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NAME

       PCLATRD  - reduce NB rows and columns of a complex Hermitian distributed matrix sub( A ) =
       A(IA:IA+N-1,JA:JA+N-1) to complex tridiagonal form by an unitary 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 )

SYNOPSIS

       SUBROUTINE PCLATRD( 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( * )

           REAL            D( * ), E( * )

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

PURPOSE

       PCLATRD reduces NB rows and columns of a complex Hermitian distributed matrix sub( A  )  =
       A(IA:IA+N-1,JA:JA+N-1) to complex tridiagonal form by an unitary 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',  PCLATRD  reduces the last NB rows and columns of a matrix, of which the
       upper triangle is supplied;
       if UPLO = 'L', PCLATRD reduces the first NB rows and columns of a  matrix,  of  which  the
       lower triangle is supplied.

       This is an auxiliary routine called by PCHETRD.

       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.

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

       A       (local input/local output) COMPLEX 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  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  unitary  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  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) REAL 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) REAL 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, 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) COMPLEX 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) COMPLEX 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 complex scalar, and v is a complex 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 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  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 Hermitian 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).