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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).