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DLAHR2(1)	    LAPACK auxiliary routine (version 3.2)	     DLAHR2(1)

NAME
       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

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' *
       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 subdiago‐
	       nal 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 ele‐
	       ments 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 reflec‐
	       tors. 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.

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 unre‐
       duced 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	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	 modi‐
       fied  element  of the upper Hessenberg matrix H, and vi denotes an ele‐
       ment of the vector defining H(i).
       This file 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	routine.   This	 function  is  not  backward  compatible  with
       LAPACK3.0.

 LAPACK auxiliary routine (versioNovember 2008			     DLAHR2(1)
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