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



NAME
       PCGGQRF  -  compute  a generalized QR factorization of an N-by-M matrix
       sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and an  N-by-P  matrix  sub(  B  )  =
       B(IB:IB+N-1,JB:JB+P-1)

SYNOPSIS
       SUBROUTINE PCGGQRF( N,  M, P, A, IA, JA, DESCA, TAUA, B, IB, JB, DESCB,
                           TAUB, WORK, LWORK, INFO )

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

           INTEGER         DESCA( * ), DESCB( * )

           COMPLEX         A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * )

PURPOSE
       PCGGQRF computes a generalized QR factorization  of  an  N-by-M  matrix
       sub(  A  )  =  A(IA:IA+N-1,JA:JA+M-1)  and  an N-by-P matrix sub( B ) =
       B(IB:IB+N-1,JB:JB+P-1):
                   sub( A ) = Q*R,        sub( B ) = Q*T*Z,

       where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix,  and
       R and T assume one of the forms:

       if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                       (  0  ) N-M                         N   M-N
                          M

       where R11 is upper triangular, and

       if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                        P-N  N                           ( T21 ) P
                                                            P

       where T12 or T21 is upper triangular.

       In  particular,  if sub( B ) is square and nonsingular, the GQR factor-
       ization of sub( A ) and sub( B ) implicitly gives the QR  factorization
       of inv( sub( B ) )* sub( A ):

                    inv( sub( B ) )*sub( A )= Z'*(inv(T)*R)

       where  inv(  sub( B ) ) denotes the inverse of the matrix sub( B ), and
       Z' denotes the conjugate transpose of matrix Z.


       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
       N       (global input) INTEGER
               The number of rows to be operated on i.e the number of rows  of
               the distributed submatrices sub( A ) and sub( B ). N >= 0.

       M       (global input) INTEGER
               The  number  of  columns  to  be  operated on i.e the number of
               columns of the distributed submatrix sub( A ).  M >= 0.

       P       (global input) INTEGER
               The number of columns to be  operated  on  i.e  the  number  of
               columns of the distributed submatrix sub( B ).  P >= 0.

       A       (local input/local output) COMPLEX pointer into the
               local  memory  to  an array of dimension (LLD_A, LOCc(JA+M-1)).
               On entry, the local pieces of  the  N-by-M  distributed  matrix
               sub( A ) which is to be factored.  On exit, the elements on and
               above the diagonal of sub( A ) contain the min(N,M) by M  upper
               trapezoidal  matrix  R  (R  is upper triangular if N >= M); the
               elements below the diagonal, with the array TAUA, represent the
               unitary matrix Q as a product of min(N,M) 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.

       TAUA    (local output) COMPLEX, array, dimension
               LOCc(JA+MIN(N,M)-1).  This  array  contains  the scalar factors
               TAUA of the elementary reflectors which represent  the  unitary
               matrix  Q.  TAUA is tied to the distributed matrix A. (see Fur-
               ther Details).   B        (local  input/local  output)  COMPLEX
               pointer  into the local memory to an array of dimension (LLD_B,
               LOCc(JB+P-1)).  On entry, the local pieces of the  N-by-P  dis-
               tributed matrix sub( B ) which is to be factored. On exit, if N
               <= P, the upper triangle of B(IB:IB+N-1,JB+P-N:JB+P-1) contains
               the N by N upper triangular matrix T; if N > P, the elements on
               and above the (N-P)-th subdiagonal contain the  N  by  P  upper
               trapezoidal  matrix  T;  the remaining elements, with the array
               TAUB, represent the unitary matrix Z as a product of elementary
               reflectors (see Further Details).  IB      (global input) INTE-
               GER 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.

       TAUB    (local output) COMPLEX, array, dimension LOCr(IB+N-1)
               This array  contains  the  scalar  factors  of  the  elementary
               reflectors  which  represent the unitary matrix Z. TAUB is tied
               to the  distributed  matrix  B  (see  Further  Details).   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 >= MAX( NB_A * ( NpA0 + MqA0 + NB_A ), MAX(
               (NB_A*(NB_A-1))/2, (PqB0 + NpB0)*NB_A ) + NB_A * NB_A, MB_B * (
               NpB0 + PqB0 + MB_B ) ), where

               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 ), NpA0   = NUMROC( N+IROFFA,
               MB_A, MYROW, IAROW, NPROW ), MqA0   = NUMROC(  M+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 ), NpB0   = NUMROC( N+IROFFB,
               MB_B, MYROW, IBROW, NPROW ), PqB0   = NUMROC(  P+ICOFFB,  NB_B,
               MYCOL, IBCOL, 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(ja) H(ja+1) . . . H(ja+k-1), where k = min(n,m).

       Each H(i) has the form

          H(i) = I - taua * v * v'

       where taua is a complex scalar, and v is a complex vector with v(1:i-1)
       = 0 and v(i) = 1; v(i+1:n) is stored on exit in
       A(ia+i:ia+n-1,ja+i-1), and taua in TAUA(ja+i-1).
       To form Q explicitly, use ScaLAPACK subroutine PCUNGQR.
       To use Q to update another matrix, use ScaLAPACK subroutine PCUNMQR.

       The matrix Z is represented as a product of elementary reflectors

          Z = H(ib)' H(ib+1)' . . . H(ib+k-1)', where k = min(n,p).

       Each H(i) has the form

          H(i) = I - taub * v * v'

       where taub is a complex scalar, and v is a  complex  vector  with  v(p-
       k+i+1:p) = 0 and v(p-k+i) = 1; conjg(v(1:p-k+i-1)) is stored on exit in
       B(ib+n-k+i-1,jb:jb+p-k+i-2), and taub in TAUB(ib+n-k+i-1).  To  form  Z
       explicitly, use ScaLAPACK subroutine PCUNGRQ.
       To use Z to update another matrix, use ScaLAPACK subroutine PCUNMRQ.


ALIGNMENT REQUIREMENTS
       The  distributed  submatrices  sub(  A  ) and sub( B ) must verify some
       alignment properties, namely the following expression should be true:

       ( MB_A.EQ.MB_B .AND. IROFFA.EQ.IROFFB .AND. IAROW.EQ.IBROW )




ScaLAPACK routine               31 October 2017                     PCGGQRF(3)