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dlahrd.f(3)				      LAPACK				      dlahrd.f(3)

NAME
       dlahrd.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dlahrd (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
	   DLAHRD reduces the first nb columns of a general rectangular matrix A so that elements
	   below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed
	   to apply the transformation to the unreduced part of A.

Function/Subroutine Documentation
   subroutine dlahrd (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)
       DLAHRD reduces the first nb columns of a general rectangular matrix A so that elements
       below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to
       apply the transformation to the unreduced part of A.

       Purpose:

	    DLAHRD 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 OBSOLETE auxiliary routine.
	    This routine will be 'deprecated' in a  future release.
	    Please use the new routine DLAHR2 instead.

       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.

	   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   h   a   a   a )
		( a   h   a   a   a )
		( a   h   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).

       Definition at line 170 of file dlahrd.f.

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

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