pdgels.f

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      SUBROUTINE pdgels( TRANS, M, N, NRHS, A, IA, JA, DESCA, B, IB, JB,
     $                   DESCB, WORK, LWORK, INFO )
*
*  -- ScaLAPACK routine (version 1.7) --
*     University of Tennessee, Knoxville, Oak Ridge National Laboratory,
*     and University of California, Berkeley.
*     May 1, 1997
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            IA, IB, INFO, JA, JB, LWORK, M, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            DESCA( * ), DESCB( * )
      DOUBLE PRECISION   A( * ), B( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  PDGELS solves overdetermined or underdetermined real linear
*  systems involving an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1),
*  or its transpose, using a QR or LQ factorization of sub( A ).  It is
*  assumed that sub( 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 || sub( B ) - sub( A )*X ||.
*
*  2. If TRANS = 'N' and m < n:  find the minimum norm solution of
*     an underdetermined system sub( A ) * X = sub( B ).
*
*  3. If TRANS = 'T' and m >= n:  find the minimum norm solution of
*     an undetermined system sub( A )**T * X = sub( 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 || sub( B ) - sub( A )**T * X ||.
*
*  where sub( B ) denotes B( IB:IB+M-1, JB:JB+NRHS-1 ) when TRANS = 'N'
*  and B( IB:IB+N-1, JB:JB+NRHS-1 ) otherwise. Several right hand side
*  vectors b and solution vectors x can be handled in a single call;
*  When TRANS = 'N', the solution vectors are stored as the columns of
*  the N-by-NRHS right hand side matrix sub( B ) and the M-by-NRHS
*  right hand side matrix sub( B ) otherwise.
*
*  Notes
*  =====
*
*  Each global data object is described by an associated description
*  vector.  This vector stores the information required to establish
*  the mapping between an object element and its corresponding process
*  and memory location.
*
*  Let A be a generic term for any 2D block cyclicly distributed array.
*  Such a global array has an associated description vector DESCA.
*  In the following comments, the character _ should be read as
*  "of the global array".
*
*  NOTATION        STORED IN      EXPLANATION
*  --------------- -------------- --------------------------------------
*  DTYPE_A(global) DESCA( DTYPE_ )The descriptor type.  In this case,
*                                 DTYPE_A = 1.
*  CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
*                                 the BLACS process grid A is distribu-
*                                 ted over. The context itself is glo-
*                                 bal, but the handle (the integer
*                                 value) may vary.
*  M_A    (global) DESCA( M_ )    The number of rows in the global
*                                 array A.
*  N_A    (global) DESCA( N_ )    The number of columns in the global
*                                 array A.
*  MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
*                                 the rows of the array.
*  NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
*                                 the columns of the array.
*  RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
*                                 row of the array A is distributed.
*  CSRC_A (global) DESCA( CSRC_ ) The process column over which the
*                                 first column of the array A is
*                                 distributed.
*  LLD_A  (local)  DESCA( LLD_ )  The leading dimension of the local
*                                 array.  LLD_A >= MAX(1,LOCr(M_A)).
*
*  Let K be the number of rows or columns of a distributed matrix,
*  and assume that its process grid has dimension p x q.
*  LOCr( K ) denotes the number of elements of K that a process
*  would receive if K were distributed over the p processes of its
*  process column.
*  Similarly, LOCc( K ) denotes the number of elements of K that a
*  process would receive if K were distributed over the q processes of
*  its process row.
*  The values of LOCr() and LOCc() may be determined via a call to the
*  ScaLAPACK tool function, NUMROC:
*          LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
*          LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
*  An upper bound for these quantities may be computed by:
*          LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
*          LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
*
*  Arguments
*  =========
*
*  TRANS   (global input) CHARACTER
*          = 'N': the linear system involves sub( A );
*          = 'T': the linear system involves sub( A )**T.
*
*  M       (global input) INTEGER
*          The number of rows to be operated on, i.e. the number of
*          rows of the distributed submatrix sub( A ). M >= 0.
*
*  N       (global input) INTEGER
*          The number of columns to be operated on, i.e. the number of
*          columns of the distributed submatrix sub( A ). N >= 0.
*
*  NRHS    (global input) INTEGER
*          The number of right hand sides, i.e. the number of columns
*          of the distributed submatrices sub( B ) and X.  NRHS >= 0.
*
*  A       (local input/local output) DOUBLE PRECISION pointer into the
*          local memory to an array of local dimension
*          ( LLD_A, LOCc(JA+N-1) ).  On entry, the M-by-N matrix A.
