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NAME

       PDGEHD2  - reduce a real general distributed matrix sub( A ) to upper Hessenberg form H by
       an orthogonal similarity transforma- tion

SYNOPSIS

       SUBROUTINE PDGEHD2( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

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

           INTEGER         DESCA( * )

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

PURPOSE

       PDGEHD2 reduces a real general distributed matrix sub( A ) to upper Hessenberg form  H  by
       an  orthogonal  similarity  transforma-  tion:   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) DOUBLE PRECISION 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 orthogonal 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) DOUBLE PRECISION 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) 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  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 real scalar, and v is a real 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 )