PDGEHD2(3)    ScaLAPACK routine of NEC Numeric Library Collection   PDGEHD2(3)



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 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
       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  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  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 )




ScaLAPACK routine               31 October 2017                     PDGEHD2(3)