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NAME

       PCGELQ2  -  compute  a LQ factorization of a complex distributed M-by-N
       matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q

SYNOPSIS

       SUBROUTINE PCGELQ2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

           INTEGER         IA, INFO, JA, LWORK, M, N

           INTEGER         DESCA( * )

           COMPLEX         A( * ), TAU( * ), WORK( * )

PURPOSE

       PCGELQ2 computes a LQ factorization of  a  complex  distributed  M-by-N
       matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = L * Q.

       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

       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.

       A       (local input/local output) COMPLEX pointer into the
               local memory to an array of  dimension  (LLD_A,  LOCc(JA+N-1)).
               On  entry,  the  local  pieces of the M-by-N distributed matrix
               sub( A ) which is to be factored. On exit, the elements on  and
               below  the diagonal of sub( A ) contain the M by min(M,N) lower
               trapezoidal matrix L (L is lower triangular if  M  <=  N);  the
               elements  above  the  diagonal, with the array TAU, repre- sent
               the unitary matrix Q as a product of elementary reflectors (see
               Further Details).  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.

       TAU     (local output) COMPLEX, array, dimension
               LOCr(IA+MIN(M,N)-1).  This array contains the scalar factors of
               the elementary reflectors.  TAU  is  tied  to  the  distributed
               matrix A.

       WORK    (local workspace/local output) COMPLEX 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 >= Nq0 + MAX( 1, Mp0 ), where

               IROFF  =  MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW =
               INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA,
               NB_A,  MYCOL,  CSRC_A,  NPCOL ), Mp0   = NUMROC( M+IROFF, MB_A,
               MYROW, IAROW, NPROW ), Nq0   = NUMROC(  N+ICOFF,  NB_A,  MYCOL,
               IACOL, NPCOL ),

               and NUMROC, INDXG2P 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    (local 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.

FURTHER DETAILS

       The matrix Q is represented as a product of elementary reflectors

          Q = H(ia+k-1)ā€™ H(ia+k-2)ā€™ . . . H(ia)ā€™, where k = min(m,n).

       Each H(i) has the form

          H(i) = I - tau * v * vā€™

       where  tau is a complex scalar, and v is a complex vector with v(1:i-1)
       =  0  and  v(i)  =  1;   conjg(v(i+1:n))   is   stored   on   exit   in
       A(ia+i-1,ja+i:ja+n-1), and tau in TAU(ia+i-1).