*          if M >= N, sub( A ) is overwritten by details of its QR
*            factorization as returned by PDGEQRF;
*          if M <  N, sub( A ) is overwritten by details of its LQ
*            factorization as returned by PDGELQF.
*
*  IA      (global input) INTEGER
*          The row index in the global array A indicating the first
*          row of sub( A ).
*
*  JA      (global input) INTEGER
*          The column index in the global array A indicating the
*          first column of sub( A ).
*
*  DESCA   (global and local input) INTEGER array of dimension DLEN_.
*          The array descriptor for the distributed matrix A.
*
*  B       (local input/local output) DOUBLE PRECISION pointer into the
*          local memory to an array of local dimension
*          (LLD_B, LOCc(JB+NRHS-1)).  On entry, this array contains the
*          local pieces of the distributed matrix B of right hand side
*          vectors, stored columnwise;
*          sub( B ) is M-by-NRHS if TRANS='N', and N-by-NRHS otherwise.
*          On exit, sub( B ) is overwritten by the solution vectors,
*          stored columnwise:  if TRANS = 'N' and M >= N, rows 1 to N
*          of sub( 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 sub( B )
*          contain the minimum norm solution vectors; if TRANS = 'T'
*          and M >= N, rows 1 to M of sub( B ) contain the minimum norm
*          solution vectors; if TRANS = 'T' and M < N, rows 1 to M of
*          sub( 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.
*
*  IB      (global input) INTEGER
*          The row index in the global array B indicating the first
*          row of sub( B ).
*
*  JB      (global input) INTEGER
*          The column index in the global array B indicating the
*          first column of sub( B ).
*
*  DESCB   (global and local input) INTEGER array of dimension DLEN_.
*          The array descriptor for the distributed matrix B.
*
*  WORK    (local workspace/local output) DOUBLE PRECISION array,
*                                                  dimension (LWORK)
*          On exit, WORK(1) returns the minimal and optimal LWORK.
*
*  LWORK   (local or global input) INTEGER
*          The dimension of the array WORK.
*          LWORK is local input and must be at least
*          LWORK >= LTAU + MAX( LWF, LWS ) where
*          If M >= N, then
*            LTAU = NUMROC( JA+MIN(M,N)-1, NB_A, MYCOL, CSRC_A, NPCOL ),
*            LWF  = NB_A * ( MpA0 + NqA0 + NB_A )
*            LWS  = MAX( (NB_A*(NB_A-1))/2, (NRHSqB0 + MpB0)*NB_A ) +
*                   NB_A * NB_A
*          Else
*            LTAU = NUMROC( IA+MIN(M,N)-1, MB_A, MYROW, RSRC_A, NPROW ),
*            LWF  = MB_A * ( MpA0 + NqA0 + MB_A )
*            LWS  = MAX( (MB_A*(MB_A-1))/2, ( NpB0 + MAX( NqA0 +
*                   NUMROC( NUMROC( N+IROFFB, MB_A, 0, 0, NPROW ),
*                   MB_A, 0, 0, LCMP ), NRHSqB0 ) )*MB_A ) +
*                   MB_A * MB_A
*          End if
*
*          where LCMP = LCM / NPROW with LCM = ILCM( NPROW, NPCOL ),
*
*          IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ),
*          IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ),
*          IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ),
*          MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ),
*          NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ),
*
*          IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ),
*          IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ),
*          IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ),
*          MpB0 = NUMROC( M+IROFFB, MB_B, MYROW, IBROW, NPROW ),
*          NpB0 = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ),
*          NRHSqB0 = NUMROC( NRHS+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ),
*
*          ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions;
*          MYROW, MYCOL, NPROW and NPCOL can be determined by calling
*          the subroutine BLACS_GRIDINFO.
*
*          If LWORK = -1, then LWORK is global input and a workspace
*          query is assumed; the routine only calculates the minimum
*          and optimal size for all work arrays. Each of these
*          values is returned in the first entry of the corresponding
*          work array, and no error message is issued by PXERBLA.
*
*  INFO    (global output) INTEGER
*          = 0:  successful exit
*          < 0:  If the i-th argument is an array and the j-entry had
*                an illegal value, then INFO = -(i*100+j), if the i-th
*                argument is a scalar and had an illegal value, then
*                INFO = -i.
