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NAME

       PDLATRD  -  reduce  NB  rows and columns of a real symmetric distributed matrix sub( A ) =
       A(IA:IA+N-1,JA:JA+N-1)  to  symmetric  tridiagonal  form  by  an   orthogonal   similarity
       transformation Q' * sub( A ) * Q,

SYNOPSIS

       SUBROUTINE PDLATRD( UPLO, N, NB, A, IA, JA, DESCA, D, E, TAU, W, IW, JW, DESCW, WORK )

           CHARACTER       UPLO

           INTEGER         IA, IW, JA, JW, N, NB

           INTEGER         DESCA( * ), DESCW( * )

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

PURPOSE

       PDLATRD  reduces  NB  rows  and  columns of a real symmetric distributed matrix sub( A ) =
       A(IA:IA+N-1,JA:JA+N-1)  to  symmetric  tridiagonal  form  by  an   orthogonal   similarity
       transformation  Q'  *  sub(  A ) * Q, and returns the matrices V and W which are needed to
       apply the transformation to the unreduced part of sub( A ).

       If UPLO = 'U', PDLATRD reduces the last NB rows and columns of  a  matrix,  of  which  the
       upper triangle is supplied;
       if  UPLO  =  'L',  PDLATRD reduces the first NB rows and columns of a matrix, of which the
       lower triangle is supplied.

       This is an auxiliary routine called by PDSYTRD.

       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.

       NB      (global input) INTEGER
               The number of rows and columns to be reduced.

       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 last NB columns
               have been reduced to tridiagonal form, with the diagonal elements overwriting  the
               diagonal elements of sub( A ); the elements above the diagonal with the array TAU,
               represent the orthogonal matrix Q as a product of elementary reflectors. If UPLO =
               'L', the first NB columns have been reduced to tridiagonal form, with the diagonal
               elements overwriting the diagonal elements of sub( A );  the  elements  below  the
               diagonal  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.

       W       (local output) DOUBLE PRECISION pointer into the local memory
               to an array of dimension (LLD_W,NB_W), This array contains the local pieces of the
               N-by-NB_W matrix W required to update the unreduced part of sub( A ).

       IW      (global input) INTEGER
               The row index in the global array W indicating the first row of sub( W ).

       JW      (global input) INTEGER
               The column index in the global array W indicating the first column of sub( W ).

       DESCW   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix W.

       WORK    (local workspace) DOUBLE PRECISION array, dimension (NB_A)

FURTHER DETAILS

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

          Q = H(n) H(n-1) . . . H(n-nb+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:n) = 0 and v(i-1) = 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(nb).

       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 elements of the vectors v together form the N-by-NB matrix V which is needed, with  W,
       to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k
       update of the form: sub( A ) := sub( A ) - V*W' - W*V'.

       The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

       if UPLO = 'U':                       if UPLO = 'L':

         (  a   a   a   v4  v5 )              (  d                  )
         (      a   a   v4  v5 )              (  1   d              )
         (          a   1   v5 )              (  v1  1   a          )
         (              d   1  )              (  v1  v2  a   a      )
         (                  d  )              (  v1  v2  a   a   a  )

       where d denotes a diagonal element of the reduced matrix, a  denotes  an  element  of  the
       original matrix that is unchanged, and vi denotes an element of the vector defining H(i).