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NAME

       PDSYTD2  -  reduce  a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an
       orthogonal similarity transformation

SYNOPSIS

       SUBROUTINE PDSYTD2( 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

       PDSYTD2 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
               >= 3*N.

               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

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