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NAME

       PSGEQPF  -  compute a QR factorization with column pivoting of a M-by-N
       distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1)

SYNOPSIS

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

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

           INTEGER         DESCA( * ), IPIV( * )

           REAL            A( * ), TAU( * ), WORK( * )

PURPOSE

       PSGEQPF  computes  a  QR factorization with column pivoting of a M-by-N
       distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1):

                              sub( A ) * P = Q * R.

       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) REAL 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
               above the diagonal of sub( A ) contain the min(M,N) by N  upper
               trapezoidal  matrix  R  (R  is upper triangular if M >= N); the
               elements below the diagonal, with the array  TAU,  repre-  sent
               the  orthogonal  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.

       IPIV    (local output) INTEGER array, dimension LOCc(JA+N-1).
               On  exit,  if  IPIV(I) = K, the local i-th column of sub( A )*P
               was the global K-th column of sub( A ). IPIV  is  tied  to  the
               distributed matrix A.

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

       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 >= MAX(3,Mp0 + Nq0) + LOCc(JA+N-1)+Nq0.

               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  ),  LOCc(JA+N-1)  = NUMROC( JA+N-1, NB_A, MYCOL,
               CSRC_A, 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    (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.

FURTHER DETAILS

       The matrix Q is represented as a product of elementary reflectors

          Q = H(1) H(2) . . . H(n)

       Each H(i) has the form

          H = I - tau * v * v’

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

       The matrix P is represented in jpvt as follows: If
          jpvt(j) = i
       then the jth column of P is the ith canonical unit vector.