dgetsls.f

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*> \brief \b DGETSLS
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGETSLS( TRANS, M, N, NRHS, A, LDA, B, LDB,
*     $                     WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGETSLS solves overdetermined or underdetermined real linear systems
*> involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
*> factorization of A.  It is assumed that A has full rank.
*>
*>
*>
*> The following options are provided:
*>
*> 1. If TRANS = 'N' and m >= n:  find the least squares solution of
*>    an overdetermined system, i.e., solve the least squares problem
*>                 minimize || B - A*X ||.
*>
*> 2. If TRANS = 'N' and m < n:  find the minimum norm solution of
*>    an underdetermined system A * X = B.
*>
*> 3. If TRANS = 'T' and m >= n:  find the minimum norm solution of
*>    an undetermined system A**T * X = B.
*>
*> 4. If TRANS = 'T' and m < n:  find the least squares solution of
*>    an overdetermined system, i.e., solve the least squares problem
*>                 minimize || B - A**T * X ||.
*>
*> 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.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N': the linear system involves A;
*>          = 'T': the linear system involves A**T.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix 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,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit,
*>          A is overwritten by details of its QR or LQ
*>          factorization as returned by DGEQR or DGELQ.
*> \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 matrix B of right hand side vectors, stored
*>          columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
*>          if TRANS = 'T'.
*>          On exit, if INFO = 0, B is overwritten by the solution
*>          vectors, stored columnwise:
*>          if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
*>          squares solution vectors.
*>          if TRANS = 'N' and m < n, rows 1 to N of B contain the
*>          minimum norm solution vectors;
*>          if TRANS = 'T' and m >= n, rows 1 to M of B contain the
*>          minimum norm solution vectors;
*>          if TRANS = 'T' and m < n, rows 1 to M of B contain the
*>          least squares solution vectors.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= MAX(1,M,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
*>          or optimal, if query was assumed) LWORK.
*>          See LWORK for details.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If LWORK = -1 or -2, then a workspace query is assumed.
*>          If LWORK = -1, the routine calculates optimal size of WORK for the
*>          optimal performance and returns this value in WORK(1).
*>          If LWORK = -2, the routine calculates minimal size of WORK and 
*>          returns this value in WORK(1).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO =  i, the i-th diagonal element of the
*>                triangular factor of A is zero, so that A does not have
*>                full rank; the least squares solution could not be
*>                computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEsolve
*
*  =====================================================================
      SUBROUTINE dgetsls( TRANS, M, N, NRHS, A, LDA, B, LDB,
     $                    WORK, LWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d0, one = 1.0d0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, TRAN
      INTEGER            I, IASCL, IBSCL, J, MAXMN, BROW,
     $                   scllen, tszo, tszm, lwo, lwm, lw1, lw2,
     $                   wsizeo, wsizem, info2
      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, SMLNUM, TQ( 5 ), WORKQ( 1 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           lsame, dlabad, dlamch, dlange
*     ..
*     .. External Subroutines ..
      EXTERNAL           dgeqr, dgemqr, dlascl, dlaset,
     $                   dtrtrs, xerbla, dgelq, dgemlq
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          dble, max, min, int
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
      info = 0
      maxmn = max( m, n )
      tran  = lsame( trans, 'T' )
*
      lquery = ( lwork.EQ.-1 .OR. lwork.EQ.-2 )
      IF( .NOT.( lsame( trans, 'N' ) .OR.
