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NAME

       PZHETRD  -  reduce  a  complex  Hermitian  matrix sub( A ) to Hermitian
       tridiagonal form T by an unitary similarity transformation

SYNOPSIS

       SUBROUTINE PZHETRD( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK,  LWORK,
                           INFO )

           CHARACTER       UPLO

           INTEGER         IA, INFO, JA, LWORK, N

           INTEGER         DESCA( * )

           DOUBLE          PRECISION D( * ), E( * )

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

PURPOSE

       PZHETRD  reduces  a  complex  Hermitian  matrix  sub(  A ) to Hermitian
       tridiagonal form T by an unitary similarity transformation: Q’ * sub( A
       ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA: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

       UPLO    (global input) CHARACTER
               Specifies whether the upper or lower  triangular  part  of  the
               Hermitian matrix sub( A ) is stored:
               = ’U’:  Upper triangular
               = ’L’:  Lower triangular

       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.

       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 Hermitian
               distributed matrix sub( A ).  If UPLO = ’U’, the leading N-by-N
               upper triangular part of sub( A ) contains the upper triangular
               part of the matrix, and its strictly lower triangular  part  is
               not  referenced.  If  UPLO  =  ’L’,  the  leading  N-by-N lower
               triangular part of sub( A ) contains the lower triangular  part
               of  the  matrix,  and its strictly upper triangular part is not
               referenced. On exit, if UPLO =  ’U’,  the  diagonal  and  first
               superdiagonal   of   sub(   A   )  are  over-  written  by  the
               corresponding elements of the tridiagonal  matrix  T,  and  the
               elements  above  the  first  superdiagonal, with the array TAU,
               represent the unitary matrix  Q  as  a  product  of  elementary
               reflectors;  if  UPLO = ’L’, the diagonal and first subdiagonal
               of sub( A ) are overwritten by the  corresponding  elements  of
               the  tridiagonal  matrix  T,  and  the elements below the first
               subdiagonal, with the array TAU, represent 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.

       D       (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1)
               The diagonal elements of  the  tridiagonal  matrix  T:  D(i)  =
               A(i,i). D is tied to the distributed matrix A.

       E       (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1)
               if   UPLO  =  ’U’,  LOCc(JA+N-2)  otherwise.  The  off-diagonal
               elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO =
               ’U’,  E(i)  =  A(i+1,i)  if  UPLO  =  ’L’.  E  is  tied  to the
               distributed matrix A.

       TAU     (local output) COMPLEX*16, array, dimension
               LOCc(JA+N-1). This array contains the scalar factors TAU of the
               elementary reflectors. 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 >= MAX( NB * ( NP +1 ), 3 * NB )

               where NB = MB_A = NB_A, NP = NUMROC( N, NB, MYROW, IAROW, NPROW
               ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, 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    (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

       If  UPLO  = ’U’, the matrix Q is represented as a product of elementary
       reflectors

          Q = H(n-1) . . . H(2) H(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(i+1:n)
       = 0 and v(i) = 1; v(1:i-1) is stored on exit in
       A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1).

       If  UPLO  = ’L’, the matrix Q is represented as a product of elementary
       reflectors

          Q = H(1) H(2) . . . H(n-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 and v(i+1) = 1; v(i+2:n) is stored on exit in
       A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1).

       The  contents  of  sub(  A  )  on exit are illustrated by the following
       examples with n = 5:

       if UPLO = ’U’:                       if UPLO = ’L’:

         (  d   e   v2  v3  v4 )              (  d                  )
         (      d   e   v3  v4 )              (  e   d              )
         (          d   e   v4 )              (  v1  e   d          )
         (              d   e  )              (  v1  v2  e   d      )
         (                  d  )              (  v1  v2  v3  e   d  )

       where d and e denote diagonal and off-diagonal elements of  T,  and  vi
       denotes an element of the vector defining H(i).

       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 .AND. IROFFA.EQ.0 ) with IROFFA =
       MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ).