Provided by: liblapack-doc_3.3.1-1_all #### NAME

```       LAPACK-3  -  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

```

#### SYNOPSIS

```       SUBROUTINE DLAHR2( 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

```       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.

```

#### 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.
K < N.

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 (LDT,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.

```

#### FURTHERDETAILS

```        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.

LAPACK auxiliary routine (version 3.3.1)   April 2011                            DLAHR2(3lapack)
```