     $    lsame( trans, 'T' ) ) ) THEN
         info = -1
      ELSE IF( m.LT.0 ) THEN
         info = -2
      ELSE IF( n.LT.0 ) THEN
         info = -3
      ELSE IF( nrhs.LT.0 ) THEN
         info = -4
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -6
      ELSE IF( ldb.LT.max( 1, m, n ) ) THEN
         info = -8
      END IF
*
      IF( info.EQ.0 ) THEN
*
*     Determine the optimum and minimum LWORK
*
       IF( m.GE.n ) THEN
         CALL dgeqr( m, n, a, lda, tq, -1, workq, -1, info2 )
         tszo = int( tq( 1 ) )
         lwo  = int( workq( 1 ) )
         CALL dgemqr( 'L', trans, m, nrhs, n, a, lda, tq,
     $                tszo, b, ldb, workq, -1, info2 )
         lwo  = max( lwo, int( workq( 1 ) ) )
         CALL dgeqr( m, n, a, lda, tq, -2, workq, -2, info2 )
         tszm = int( tq( 1 ) )
         lwm  = int( workq( 1 ) )
         CALL dgemqr( 'L', trans, m, nrhs, n, a, lda, tq,
     $                tszm, b, ldb, workq, -1, info2 )
         lwm = max( lwm, int( workq( 1 ) ) )
         wsizeo = tszo + lwo
         wsizem = tszm + lwm
       ELSE
         CALL dgelq( m, n, a, lda, tq, -1, workq, -1, info2 )
         tszo = int( tq( 1 ) )
         lwo  = int( workq( 1 ) )
         CALL dgemlq( 'L', trans, n, nrhs, m, a, lda, tq,
     $                tszo, b, ldb, workq, -1, info2 )
         lwo  = max( lwo, int( workq( 1 ) ) )
         CALL dgelq( m, n, a, lda, tq, -2, workq, -2, info2 )
         tszm = int( tq( 1 ) )
         lwm  = int( workq( 1 ) )
         CALL dgemlq( 'L', trans, n, nrhs, m, a, lda, tq,
     $                tszm, b, ldb, workq, -1, info2 )
         lwm  = max( lwm, int( workq( 1 ) ) )
         wsizeo = tszo + lwo
         wsizem = tszm + lwm
       END IF
*
       IF( ( lwork.LT.wsizem ).AND.( .NOT.lquery ) ) THEN
          info = -10
       END IF
*
       work( 1 ) = dble( wsizeo )
*
      END IF
*
      IF( info.NE.0 ) THEN
        CALL xerbla( 'DGETSLS', -info )
        RETURN
      END IF
      IF( lquery ) THEN
        IF( lwork.EQ.-2 ) work( 1 ) = dble( wsizem )
        RETURN
      END IF
      IF( lwork.LT.wsizeo ) THEN
        lw1 = tszm
        lw2 = lwm
      ELSE
        lw1 = tszo
        lw2 = lwo
      END IF
*
*     Quick return if possible
*
      IF( min( m, n, nrhs ).EQ.0 ) THEN
           CALL dlaset( 'FULL', max( m, n ), nrhs, zero, zero,
     $                  b, ldb )
           RETURN
      END IF
*
*     Get machine parameters
*
       smlnum = dlamch( 'S' ) / dlamch( 'P' )
       bignum = one / smlnum
       CALL dlabad( smlnum, bignum )
*
*     Scale A, B if max element outside range [SMLNUM,BIGNUM]
*
      anrm = dlange( 'M', m, n, a, lda, work )
      iascl = 0
      IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL dlascl( 'g', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
         IASCL = 1
.GT.      ELSE IF( ANRMBIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL DLASCL( 'g', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
         IASCL = 2
.EQ.      ELSE IF( ANRMZERO ) THEN
*
*        Matrix all zero. Return zero solution.
