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NAME

       PDSYTRD  -  reduce  a  real  symmetric  matrix  sub(  A  ) to symmetric
       tridiagonal form T by an orthogonal similarity transformation

SYNOPSIS

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

           CHARACTER       UPLO

           INTEGER         IA, INFO, JA, LWORK, N

           INTEGER         DESCA( * )

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

PURPOSE

       PDSYTRD  reduces  a  real  symmetric  matrix  sub(  A  )  to  symmetric
       tridiagonal  form  T  by  an orthogonal 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
               symmetric 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) 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  symmetric
               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  orthogonal  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 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.

       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) DOUBLE PRECISION, 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) 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 >= 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 real scalar, and v is a real 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 real scalar, and v is a real 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 ).