 PDORMBR(1) VECT = 'Q', PDORMBR overwrites the general real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L' SIDE = 'R' TRANS = 'N'

## SYNOPSIS

SUBROUTINE PDORMBR(
VECT, SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC, JC, DESCC, WORK, LWORK, INFO )

CHARACTER SIDE, TRANS, VECT

INTEGER IA, IC, INFO, JA, JC, K, LWORK, M, N

INTEGER DESCA( * ), DESCC( * )

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

## PURPOSE

If VECT = 'Q', PDORMBR overwrites the general real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T': Q**T * sub( C ) sub( C ) * Q**T

If VECT = 'P', PDORMBR overwrites sub( C ) with

SIDE = 'L'           SIDE = 'R'
TRANS = 'N': P * sub( C ) sub( C ) * P
TRANS = 'T': P**T * sub( C ) sub( C ) * P**T

Here Q and P**T are the orthogonal distributed matrices determined by PDGEBRD when reducing a real distributed matrix A(IA:*,JA:*) to bidiagonal form: A(IA:*,JA:*) = Q * B * P**T. Q and P**T are defined as products of elementary reflectors H(i) and G(i) respectively.

Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the order of the orthogonal matrix Q or P**T that is applied.

If VECT = 'Q', A(IA:*,JA:*) is assumed to have been an NQ-by-K matrix:
if nq >= k, Q = H(1) H(2) . . . H(k);
if nq < k, Q = H(1) H(2) . . . H(nq-1).

If VECT = 'P', A(IA:*,JA:*) is assumed to have been a K-by-NQ matrix:
if k < nq, P = G(1) G(2) . . . G(k);
if k >= nq, P = G(1) G(2) . . . G(nq-1).

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

VECT (global input) CHARACTER
= 'Q': apply Q or Q**T;
= 'P': apply P or P**T.
SIDE (global input) CHARACTER

= 'L': apply Q, Q**T, P or P**T from the Left;
= 'R': apply Q, Q**T, P or P**T from the Right.
TRANS (global input) CHARACTER

= 'N': No transpose, apply Q or P;
= 'T': Transpose, apply Q**T or P**T.
M (global input) INTEGER
The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). 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( C ). N >= 0.
K (global input) INTEGER
If VECT = 'Q', the number of columns in the original distributed matrix reduced by PDGEBRD. If VECT = 'P', the number of rows in the original distributed matrix reduced by PDGEBRD. K >= 0.
A (local input) DOUBLE PRECISION pointer into the local memory
to an array of dimension (LLD_A,LOCc(JA+MIN(NQ,K)-1)) if VECT='Q', and (LLD_A,LOCc(JA+NQ-1)) if VECT = 'P'. NQ = M if SIDE = 'L', and NQ = N otherwise. The vectors which define the elementary reflectors H(i) and G(i), whose products determine the matrices Q and P, as returned by PDGEBRD. If VECT = 'Q', LLD_A >= max(1,LOCr(IA+NQ-1)); if VECT = 'P', LLD_A >= max(1,LOCr(IA+MIN(NQ,K)-1)).
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 input) DOUBLE PRECISION array, dimension
LOCc(JA+MIN(NQ,K)-1) if VECT = 'Q', LOCr(IA+MIN(NQ,K)-1) if VECT = 'P', TAU(i) must contain the scalar factor of the elementary reflector H(i) or G(i), which determines Q or P, as returned by PDGEBRD in its array argument TAUQ or TAUP. TAU is tied to the distributed matrix A.
C (local input/local output) DOUBLE PRECISION pointer into the
local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, if VECT='Q', sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q; if VECT='P, sub( C ) is overwritten by P*sub( C ) or P'*sub( C ) or sub( C )*P or sub( C )*P'.
IC (global input) INTEGER
The row index in the global array C indicating the first row of sub( C ).
JC (global input) INTEGER
The column index in the global array C indicating the first column of sub( C ).
DESCC (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix C.
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 If SIDE = 'L', NQ = M; if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC; else IAA=IA+1; JAA=JA; MI=M-1; NI=N; ICC=IC+1; JCC=JC; end if else if SIDE = 'R', NQ = N; if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC; else IAA=IA; JAA=JA+1; MI=M; NI=N-1; ICC=IC; JCC=JC+1; end if end if

If VECT = 'Q', If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 ) )*NB_A ) + NB_A * NB_A end if else if VECT <> 'Q', if SIDE = 'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC( MI+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if end if

where LCMP = LCM / NPROW, LCMQ = LCM / NPCOL, with LCM = ICLM( NPROW, NPCOL ),

IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW = INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JAA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( MI+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), NpA0 = NUMROC( NI+IROFFA, MB_A, MYROW, IAROW, NPROW ),

IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ),

INDXG2P and NUMROC 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.

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

The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true:

If VECT = 'Q', If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC ) else If SIDE = 'L', ( MB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) end if