs, b, ldb, info )
      ELSE IF( ibscl.EQ.2 ) THEN
         CALL slascl( 'G', 0, 0, bignum, bnrm, n, nrhs, b, ldb, info )
      END IF
*
   10 CONTINUE
      work( 1 ) = maxwrk
      iwork( 1 ) = liwork
      RETURN
*
*     End of SGELSD
*
      END