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## NAME

```       PCUNMBR   -   VECT  =  ’Q’,  PCUNMBR  overwrites  the  general  complex
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 PCUNMBR( 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( * )

COMPLEX         A( * ), C( * ), TAU( * ), WORK( * )

```

## PURPOSE

```       If VECT = ’Q’, PCUNMBR overwrites the general complex distributed M-by-
N  matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = ’C’:      Q**H
* sub( C )       sub( C ) * Q**H

If VECT = ’P’, PCUNMBR overwrites sub( C ) with

SIDE = ’L’           SIDE = ’R’
TRANS = ’N’:      P * sub( C )          sub( C ) * P
TRANS = ’C’:      P**H * sub( C )       sub( C ) * P**H

Here Q and P**H are the  unitary  distributed  matrices  determined  by
PCGEBRD  when  reducing  a  complex  distributed matrix A(IA:*,JA:*) to
bidiagonal form: A(IA:*,JA:*) = Q * B * P**H. Q and P**H 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 unitary matrix Q or P**H 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**H;
= ’P’: apply P or P**H.

SIDE    (global input) CHARACTER
= ’L’: apply Q, Q**H, P or P**H from the Left;
= ’R’: apply Q, Q**H, P or P**H from the Right.

TRANS   (global input) CHARACTER
= ’N’:  No transpose, apply Q or P;
= ’C’:  Conjugate transpose, apply Q**H or P**H.

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 PCGEBRD.   If  VECT  =  ’P’,  the
number  of  rows  in the original distributed matrix reduced by
PCGEBRD.  K >= 0.

A       (local input) COMPLEX 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 PCGEBRD.  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) COMPLEX 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) COMPLEX 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) 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 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
```