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NAME

       PCGEHD2  - reduce a complex general distributed matrix sub( A ) to upper Hessenberg form H
       by an unitary similarity transformation

SYNOPSIS

       SUBROUTINE PCGEHD2( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

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

           INTEGER         DESCA( * )

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

PURPOSE

       PCGEHD2 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 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  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 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 general distributed matrix sub( A ) to  be
               reduced.  On  exit,  the  upper triangle and the first subdiagonal of sub( A ) are
               overwritten 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 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 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 )