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SUBROUTINE <a name="DLAHRD.1"></a><a href="dlahrd.f.html#DLAHRD.1">DLAHRD</a>( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> INTEGER K, LDA, LDT, LDY, N, NB
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
$ Y( LDY, NB )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DLAHRD.18"></a><a href="dlahrd.f.html#DLAHRD.1">DLAHRD</a> reduces the first NB columns of a real general n-by-(n-k+1)
</span><span class="comment">*</span><span class="comment"> matrix A so that elements below the k-th subdiagonal are zero. The
</span><span class="comment">*</span><span class="comment"> reduction is performed by an orthogonal similarity transformation
</span><span class="comment">*</span><span class="comment"> Q' * A * Q. The routine returns the matrices V and T which determine
</span><span class="comment">*</span><span class="comment"> Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This is an OBSOLETE auxiliary routine.
</span><span class="comment">*</span><span class="comment"> This routine will be 'deprecated' in a future release.
</span><span class="comment">*</span><span class="comment"> Please use the new routine <a name="DLAHR2.26"></a><a href="dlahr2.f.html#DLAHR2.1">DLAHR2</a> instead.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The order of the matrix A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> K (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The offset for the reduction. Elements below the k-th
</span><span class="comment">*</span><span class="comment"> subdiagonal in the first NB columns are reduced to zero.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> NB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of columns to be reduced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1)
</span><span class="comment">*</span><span class="comment"> On entry, the n-by-(n-k+1) general matrix A.
</span><span class="comment">*</span><span class="comment"> On exit, the elements on and above the k-th subdiagonal in
</span><span class="comment">*</span><span class="comment"> the first NB columns are overwritten with the corresponding
</span><span class="comment">*</span><span class="comment"> elements of the reduced matrix; the elements below the k-th
</span><span class="comment">*</span><span class="comment"> subdiagonal, with the array TAU, represent the matrix Q as a
</span><span class="comment">*</span><span class="comment"> product of elementary reflectors. The other columns of A are
</span><span class="comment">*</span><span class="comment"> unchanged. See Further Details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDA (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array A. LDA >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TAU (output) DOUBLE PRECISION array, dimension (NB)
</span><span class="comment">*</span><span class="comment"> The scalar factors of the elementary reflectors. See Further
</span><span class="comment">*</span><span class="comment"> Details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> T (output) DOUBLE PRECISION array, dimension (LDT,NB)
</span><span class="comment">*</span><span class="comment"> The upper triangular matrix T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDT (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array T. LDT >= NB.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Y (output) DOUBLE PRECISION array, dimension (LDY,NB)
</span><span class="comment">*</span><span class="comment"> The n-by-nb matrix Y.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDY (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array Y. LDY >= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The matrix Q is represented as a product of nb elementary reflectors
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q = H(1) H(2) . . . H(nb).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Each H(i) has the form
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> H(i) = I - tau * v * v'
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where tau is a real scalar, and v is a real vector with
</span><span class="comment">*</span><span class="comment"> v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
</span><span class="comment">*</span><span class="comment"> A(i+k+1:n,i), and tau in TAU(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The elements of the vectors v together form the (n-k+1)-by-nb matrix
</span><span class="comment">*</span><span class="comment"> V which is needed, with T and Y, to apply the transformation to the
</span><span class="comment">*</span><span class="comment"> unreduced part of the matrix, using an update of the form:
</span><span class="comment">*</span><span class="comment"> A := (I - V*T*V') * (A - Y*V').
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The contents of A on exit are illustrated by the following example
</span><span class="comment">*</span><span class="comment"> with n = 7, k = 3 and nb = 2:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ( a h a a a )
</span><span class="comment">*</span><span class="comment"> ( a h a a a )
</span><span class="comment">*</span><span class="comment"> ( a h a a a )
</span><span class="comment">*</span><span class="comment"> ( h h a a a )
</span><span class="comment">*</span><span class="comment"> ( v1 h a a a )
</span><span class="comment">*</span><span class="comment"> ( v1 v2 a a a )
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