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NAME

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

SYNOPSIS

       SUBROUTINE PDGEHRD( 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

       PDGEHRD  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+ILO-2 and JA+IHI: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*NB + NB*MAX( IHIP+1, IHLP+INLQ )

               where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD(
               JA-1,  NB  ),  IOFF = MOD( IA+ILO-2, NB ), IAROW = INDXG2P( IA,
               NB, MYROW, RSRC_A, NPROW ),  IHIP  =  NUMROC(  IHI+IROFFA,  NB,
               MYROW,  IAROW,  NPROW  ), ILROW = INDXG2P( IA+ILO-1, NB, MYROW,
               RSRC_A, NPROW ), IHLP  =  NUMROC(  IHI-ILO+IOFF+1,  NB,  MYROW,
               ILROW,  NPROW  ), ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, CSRC_A,
               NPCOL ), INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL,  ILCOL,  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.

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 follow-
       ing 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 )