*
         CALL DLASET( 'f', MAXMN, NRHS, ZERO, ZERO, B, LDB )
         GO TO 50
      END IF
*
      BROW = M
      IF ( TRAN ) THEN
        BROW = N
      END IF
      BNRM = DLANGE( 'm', BROW, NRHS, B, LDB, WORK )
      IBSCL = 0
.GT..AND..LT.      IF( BNRMZERO  BNRMSMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL DLASCL( 'g', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB,
     $                INFO )
         IBSCL = 1
.GT.      ELSE IF( BNRMBIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL DLASCL( 'g', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB,
     $                INFO )
         IBSCL = 2
      END IF
*
.GE.      IF ( MN ) THEN
*
*        compute QR factorization of A
*
        CALL DGEQR( M, N, A, LDA, WORK( LW2+1 ), LW1,
     $              WORK( 1 ), LW2, INFO )
.NOT.        IF ( TRAN ) THEN
*
*           Least-Squares Problem min || A * X - B ||
*
*           B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
*
          CALL DGEMQR( 'l' , 't', M, NRHS, N, A, LDA,
     $                 WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
     $                 INFO )
*
*           B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
*
          CALL DTRTRS( 'u', 'n', 'n', N, NRHS,
     $                  A, LDA, B, LDB, INFO )
.GT.          IF( INFO0 ) THEN
            RETURN
          END IF
          SCLLEN = N
        ELSE
*
*           Overdetermined system of equations A**T * X = B
*
*           B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
*
            CALL DTRTRS( 'u', 't', 'n', N, NRHS,
     $                   A, LDA, B, LDB, INFO )
*
.GT.            IF( INFO0 ) THEN
               RETURN
            END IF
*
*           B(N+1:M,1:NRHS) = ZERO
*
            DO 20 J = 1, NRHS
               DO 10 I = N + 1, M
                  B( I, J ) = ZERO
   10          CONTINUE
   20       CONTINUE
*
*           B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS)
*
            CALL DGEMQR( 'l', 'n', M, NRHS, N, A, LDA,
     $                   WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
     $                   INFO )
*
            SCLLEN = M
*
         END IF
*
      ELSE
*
*        Compute LQ factorization of A
*
         CALL DGELQ( M, N, A, LDA, WORK( LW2+1 ), LW1,
     $               WORK( 1 ), LW2, INFO )
*
*        workspace at least M, optimally M*NB.
*
.NOT.         IF( TRAN ) THEN
*
*           underdetermined system of equations A * X = B
*
*           B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
*
            CALL DTRTRS( 'l', 'n', 'n', M, NRHS,
     $                   A, LDA, B, LDB, INFO )
*
.GT.            IF( INFO0 ) THEN
               RETURN
            END IF
*
*           B(M+1:N,1:NRHS) = 0
*
            DO 40 J = 1, NRHS
               DO 30 I = M + 1, N
                  B( I, J ) = ZERO
   30          CONTINUE
   40       CONTINUE
*
*           B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS)
*
            CALL DGEMLQ( 'l', 't', N, NRHS, M, A, LDA,
     $                   WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
     $                   INFO )
*
*           workspace at least NRHS, optimally NRHS*NB
*
            SCLLEN = N
*
         ELSE
*
*           overdetermined system min || A**T * X - B ||
*
*           B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
*
            CALL DGEMLQ( 'l', 'n', N, NRHS, M, A, LDA,
     $                   WORK( LW2+1 ), LW1, B, LDB, WORK( 1 ), LW2,
     $                   INFO )
*
*           workspace at least NRHS, optimally NRHS*NB
*
*           B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
*
            CALL DTRTRS( 'lower', 'transpose', 'non-unit', M, NRHS,
     $                   A, LDA, B, LDB, INFO )
*
.GT.            IF( INFO0 ) THEN
               RETURN
            END IF
*
            SCLLEN = M
*
         END IF
*
      END IF
*
*     Undo scaling
*
.EQ.      IF( IASCL1 ) THEN
        CALL DLASCL( 'g', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
     $               INFO )
.EQ.      ELSE IF( IASCL2 ) THEN
        CALL DLASCL( 'g', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
     $               INFO )
      END IF
.EQ.      IF( IBSCL1 ) THEN
        CALL DLASCL( 'g', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
     $               INFO )
.EQ.      ELSE IF( IBSCL2 ) THEN
        CALL DLASCL( 'g', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
     $               INFO )
      END IF
*
   50 CONTINUE
      WORK( 1 ) = DBLE( TSZO + LWO )
      RETURN
*
*     End of DGETSLS
*
      END