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dlahr2.f(3)				      LAPACK				      dlahr2.f(3)

NAME
       dlahr2.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dlahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, 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.

Function/Subroutine Documentation
   subroutine dlahr2 (integerN, integerK, integerNB, double precision, dimension( lda, * )A,
       integerLDA, double precision, dimension( nb )TAU, double precision, dimension( ldt, nb )T,
       integerLDT, double precision, dimension( ldy, nb )Y, integerLDY)
       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.

       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:
	   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 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

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

	   TAU

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

	   T

		     T is DOUBLE PRECISION 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 DOUBLE PRECISION 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.

       Date:
	   September 2012

       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.

       Definition at line 182 of file dlahr2.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      dlahr2.f(3)
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