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slahrd

 NAME
      SLAHRD - reduce the first NB columns of a real general n-
      by-(n-k+1) matrix A so that elements below the k-th subdiag-
      onal are zero

 SYNOPSIS
      SUBROUTINE SLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )

          INTEGER        K, LDA, LDT, LDY, N, NB

          REAL           A( LDA, * ), T( LDT, NB ), TAU( NB ), Y(
                         LDY, NB )

 PURPOSE
      SLAHRD reduces the first NB columns of a real general n-by-
      (n-k+1) matrix A so that elements below the k-th subdiagonal
      are zero. The reduction is performed by an orthogonal simi-
      larity 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 SGEHRD.

 ARGUMENTS
      N       (input) INTEGER
              The order of the matrix A.

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

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

      A       (input/output) REAL array, dimension (LDA,N-K+1)
              On entry, the n-by-(n-k+1) general matrix A.  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 matrix; the
              elements below the k-th subdiagonal, with the array
              TAU, represent the matrix Q as a product of elemen-
              tary reflectors. The other columns of A are
              unchanged. See Further Details.  LDA     (input)
              INTEGER The leading dimension of the array A.  LDA
              >= max(1,N).

      TAU     (output) REAL array, dimension (NB)
              The scalar factors of the elementary reflectors. See
              Further Details.

      T       (output) REAL array, dimension (NB,NB)
              The upper triangular matrix T.

      LDT     (input) INTEGER
              The leading dimension of the array T.  LDT >= NB.

      Y       (output) REAL array, dimension (LDY,NB)
              The n-by-nb matrix Y.

      LDY     (input) INTEGER
              The leading dimension of the array Y. LDY >= N.

 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(i+k+1:n,i), and tau in TAU(i).

      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 := (I - V*T*V') * (A - Y*V').

      The contents of A 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, h
      denotes a modified element of the upper Hessenberg matrix H,
      and vi denotes an element of the vector defining H(i).