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SUBROUTINE <a name="DLABRD.1"></a><a href="dlabrd.f.html#DLABRD.1">DLABRD</a>( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, 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 LDA, LDX, LDY, M, N, NB
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
$ TAUQ( * ), X( LDX, * ), Y( LDY, * )
<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="DLABRD.19"></a><a href="dlabrd.f.html#DLABRD.1">DLABRD</a> reduces the first NB rows and columns of a real general
</span><span class="comment">*</span><span class="comment"> m by n matrix A to upper or lower bidiagonal form by an orthogonal
</span><span class="comment">*</span><span class="comment"> transformation Q' * A * P, and returns the matrices X and Y which
</span><span class="comment">*</span><span class="comment"> are needed to apply the transformation to the unreduced part of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
</span><span class="comment">*</span><span class="comment"> bidiagonal form.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This is an auxiliary routine called by <a name="DGEBRD.27"></a><a href="dgebrd.f.html#DGEBRD.1">DGEBRD</a>
</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"> M (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of rows in the matrix A.
</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 number of columns in the matrix A.
</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 leading rows and columns of A 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)
</span><span class="comment">*</span><span class="comment"> On entry, the m by n general matrix to be reduced.
</span><span class="comment">*</span><span class="comment"> On exit, the first NB rows and columns of the matrix are
</span><span class="comment">*</span><span class="comment"> overwritten; the rest of the array is unchanged.
</span><span class="comment">*</span><span class="comment"> If m >= n, elements on and below the diagonal in the first NB
</span><span class="comment">*</span><span class="comment"> columns, with the array TAUQ, represent the orthogonal
</span><span class="comment">*</span><span class="comment"> matrix Q as a product of elementary reflectors; and
</span><span class="comment">*</span><span class="comment"> elements above the diagonal in the first NB rows, with the
</span><span class="comment">*</span><span class="comment"> array TAUP, represent the orthogonal matrix P as a product
</span><span class="comment">*</span><span class="comment"> of elementary reflectors.
</span><span class="comment">*</span><span class="comment"> If m < n, elements below the diagonal in the first NB
</span><span class="comment">*</span><span class="comment"> columns, with the array TAUQ, represent the orthogonal
</span><span class="comment">*</span><span class="comment"> matrix Q as a product of elementary reflectors, and
</span><span class="comment">*</span><span class="comment"> elements on and above the diagonal in the first NB rows,
</span><span class="comment">*</span><span class="comment"> with the array TAUP, represent the orthogonal matrix P as
</span><span class="comment">*</span><span class="comment"> a product of elementary reflectors.
</span><span class="comment">*</span><span class="comment"> 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,M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> D (output) DOUBLE PRECISION array, dimension (NB)
</span><span class="comment">*</span><span class="comment"> The diagonal elements of the first NB rows and columns of
</span><span class="comment">*</span><span class="comment"> the reduced matrix. D(i) = A(i,i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> E (output) DOUBLE PRECISION array, dimension (NB)
</span><span class="comment">*</span><span class="comment"> The off-diagonal elements of the first NB rows and columns of
</span><span class="comment">*</span><span class="comment"> the reduced matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TAUQ (output) DOUBLE PRECISION array dimension (NB)
</span><span class="comment">*</span><span class="comment"> The scalar factors of the elementary reflectors which
</span><span class="comment">*</span><span class="comment"> represent the orthogonal matrix Q. See Further Details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TAUP (output) DOUBLE PRECISION array, dimension (NB)
</span><span class="comment">*</span><span class="comment"> The scalar factors of the elementary reflectors which
</span><span class="comment">*</span><span class="comment"> represent the orthogonal matrix P. See Further Details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> X (output) DOUBLE PRECISION array, dimension (LDX,NB)
</span><span class="comment">*</span><span class="comment"> The m-by-nb matrix X required to update the unreduced part
</span><span class="comment">*</span><span class="comment"> of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDX (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array X. LDX >= M.
</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 required to update the unreduced part
</span><span class="comment">*</span><span class="comment"> of A.
</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 matrices Q and P are represented as products of elementary
</span><span class="comment">*</span><span class="comment"> reflectors:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Each H(i) and G(i) has the form:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where tauq and taup are real scalars, and v and u are real vectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
</span><span class="comment">*</span><span class="comment"> A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
</span><span class="comment">*</span><span class="comment"> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
</span><span class="comment">*</span><span class="comment"> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
</span><span class="comment">*</span><span class="comment"> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The elements of the vectors v and u together form the m-by-nb matrix
</span><span class="comment">*</span><span class="comment"> V and the nb-by-n matrix U' which are needed, with X and Y, to apply
</span><span class="comment">*</span><span class="comment"> the transformation to the unreduced part of the matrix, using a block
</span><span class="comment">*</span><span class="comment"> update of the form: A := A - V*Y' - X*U'.
</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 examples
</span><span class="comment">*</span><span class="comment"> with nb = 2:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
</span><span class="comment">*</span><span class="comment"> ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
</span><span class="comment">*</span><span class="comment"> ( v1 v2 a a a ) ( v1 1 a a a a )
</span><span class="comment">*</span><span class="comment"> ( v1 v2 a a a ) ( v1 v2 a a a a )
</span><span class="comment">*</span><span class="comment"> ( v1 v2 a a a ) ( v1 v2 a a a a )
</span><span class="comment">*</span><span class="comment"> ( v1 v2 a a a )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where a denotes an element of the original matrix which is unchanged,
</span><span class="comment">*</span><span class="comment"> vi denotes an element of the vector defining H(i), and ui an element
</span><span class="comment">*</span><span class="comment"> of the vector defining G(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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