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
     $                   lld_, mb_, m_, nb_, n_, rsrc_
      parameter( block_cyclic_2d = 1, dlen_ = 9, dtype_ = 1,
     $                     ctxt_ = 2, m_ = 3, n_ = 4, mb_ = 5, nb_ = 6,
     $                     rsrc_ = 7, csrc_ = 8, lld_ = 9 )
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, TPSD
      INTEGER            BROW, IACOL, IAROW, IASCL, IBCOL, IBROW, IBSCL,
     $                   icoffa, icoffb, ictxt, ipw, iroffa, iroffb,
     $                   lcm, lcmp, ltau, lwf, lwmin, lws, mpa0, mpb0,
     $                   mycol, myrow, npb0, npcol, nprow, nqa0,
     $                   nrhsqb0, scllen
      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, SMLNUM
*     ..
*     .. Local Arrays ..
      INTEGER            IDUM1( 2 ), IDUM2( 2 )
      DOUBLE PRECISION   RWORK( 1 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILCM
      INTEGER            INDXG2P, NUMROC
      DOUBLE PRECISION   PDLAMCH, PDLANGE
      EXTERNAL           ilcm, indxg2p, lsame, numroc, pdlamch,
     $                   pdlange
*     ..
*     .. External Subroutines ..
      EXTERNAL           blacs_gridinfo, chk1mat, pchk2mat, pdgelqf,
     $                   pdgeqrf, pdlabad, pdlascl, pdlaset,
     $                   pdormlq, pdormqr, pdtrsm, pxerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          dble, ichar, max, min, mod
*     ..
*     .. Executable Statements ..
*
*     Get grid parameters
*
      ictxt = desca( ctxt_ )
      CALL blacs_gridinfo( ictxt, nprow, npcol, myrow, mycol )
*
*     Test the input parameters
*
      info = 0
      IF( nprow.EQ.-1 ) THEN
         info = -( 800 + ctxt_ )
      ELSE
         CALL chk1mat( m, 2, n, 3, ia, ja, desca, 8, info )
         IF ( m .GE. n ) THEN
            CALL chk1mat( m, 2, nrhs, 4, ib, jb, descb, 12, info )
         ELSE
            CALL chk1mat( n, 3, nrhs, 4, ib, jb, descb, 12, info )
         ENDIF
         IF( info.EQ.0 ) THEN
            iroffa = mod( ia-1, desca( mb_ ) )
            icoffa = mod( ja-1, desca( nb_ ) )
            iroffb = mod( ib-1, descb( mb_ ) )
            icoffb = mod( jb-1, descb( nb_ ) )
            iarow = indxg2p( ia, desca( mb_ ), myrow, desca( rsrc_ ),
     $                       nprow )
            iacol = indxg2p( ia, desca( nb_ ), mycol, desca( csrc_ ),
     $                       npcol )
            mpa0 = numroc( m+iroffa, desca( mb_ ), myrow, iarow, nprow )
            nqa0 = numroc( n+icoffa, desca( nb_ ), mycol, iacol, npcol )
*
            ibrow = indxg2p( ib, descb( mb_ ), myrow, descb( rsrc_ ),
     $                       nprow )
            ibcol = indxg2p( ib, descb( nb_ ), mycol, descb( csrc_ ),
     $                       npcol )
            nrhsqb0 = numroc( nrhs+icoffb, descb( nb_ ), mycol, ibcol,
     $                        npcol )
            IF( m.GE.n ) THEN
               mpb0 = numroc( m+iroffb, descb( mb_ ), myrow, ibrow,
     $                        nprow )
               ltau = numroc( ja+min(m,n)-1, desca( nb_ ), mycol,
     $                        desca( csrc_ ), npcol )
               lwf  = desca( nb_ ) * ( mpa0 + nqa0 + desca( nb_ ) )
               lws = max( ( desca( nb_ )*( desca( nb_ ) - 1 ) ) / 2,
     $               ( mpb0 + nrhsqb0 ) * desca( nb_ ) ) +
     $               desca( nb_ )*desca( nb_ )
            ELSE
               lcm = ilcm( nprow, npcol )
               lcmp = lcm / nprow
               npb0 = numroc( n+iroffb, descb( mb_ ), myrow, ibrow,
     $                        nprow )
               ltau = numroc( ia+min(m,n)-1, desca( mb_ ), myrow,
     $                        desca( rsrc_ ), nprow )
               lwf  = desca( mb_ ) * ( mpa0 + nqa0 + desca( mb_ ) )
               lws  = max( ( desca( mb_ )*( desca( mb_ ) - 1 ) ) / 2,
     $                ( npb0 + max( nqa0 + numroc( numroc( n+iroffb,
     $                desca( mb_ ), 0, 0, nprow ), desca( mb_ ), 0, 0,
     $                lcmp ), nrhsqb0 ) )*desca( mb_ ) ) +
     $                desca( mb_ ) * desca( mb_ )
            END IF
            lwmin = ltau + max( lwf, lws )
            work( 1 ) = dble( lwmin )
            lquery = ( lwork.EQ.-1 )
*
            tpsd = .true.
