# man pzggqrf (Fonctions bibliothèques) - 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)

## NAME

PZGGQRF - 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 PZGGQRF(
- 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*16 A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * )

## PURPOSE

PZGGQRF 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
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 conjugate 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) COMPLEX*16 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*16, 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 Further Details). B (local input/local output) COMPLEX*16 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 unitary 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) COMPLEX*16, 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*16 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 PZUNGQR.

To use Q to update another matrix, use ScaLAPACK subroutine PZUNMQR.

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 PZUNGRQ.

To use Z to update another matrix, use ScaLAPACK subroutine PZUNMRQ.

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 )