NAME

       PDLAHRD  -  reduce  the first NB columns of a real general N-by-(N-K+1) distributed matrix
       A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero

SYNOPSIS

       SUBROUTINE PDLAHRD( N, K, NB, A, IA, JA, DESCA, TAU, T, Y, IY, JY, DESCY, WORK )

           INTEGER         IA, IY, JA, JY, K, N, NB

           INTEGER         DESCA( * ), DESCY( * )

           DOUBLE          PRECISION A( * ), T( * ), TAU( * ), WORK( * ), Y( * )

PURPOSE

       PDLAHRD reduces the first NB columns of a real  general  N-by-(N-K+1)  distributed  matrix
       A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero. The reduction
       is performed by an orthogo- nal similarity transformation Q' * A * Q. The routine  returns
       the  matrices  V  and  T  which  determine Q as a block reflector I - V*T*V', and also the
       matrix Y = A * V * T.

       This is an auxiliary routine called by PDGEHRD. In the following comments sub( A ) denotes
       A(IA:IA+N-1,JA:JA+N-1).

ARGUMENTS

       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.

       K       (global input) INTEGER
               The offset for the reduction. Elements below the k-th subdiagonal in the first  NB
               columns are reduced to zero.

       NB      (global input) INTEGER
               The number of 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-K)). On entry, this
               array contains the the local pieces of the N-by-(N-K+1) general distributed matrix
               A(IA:IA+N-1,JA:JA+N-K). On exit, the elements on and above the k-th subdiagonal in
               the first NB columns are  overwritten  with  the  corresponding  elements  of  the
               reduced  distributed  matrix;  the  elements  below the k-th subdiagonal, with the
               array TAU, represent the matrix Q as a product of elementary reflectors. The other
               columns   of  A(IA:IA+N-1,JA:JA+N-K)  are  unchanged.  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.

       TAU     (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-2)
               The scalar factors of the elementary reflectors (see Further Details). TAU is tied
               to the distributed matrix A.

       T       (local output) DOUBLE PRECISION array, dimension (NB_A,NB_A)
               The upper triangular matrix T.

       Y       (local output) DOUBLE PRECISION pointer into the local memory
               to an array of dimension (LLD_Y,NB_A). On exit,  this  array  contains  the  local
               pieces of the N-by-NB distributed matrix Y. LLD_Y >= LOCr(IA+N-1).

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

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

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

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

FURTHER DETAILS

       The matrix Q is represented as a product of nb 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+k-1)  =  0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(ia+i+k:ia+n-1,ja+i-1), and
       tau in TAU(ja+i-1).

       The elements of the vectors v together form the (n-k+1)-by-nb matrix V  which  is  needed,
       with  T  and  Y, to apply the transformation to the unreduced part of the matrix, using an
       update of the form: A(ia:ia+n-1,ja:ja+n-k) := (I-V*T*V')*(A(ia:ia+n-1,ja:ja+n-k)-Y*V').

       The contents of A(ia:ia+n-1,ja:ja+n-k) on exit are illustrated by  the  following  example
       with n = 7, k = 3 and nb = 2:

          ( a   h   a   a   a )
          ( a   h   a   a   a )
          ( a   h   a   a   a )
          ( h   h   a   a   a )
          ( v1  h   a   a   a )
          ( v1  v2  a   a   a )
          ( v1  v2  a   a   a )

       where a denotes an element of the original matrix
       A(ia:ia+n-1,ja:ja+n-k), h denotes a modified element of the upper Hessenberg matrix H, and
       vi denotes an element of the vector defining H(i).