            IF( lsame( trans, 'N' ) )
     $         tpsd = .false.
*
            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( m.GE.n .AND. iroffa.NE.iroffb ) THEN
               info = -10
            ELSE IF( m.GE.n .AND. iarow.NE.ibrow ) THEN
               info = -10
            ELSE IF( m.LT.n .AND. icoffa.NE.iroffb ) THEN
               info = -10
            ELSE IF( m.GE.n .AND. desca( mb_ ).NE.descb( mb_ ) ) THEN
               info = -( 1200 + mb_ )
            ELSE IF( m.LT.n .AND. desca( nb_ ).NE.descb( mb_ ) ) THEN
               info = -( 1200 + mb_ )
            ELSE IF( ictxt.NE.descb( ctxt_ ) ) THEN
               info = -( 1200 + ctxt_ )
            ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
               info = -14
            END IF
         END IF
*
         IF( .NOT.tpsd ) THEN
            idum1( 1 ) = ichar( 'N' )
         ELSE
            idum1( 1 ) = ichar( 'T' )
         END IF
         idum2( 1 ) = 1
         IF( lwork.EQ.-1 ) THEN
            idum1( 2 ) = -1
         ELSE
            idum1( 2 ) = 1
         END IF
         idum2( 2 ) = 14
         CALL pchk2mat( m, 2, n, 3, ia, ja, desca, 8, n, 3, nrhs, 4,
     $                  ib, jb, descb, 12, 2, idum1, idum2, info )
      END IF
*
      IF( info.NE.0 ) THEN
         CALL pxerbla( ictxt, 'PDGELS', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( min( m, n, nrhs ).EQ.0 ) THEN
         CALL pdlaset( 'Full', max( m, n ), nrhs, zero, zero, b,
     $                 ib, jb, descb )
         RETURN
      END IF
*
*     Get machine parameters
*
      smlnum = pdlamch( ictxt, 'S' )
      smlnum = smlnum / pdlamch( ictxt, 'P' )
      bignum = one / smlnum
      CALL pdlabad( ictxt, smlnum, bignum )
*
*     Scale A, B if max entry outside range [SMLNUM,BIGNUM]
*
      anrm = pdlange( 'M', m, n, a, ia, ja, desca, rwork )
      iascl = 0
      IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL pdlascl( 'G', anrm, smlnum, m, n, a, ia, ja, desca,
     $                 info )
         iascl = 1
      ELSE IF( anrm.GT.bignum ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL pdlascl( 'G', anrm, bignum, m, n, a, ia, ja, desca,
     $                 info )
         iascl = 2
      ELSE IF( anrm.EQ.zero ) THEN
*
*        Matrix all zero. Return zero solution.
*
         CALL pdlaset( 'F', max( m, n ), nrhs, zero, zero, b, ib, jb,
     $                 descb )
         GO TO 10
      END IF
*
      brow = m
      IF( tpsd )
     $   brow = n
*
      bnrm = pdlange( 'M', brow, nrhs, b, ib, jb, descb, rwork )
*
      ibscl = 0
      IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL pdlascl( 'G', bnrm, smlnum, brow, nrhs, b, ib, jb,
     $                 descb, info )
         ibscl = 1
      ELSE IF( bnrm.GT.bignum ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL pdlascl( 'G', bnrm, bignum, brow, nrhs, b, ib, jb,
     $                 descb, info )
         ibscl = 2
      END IF
*
      ipw = ltau + 1
*
      IF( m.GE.n ) THEN
*
*        compute QR factorization of A
*
         CALL pdgeqrf( m, n, a, ia, ja, desca, work, work( ipw ),
     $                 lwork-ltau, info )
*
*        workspace at least N, optimally N*NB
*
         IF( .NOT.tpsd ) THEN
*
*           Least-Squares Problem min || A * X - B ||
*
*           B(IB:IB+M-1,JB:JB+NRHS-1) := Q' * B(IB:IB+M-1,JB:JB+NRHS-1)
*
            CALL pdormqr( 'Left', 'Transpose', m, nrhs, n, a, ia, ja,
     $                    desca, work, b, ib, jb, descb, work( ipw ),
     $                    lwork-ltau, info )
*
*           workspace at least NRHS, optimally NRHS*NB
*
*           B(IB:IB+N-1,JB:JB+NRHS-1) := inv(R) *
