```      SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
\$                  INFO )
*
*  -- LAPACK driver routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
CHARACTER          TRANS
INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS
*     ..
*     .. Array Arguments ..
DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGELS solves overdetermined or underdetermined real linear systems
*  involving an M-by-N matrix A, or its transpose, using a QR or 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.
*
*  Arguments
*  =========
*
*  TRANS   (input) CHARACTER*1
*          = 'N': the linear system involves A;
*          = 'T': the linear system involves A**T.
*
*  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) DOUBLE PRECISION array, dimension (LDA,N)
*          On entry, the M-by-N matrix A.
*          On exit,
*            if M >= N, A is overwritten by details of its QR
*                       factorization as returned by DGEQRF;
*            if M <  N, A is overwritten by details of its LQ
*                       factorization as returned by DGELQF.
*
*  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 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; the residual sum of squares for the
*          solution in each column is given by the sum of squares of
*          elements N+1 to M in that column;
*          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; the residual sum of squares
*          for the solution in each column is given by the sum of
*          squares of elements M+1 to N in that column.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B. LDB >= MAX(1,M,N).
*
*  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 >= max( 1, MN + max( MN, NRHS ) ).
*          For optimal performance,
*          LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
*          where MN = min(M,N) and NB is the optimum block size.
*
*          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.
*
*  INFO    (output) 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.
*
*  =====================================================================
*
*     .. Parameters ..
DOUBLE PRECISION   ZERO, ONE
PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
LOGICAL            LQUERY, TPSD
INTEGER            BROW, I, IASCL, IBSCL, J, MN, NB, SCLLEN, WSIZE
DOUBLE PRECISION   ANRM, BIGNUM, BNRM, SMLNUM
*     ..
*     .. Local Arrays ..
DOUBLE PRECISION   RWORK( 1 )
*     ..
*     .. External Functions ..
LOGICAL            LSAME
INTEGER            ILAENV
DOUBLE PRECISION   DLAMCH, DLANGE
EXTERNAL           LSAME, ILAENV, DLABAD, DLAMCH, DLANGE
*     ..
*     .. External Subroutines ..
EXTERNAL           DGELQF, DGEQRF, DLASCL, DLASET, DORMLQ, DORMQR,
\$                   DTRTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          DBLE, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
INFO = 0
MN = MIN( M, N )
LQUERY = ( LWORK.EQ.-1 )
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
ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND. .NOT.LQUERY )
\$          THEN
INFO = -10
END IF
*
*     Figure out optimal block size
*
IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN
*
TPSD = .TRUE.
IF( LSAME( TRANS, 'N' ) )
\$      TPSD = .FALSE.
*
IF( M.GE.N ) THEN
NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
IF( TPSD ) THEN
NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LN', M, NRHS, N,
\$              -1 ) )
ELSE
NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N,
\$              -1 ) )
END IF
ELSE
NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
IF( TPSD ) THEN
NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M,
\$              -1 ) )
ELSE
NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LN', N, NRHS, M,
\$              -1 ) )
END IF
END IF
*
WSIZE = MAX( 1, MN+MAX( MN, NRHS )*NB )
WORK( 1 ) = DBLE( WSIZE )
*
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGELS ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
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
*
*     Scale A, B if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', M, N, A, LDA, RWORK )
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
ELSE IF( ANRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
CALL DLASCL( '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 DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
GO TO 50
END IF
*
BROW = M
IF( TPSD )
\$   BROW = N
BNRM = DLANGE( 'M', BROW, NRHS, B, LDB, RWORK )
IBSCL = 0
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB,
\$                INFO )
IBSCL = 1
ELSE IF( BNRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB,
\$                INFO )
IBSCL = 2
END IF
*
IF( M.GE.N ) THEN
*
*        compute QR factorization of A
*
CALL DGEQRF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN,
\$                INFO )
*
*        workspace at least N, optimally N*NB
*
IF( .NOT.TPSD ) THEN
*
*           Least-Squares Problem min || A * X - B ||
*
*           B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS)
*
CALL DORMQR( 'Left', 'Transpose', M, NRHS, N, A, LDA,
\$                   WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
\$                   INFO )
*
*           workspace at least NRHS, optimally NRHS*NB
*
*           B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
*
CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', N, NRHS,
\$                   A, LDA, B, LDB, INFO )
*
IF( INFO.GT.0 ) THEN
RETURN
END IF
*
SCLLEN = N
*
ELSE
*
*           Overdetermined system of equations A' * X = B
*
*           B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS)
*
CALL DTRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS,
\$                   A, LDA, B, LDB, INFO )
*
IF( INFO.GT.0 ) 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 DORMQR( 'Left', 'No transpose', M, NRHS, N, A, LDA,
\$                   WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
\$                   INFO )
*
*           workspace at least NRHS, optimally NRHS*NB
*
SCLLEN = M
*
END IF
*
ELSE
*
*        Compute LQ factorization of A
*
CALL DGELQF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN,
\$                INFO )
*
*        workspace at least M, optimally M*NB.
*
IF( .NOT.TPSD ) THEN
*
*           underdetermined system of equations A * X = B
*
*           B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
*
CALL DTRTRS( 'Lower', 'No transpose', 'Non-unit', M, NRHS,
\$                   A, LDA, B, LDB, INFO )
*
IF( INFO.GT.0 ) 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,:)' * B(1:M,1:NRHS)
*
CALL DORMLQ( 'Left', 'Transpose', N, NRHS, M, A, LDA,
\$                   WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
\$                   INFO )
*
*           workspace at least NRHS, optimally NRHS*NB
*
SCLLEN = N
*
ELSE
*
*           overdetermined system min || A' * X - B ||
*
*           B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
*
CALL DORMLQ( 'Left', 'No transpose', N, NRHS, M, A, LDA,
\$                   WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
\$                   INFO )
*
*           workspace at least NRHS, optimally NRHS*NB
*
*           B(1:M,1:NRHS) := inv(L') * B(1:M,1:NRHS)
*
CALL DTRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS,
\$                   A, LDA, B, LDB, INFO )
*
IF( INFO.GT.0 ) THEN
RETURN
END IF
*
SCLLEN = M
*
END IF
*
END IF
*
*     Undo scaling
*
IF( IASCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
\$                INFO )
ELSE IF( IASCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
\$                INFO )
END IF
IF( IBSCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
\$                INFO )
ELSE IF( IBSCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
\$                INFO )
END IF
*
50 CONTINUE
WORK( 1 ) = DBLE( WSIZE )
*
RETURN
*
*     End of DGELS
*
END

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