slahr2.f - Man Page

subroutine slahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
SLAHR2 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.

SLAHR2 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.  

Purpose:

 SLAHR2 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 SGEHRD.

Parameters

N

          N is INTEGER
          The order of the matrix A.

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.

NB

          NB is INTEGER
          The number of columns to be reduced.

A

          A is 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 elementary reflectors. The other columns of A are
          unchanged. See Further Details.

LDA

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

TAU

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

T

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

LDT

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

Y

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

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 SLAHRD
  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 SLAHRD routine. (This
  subroutine is not backward compatible with LAPACK-3.0's SLAHRD.)

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.

Definition at line 180 of file slahr2.f.