*                                        B(IB:IB+N-1,JB:JB+NRHS-1)
*
            CALL pdtrsm( 'Left', 'Upper', 'No transpose', 'Non-unit', n,
     $                   nrhs, one, a, ia, ja, desca, b, ib, jb, descb )
*
            scllen = n
*
         ELSE
*
*           Overdetermined system of equations sub( A )' * X = sub( B )
*
*           sub( B ) := inv(R') * sub( B )
*
            CALL pdtrsm( 'Left', 'Upper', 'Transpose', 'Non-unit', n,
     $                   nrhs, one, a, ia, ja, desca, b, ib, jb, descb )
*
*           B(IB+N:IB+M-1,JB:JB+NRHS-1) = ZERO
*
            CALL pdlaset( 'All', m-n, nrhs, zero, zero, b, ib+n, jb,
     $                    descb )
*
*           B(IB:IB+M-1,JB:JB+NRHS-1) := Q(1:N,:) *
*                                        B(IB:IB+N-1,JB:JB+NRHS-1)
*
            CALL pdormqr( 'Left', 'No transpose', m, nrhs, n, a, ia, ja,
     $                    desca, work, b, ib, jb, descb, work( ipw ),
     $                    lwork-ltau, info )
*
*           workspace at least NRHS, optimally NRHS*NB
*
            scllen = m
*
         END IF
*
      ELSE
*
*        Compute LQ factorization of sub( A )
*
         CALL pdgelqf( m, n, a, ia, ja, desca, work, work( ipw ),
     $                 lwork-ltau, info )
*
*        workspace at least M, optimally M*NB.
*
         IF( .NOT.tpsd ) THEN
*
*           underdetermined system of equations sub( A ) * X = sub( B )
*
*           B(IB:IB+M-1,JB:JB+NRHS-1) := inv(L) *
*                                        B(IB:IB+M-1,JB:JB+NRHS-1)
*
            CALL pdtrsm( 'Left', 'Lower', 'No transpose', 'Non-unit', m,
     $                   nrhs, one, a, ia, ja, desca, b, ib, jb, descb )
*
*           B(IB+M:IB+N-1,JB:JB+NRHS-1) = 0
*
            CALL pdlaset( 'All', n-m, nrhs, zero, zero, b, ib+m, jb,
     $                    descb )
*
*           B(IB:IB+N-1,JB:JB+NRHS-1) := Q(1:N,:)' *
*                                        B(IB:IB+M-1,JB:JB+NRHS-1)
*
            CALL pdormlq( 'Left', 'Transpose', n, nrhs, m, a, ia, ja,
     $                    desca, work, b, ib, jb, descb, work( ipw ),
     $                    lwork-ltau, info )
*
*           workspace at least NRHS, optimally NRHS*NB
*
            scllen = n
*
         ELSE
*
*           overdetermined system min || A' * X - B ||
*
*           B(IB:IB+N-1,JB:JB+NRHS-1) := Q * B(IB:IB+N-1,JB:JB+NRHS-1)
*
            CALL pdormlq( 'Left', 'No transpose', n, nrhs, m, a, ia, ja,
     $                    desca, work, b, ib, jb, descb, work( ipw ),
     $                    lwork-ltau, info )
*
*           workspace at least NRHS, optimally NRHS*NB
*
*           B(IB:IB+M-1,JB:JB+NRHS-1) := inv(L') *
*                                        B(IB:IB+M-1,JB:JB+NRHS-1)
*
            CALL pdtrsm( 'Left', 'Lower', 'Transpose', 'Non-unit', m,
     $                   nrhs, one, a, ia, ja, desca, b, ib, jb, descb )
*
            scllen = m
*
         END IF
*
      END IF
*
*     Undo scaling
*
      IF( iascl.EQ.1 ) THEN
         CALL pdlascl( 'G', anrm, smlnum, scllen, nrhs, b, ib, jb,
     $                 descb, info )
      ELSE IF( iascl.EQ.2 ) THEN
         CALL pdlascl( 'G', anrm, bignum, scllen, nrhs, b, ib, jb,
     $                 descb, info )
      END IF
      IF( ibscl.EQ.1 ) THEN
         CALL pdlascl( 'G', smlnum, bnrm, scllen, nrhs, b, ib, jb,
     $                 descb, info )
      ELSE IF( ibscl.EQ.2 ) THEN
         CALL pdlascl( 'G', bignum, bnrm, scllen, nrhs, b, ib, jb,
     $                 descb, info )
      END IF
*
   10 CONTINUE
*
      work( 1 ) = dble( lwmin )
*
      RETURN
*
*     End of PDGELS
*
      END