Loading...
Searching...
No Matches

## dlahr2()

subroutine dlahr2 | ( | integer | n,
|  | integer | k,
|  | integer | nb,
|  | double precision, dimension( lda, * ) | a,
|  | integer | lda,
|  | double precision, dimension( nb ) | tau,
|  | double precision, dimension( ldt, nb ) | t,
|  | integer | ldt,
|  | double precision, dimension( ldy, nb ) | y,
|  | integer | ldy )

DLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

Download DLAHR2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> DLAHR2 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**T * A * Q. The routine returns the matrices V and T which determine
      !> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
      !>
      !> This is an auxiliary routine called by DGEHRD.
      !>

Parameters

  [in]

  N

      !>          N is INTEGER
      !>          The order of the matrix A.
      !>

  [in]

  K

      !>          K is INTEGER
      !>          The offset for the reduction. Elements below the k-th
      !>          subdiagonal in the first NB columns are reduced to zero.
      !>          K < N.
      !>

  [in]

  NB

      !>          NB is INTEGER
      !>          The number of columns to be reduced.
      !>

  [in,out]

  A

      !>          A is 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.
      !>

  [in]

  LDA

      !>          LDA is INTEGER
      !>          The leading dimension of the array A.  LDA >= max(1,N).
      !>

  [out]

  TAU

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

  [out]

  T

      !>          T is DOUBLE PRECISION array, dimension (LDT,NB)
      !>          The upper triangular matrix T.
      !>

  [in]

  LDT

      !>          LDT is INTEGER
      !>          The leading dimension of the array T.  LDT >= NB.
      !>

  [out]

  Y

      !>          Y is DOUBLE PRECISION array, dimension (LDY,NB)
      !>          The n-by-nb matrix Y.
      !>

  [in]

  LDY

      !>          LDY is INTEGER
      !>          The leading dimension of the array Y. LDY >= N.
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.

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**T
      !>
      !>  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**T) * (A - Y*V**T).
      !>
      !>  The contents of A on exit are illustrated by the following example
      !>  with n = 7, k = 3 and nb = 2:
      !>
      !>     ( a   a   a   a   a )
      !>     ( a   a   a   a   a )
      !>     ( a   a   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).
      !>
      !>  This subroutine is a slight modification of LAPACK-3.0's DLAHRD
      !>  incorporating improvements proposed by Quintana-Orti and Van de
      !>  Gejin. Note that the entries of A(1:K,2:NB) differ from those
      !>  returned by the original LAPACK-3.0's DLAHRD routine. (This
      !>  subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
      !>

References:
  Gregorio Quintana-Orti and Robert van de Geijn, "Improving the performance of reduction to Hessenberg form," ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.