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DLAHRD(3F)							    DLAHRD(3F)


NAME    [Toc]    [Back]

     DLAHRD - reduce 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    [Toc]    [Back]

     SUBROUTINE	DLAHRD(	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    [Toc]    [Back]

     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' * 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    [Toc]    [Back]

     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.

     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	(NB,NB)
	     The upper triangular matrix T.






									Page 1






DLAHRD(3F)							    DLAHRD(3F)



     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
     unreduced 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   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).
DLAHRD(3F)							    DLAHRD(3F)


NAME    [Toc]    [Back]

     DLAHRD - reduce 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    [Toc]    [Back]

     SUBROUTINE	DLAHRD(	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    [Toc]    [Back]

     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' * 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    [Toc]    [Back]

     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.

     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	(NB,NB)
	     The upper triangular matrix T.






									Page 1






DLAHRD(3F)							    DLAHRD(3F)



     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
     unreduced 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   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).


									PPPPaaaaggggeeee 2222
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