DLAHRD (3lapack)

reduce 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

SYNOPSIS

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

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

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

PURPOSE

  DLAHRD 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 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 DGEHRD.

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) DOUBLE PRECISION 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 elementary 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) DOUBLE PRECISION array, dimension (NB)
          The scalar factors of the elementary reflectors. See Further
          Details.

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

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

  Y       (output) DOUBLE PRECISION 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).