PZLARZ(3)     ScaLAPACK routine of NEC Numeric Library Collection    PZLARZ(3)



NAME
       PZLARZ  -  applie  a complex elementary reflector Q to a complex M-by-N
       distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from  either  the
       left or the right

SYNOPSIS
       SUBROUTINE PZLARZ( SIDE,  M,  N, L, V, IV, JV, DESCV, INCV, TAU, C, IC,
                          JC, DESCC, WORK )

           CHARACTER      SIDE

           INTEGER        IC, INCV, IV, JC, JV, L, M, N

           INTEGER        DESCC( * ), DESCV( * )

           COMPLEX*16     C( * ), TAU( * ), V( * ), WORK( * )

PURPOSE
       PZLARZ applies a complex elementary reflector Q  to  a  complex  M-by-N
       distributed  matrix  sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the
       left or the right. Q is represented in the form
             Q = I - tau * v * v'

       where tau is a complex scalar and v is a complex vector.

       If tau = 0, then Q is taken to be the unit matrix.

       Q is a product of k elementary reflectors as returned by PZTZRZF.


       Notes
       =====

       Each global data object is described by an associated description  vec-
       tor.  This vector stores the information required to establish the map-
       ping 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 col-
       umn.
       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

       Because vectors may be viewed as a subclass of matrices, a  distributed
       vector is considered to be a distributed matrix.

       Restrictions
       ============

       If  SIDE  =  'Left' and INCV = 1, then the row process having the first
       entry   V(IV,JV)   must   also   own   C(IC+M-L,JC:JC+N-1).   Moreover,
       MOD(IV-1,MB_V)  must  be equal to MOD(IC+N-L-1,MB_C), if INCV=M_V, only
       the last equality must be satisfied.

       If SIDE = 'Right' and INCV = M_V then the  column  process  having  the
       first   entry   V(IV,JV)   must   also   own   C(IC:IC+M-1,JC+N-L)  and
       MOD(JV-1,NB_V) must be equal to MOD(JC+N-L-1,NB_C), if INCV  =  1  only
       the last equality must be satisfied.


ARGUMENTS
       SIDE    (global input) CHARACTER
               = 'L': form  Q * sub( C ),
               = 'R': form  sub( C ) * Q.

       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.

       L       (global input) INTEGER
               The  columns  of  the distributed submatrix sub( A ) containing
               the meaningful part of the Householder reflectors.  If  SIDE  =
               'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

       V       (local input) COMPLEX*16 pointer into the local memory
               to  an array of dimension (LLD_V,*) containing the local pieces
               of the  distributed  vectors  V  representing  the  Householder
               transformation Q, V(IV:IV+L-1,JV) if SIDE = 'L' and INCV = 1,
               V(IV,JV:JV+L-1) if SIDE = 'L' and INCV = M_V,
               V(IV:IV+L-1,JV) if SIDE = 'R' and INCV = 1,
               V(IV,JV:JV+L-1) if SIDE = 'R' and INCV = M_V,

               The vector v in the representation of Q. V is not used if TAU =
               0.

       IV      (global input) INTEGER
               The row index in the global array V indicating the first row of
               sub( V ).

       JV      (global input) INTEGER
               The  column  index  in  the global array V indicating the first
               column of sub( V ).

       DESCV   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix V.

       INCV    (global input) INTEGER
               The global increment for the elements of V. Only two values  of
               INCV  are  supported  in  this version, namely 1 and M_V.  INCV
               must not be zero.

       TAU     (local input) COMPLEX*16, array, dimension  LOCc(JV) if
               INCV = 1, and  LOCr(IV)  otherwise.  This  array  contains  the
               Householder scalars related to the Householder vectors.  TAU is
               tied to the distributed matrix V.

       C       (local input/local output) COMPLEX*16 pointer into the
               local memory to an array of dimension (LLD_C,  LOCc(JC+N-1)  ),
               containing  the  local pieces of sub( C ). On exit, sub( C ) is
               overwritten by the Q * sub( C ) if SIDE = 'L', or sub( C ) *  Q
               if SIDE = 'R'.

       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) COMPLEX*16 array, dimension (LWORK)
               If  INCV  =  1,  if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0
               else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE =  'R',
               LWORK   >=  NqC0  +  MAX(  MAX(  1,  MpC0  ),  NUMROC(  NUMROC(
               N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV
               = M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ), NUM-
               ROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP )  )  else
               if  SIDE  =  'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >=
               NqC0 + MAX( 1, MpC0 ) end if end if end if

               where LCM is the least common multiple of NPROW and  NPCOL  and
               LCM  =  ILCM(  NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM /
               NPCOL,

               IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW =
               INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC,
               NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 =  NUMROC(  M+IROFFC,  MB_C,
               MYROW,  ICROW,  NPROW  ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL,
               ICCOL, NPCOL ),

               ILCM, INDXG2P and NUMROC are ScaLAPACK tool  functions;  MYROW,
               MYCOL, NPROW and NPCOL can be determined by calling the subrou-
               tine BLACS_GRIDINFO.


ALIGNMENT REQUIREMENTS
       The distributed submatrices V(IV:*,  JV:*)  and  C(IC:IC+M-1,JC:JC+N-1)
       must verify some alignment properties, namely the following expressions
       should be true:

       MB_V = NB_V,

       If INCV = 1, If SIDE = 'Left', (  MB_V.EQ.MB_C  .AND.  IROFFV.EQ.IROFFC
       .AND.  IVROW.EQ.ICROW  )  If  SIDE  =  'Right',  (  MB_V.EQ.NB_A  .AND.
       MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV =  M_V,  If  SIDE  =
       'Left',  (  MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If
       SIDE  =  'Right',   (   NB_V.EQ.NB_C   .AND.   ICOFFV.EQ.ICOFFC   .AND.
       IVCOL.EQ.ICCOL ) end if



ScaLAPACK routine               31 October 2017                      PZLARZ(3)