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NAME

       PSGELS  -  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 PSGELS( 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( * )

           REAL           A( * ), B( * ), WORK( * )

PURPOSE

       PSGELS  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) REAL 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 PSGEQRF; if M <  N, sub(  A  )  is  overwritten  by
               details of its LQ factorization as returned by PSGELQF.

       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) REAL 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) REAL 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.