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NAME

       PDGELS  -  solve overdetermined or underdetermined real linear systems involving an M-by-N
       matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1),

SYNOPSIS

       SUBROUTINE PDGELS( TRANS, M, N, NRHS, A, IA, JA, DESCA, B, IB,  JB,  DESCB,  WORK,  LWORK,
                          INFO )

           CHARACTER      TRANS

           INTEGER        IA, IB, INFO, JA, JB, LWORK, M, N, NRHS

           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.