PZGEHD2(3)    ScaLAPACK routine of NEC Numeric Library Collection   PZGEHD2(3)



NAME
       PZGEHD2 - reduce a complex general distributed matrix sub( A ) to upper
       Hessenberg form H by an unitary similarity transformation

SYNOPSIS
       SUBROUTINE PZGEHD2( N, ILO, IHI, A, IA, JA, DESCA,  TAU,  WORK,  LWORK,
                           INFO )

           INTEGER         IA, IHI, ILO, INFO, JA, LWORK, N

           INTEGER         DESCA( * )

           COMPLEX*16      A( * ), TAU( * ), WORK( * )

PURPOSE
       PZGEHD2  reduces a complex general distributed matrix sub( A ) to upper
       Hessenberg form H by an unitary similarity transformation: Q' * sub(  A
       ) * Q = H, where
       sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1).


       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


ARGUMENTS
       N       (global input) INTEGER
               The  number  of  rows  and  columns to be operated on, i.e. the
               order of the distributed submatrix sub( A ). N >= 0.

       ILO     (global input) INTEGER
               IHI     (global input) INTEGER It is assumed that sub( A  )  is
               already  upper triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1
               and columns JA:JA+JLO-2 and JA+JHI:JA+N-1. See Further Details.
               If N > 0,

       A       (local input/local output) COMPLEX*16 pointer into the
               local memory to an array of dimension (LLD_A,LOCc(JA+N-1)).  On
               entry, this array contains the local pieces of the N-by-N  gen-
               eral  distributed  matrix  sub( A ) to be reduced. On exit, the
               upper triangle and the first subdiagonal of sub( A ) are  over-
               written  with the upper Hessenberg matrix H, and the ele- ments
               below the first subdiagonal, with the array  TAU,  repre-  sent
               the unitary matrix Q as a product of 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.

       TAU     (local output) COMPLEX*16 array, dimension LOCc(JA+N-2)
               The scalar factors of the elementary  reflectors  (see  Further
               Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are set
               to zero. TAU is tied to the distributed matrix A.

       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 >= NB + MAX( NpA0, NB )

               where NB = MB_A = NB_A, IROFFA =  MOD(  IA-1,  NB  ),  IAROW  =
               INDXG2P(  IA,  NB,  MYROW,  RSRC_A,  NPROW  ),  NpA0  = NUMROC(
               IHI+IROFFA, NB, MYROW, IAROW, NPROW ),

               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    (local 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  (ihi-ilo)  elementary
       reflectors

          Q = H(ilo) H(ilo+1) . . . H(ihi-1).

       Each H(i) has the form

          H(i) = I - tau * v * v'

       where  tau is a complex scalar, and v is a complex vector with v(1:i) =
       0, v(i+1) = 1 and v(ihi+1:n) = 0;  v(i+2:ihi)  is  stored  on  exit  in
       A(ia+ilo+i:ia+ihi-1,ja+ilo+i-2), and tau in TAU(ja+ilo+i-2).

       The  contents  of  A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follo-
       wing example, with n = 7, ilo = 2 and ihi = 6:

       on entry                         on exit

       ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
       (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
       (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
       (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
       (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
       (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
       (                         a )    (                          a )

       where a denotes an element of the original matrix sub( A ), h denotes a
       modified  element  of  the upper Hessenberg matrix H, and vi denotes an
       element of the vector defining H(ja+ilo+i-2).


ALIGNMENT REQUIREMENTS
       The distributed submatrix sub( A ) must verify some  alignment  proper-
       ties, namely the following expression should be true:
       ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )




ScaLAPACK routine               31 October 2017                     PZGEHD2(3)