SLAHRD(l)								 )								 SLAHRD(l)

NAME

       SLAHRD - 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

       SUBROUTINE SLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )

	   INTEGER	  K, LDA, LDT, LDY, N, NB

	   REAL 	  A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )

PURPOSE

       SLAHRD  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 SGEHRD.

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.

       NB      (input) INTEGER
	       The number of columns to be reduced.

       A       (input/output) REAL 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) REAL array, dimension (NB)
	       The scalar factors of the elementary reflectors. See Further Details.

       T       (output) REAL array, dimension (LDT,NB)
	       The upper triangular matrix T.

       LDT     (input) INTEGER
	       The leading dimension of the array T.  LDT >= NB.

       Y       (output) REAL 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 ele-
       ment of the vector defining H(i).

LAPACK version 3.0						   15 June 2000 							 SLAHRD(l)