Provided by: scalapack-doc_1.5-10_all #### NAME

```       PDGGQRF  -  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 PDGGQRF( 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( * )

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

```

#### PURPOSE

```       PDGGQRF  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 orthogonal matrix, Z is a P-by-P  orthogonal  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 factorization 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
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

```       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) DOUBLE PRECISION 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 orthogonal 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) DOUBLE PRECISION, array, dimension
LOCc(JA+MIN(N,M)-1). This array contains the scalar factors TAUA of the elementary
reflectors  which  represent  the  orthogonal  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+P-1)).  On entry, the local pieces of the N-by-P distributed 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
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 LOCr(IB+N-1)
This array  contains  the  scalar  factors  of  the  elementary  reflectors  which
represent the orthogonal unitary 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( 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.

```

#### FURTHERDETAILS

```       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 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:ia+n-1,ja+i-1), and taua in TAUA(ja+i-1).
To form Q explicitly, use ScaLAPACK subroutine PDORGQR.
To use Q to update another matrix, use ScaLAPACK subroutine PDORMQR.

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 real scalar, and v is a real vector with
v(p-k+i+1:p) = 0 and v(p-k+i) = 1; 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
PDORGRQ.
To use Z to update another matrix, use ScaLAPACK subroutine PDORMRQ.

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 )
```