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## NAME

```       PDGGRQF  -  compute  a generalized RQ factorization of an M-by-N matrix
sub( A ) = A(IA:IA+M-1,JA:JA+N-1)

```

## SYNOPSIS

```       SUBROUTINE PDGGRQF( M, P, N, 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( * )

DOUBLE          PRECISION  A(  *  ),  B( * ), TAUA( * ), TAUB( * ),
WORK( * )

```

## PURPOSE

```       PDGGRQF computes a generalized RQ factorization  of  an  M-by-N  matrix
sub(  A  )  =  A(IA:IA+M-1,JA:JA+N-1)  and  a  P-by-N matrix sub( B ) =
B(IB:IB+P-1,JB:JB+N-1):

sub( A ) = R*Q,        sub( B ) = Z*T*Q,

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

if M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
N-M  M                           ( R21 ) N
N

where R12 or R21 is upper triangular, and

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

where T11 is upper triangular.

In  particular,  if  sub(  B  )  is  square  and  nonsingular,  the GRQ
factorization of sub( A  )  and  sub(  B  )  implicitly  gives  the  RQ
factorization of sub( A )*inv( sub( B ) ):

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

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

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.

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

N       (global input) INTEGER
The  number  of  columns  to  be  operated on i.e the number of
columns of the distributed submatrices sub( A ) and sub(  B  ).
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, if M <= N, the upper
triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) contains the M  by  M
upper triangular matrix R; if M >= N, the elements on and above
the (M-N)-th subdiagonal contain the M by N  upper  trapezoidal
matrix   R;  the  remaining  elements,  with  the  array  TAUA,
represent 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.

TAUA    (local output) DOUBLE PRECISION, array, dimension LOCr(IA+M-1)
This array  contains  the  scalar  factors  of  the  elementary
reflectors  which  represent  the  orthogonal unitary matrix Q.
TAUA is tied to the distributed matrix A (see Further Details).

B       (local input/local output) DOUBLE PRECISION pointer into the
local  memory  to  an array of dimension (LLD_B, LOCc(JB+N-1)).
On entry, the local pieces of  the  P-by-N  distributed  matrix
sub( B ) which is to be factored.  On exit, the elements on and
above the diagonal of sub( B ) contain the min(P,N) by N  upper
trapezoidal  matrix  T  (T  is upper triangular if P >= N); the
elements below the diagonal, with the array TAUB, represent the
orthogonal  matrix Z as a product of elementary reflectors (see
Further Details).  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.

TAUB    (local output) DOUBLE PRECISION, array, dimension
LOCc(JB+MIN(P,N)-1). This array  contains  the  scalar  factors
TAUB   of   the   elementary  reflectors  which  represent  the
orthogonal matrix Z. TAUB is tied to the distributed  matrix  B
(see  Further Details).  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 >= MAX( MB_A * ( MpA0 + NqA0 + MB_A  ),  MAX(
(MB_A*(MB_A-1))/2, (PpB0 + NqB0)*MB_A ) + MB_A * MB_A, NB_B * (
PpB0 + NqB0 + NB_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 ), 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 ), PpB0    =  NUMROC(  P+IROFFB,
MB_B,  MYROW,  IBROW, NPROW ), NqB0   = NUMROC( N+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.

```

## FURTHERDETAILS

```       The matrix Q is represented as a product of elementary reflectors

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

Each H(i) has the form

H(i) = I - taua * v * v’

where taua is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored  on  exit  in
A(ia+m-k+i-1,ja:ja+n-k+i-2),  and  taua in TAUA(ia+m-k+i-1).  To form Q
explicitly, use ScaLAPACK subroutine PDORGRQ.
To use Q to update another matrix, use ScaLAPACK subroutine PDORMRQ.

The matrix Z is represented as a product of elementary reflectors

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

Each H(i) has the form

H(i) = I - taub * v * v’

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

Alignment requirements
======================

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

( NB_A.EQ.NB_B .AND. ICOFFA.EQ.ICOFFB .AND. IACOL.EQ.IBCOL )
```