slahr2()

subroutine slahr2	(	integer	n,
		integer	k,
		integer	nb,
		real, dimension( lda, * )	a,
		integer	lda,
		real, dimension( nb )	tau,
		real, dimension( ldt, nb )	t,
		integer	ldt,
		real, dimension( ldy, nb )	y,
		integer	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.

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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

[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 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. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !>
[out]	TAU	!> TAU is REAL array, dimension (NB) !> The scalar factors of the elementary reflectors. See Further !> Details. !>
[out]	T	!> T is REAL 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 REAL 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 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.