PDGELQF(3)    ScaLAPACK routine of NEC Numeric Library Collection   PDGELQF(3)



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

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

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

           INTEGER         DESCA( * )

           DOUBLE          PRECISION A( * ), TAU( * ), WORK( * )

PURPOSE
       PDGELQF computes a LQ factorization of a real 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 vec-
       tor.  This vector stores the information required to establish the map-
       ping 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  col-
       umn.
       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) DOUBLE PRECISION 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  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.

       TAU     (local output) DOUBLE PRECISION 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) 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 >= MB_A * ( Mp0 + Nq0 + MB_A ), 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    (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(ia+k-1) H(ia+k-2) . . . H(ia), where k = min(m,n).

       Each H(i) has the form

          H(i) = 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:n) is stored on exit in A(ia+i-1,ja+i:ja+n-1), and  tau
       in TAU(ia+i-1).




ScaLAPACK routine               31 October 2017                     PDGELQF(3)