NAME

       PSLAHRD   -   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 PSLAHRD( 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( * )

           REAL            A( * ), T( * ), TAU( * ), WORK( * ), Y( * )

PURPOSE

       PSLAHRD  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  PSGEHRD.  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) REAL 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) REAL 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) REAL array, dimension (NB_A,NB_A)
               The upper triangular matrix T.

       Y       (local output) REAL 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) REAL 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).