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version 1.8, 2011/07/22 07:38:06 version 1.9, 2011/11/21 20:42:54
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   *> \brief \b DLABRD
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLABRD + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlabrd.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlabrd.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlabrd.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
   *                          LDY )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            LDA, LDX, LDY, M, N, NB
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
   *      $                   TAUQ( * ), X( LDX, * ), Y( LDY, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLABRD reduces the first NB rows and columns of a real general
   *> m by n matrix A to upper or lower bidiagonal form by an orthogonal
   *> transformation Q**T * A * P, and returns the matrices X and Y which
   *> are needed to apply the transformation to the unreduced part of A.
   *>
   *> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
   *> bidiagonal form.
   *>
   *> This is an auxiliary routine called by DGEBRD
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of rows in the matrix A.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of columns in the matrix A.
   *> \endverbatim
   *>
   *> \param[in] NB
   *> \verbatim
   *>          NB is INTEGER
   *>          The number of leading rows and columns of A to be reduced.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   *>          On entry, the m by n general matrix to be reduced.
   *>          On exit, the first NB rows and columns of the matrix are
   *>          overwritten; the rest of the array is unchanged.
   *>          If m >= n, elements on and below the diagonal in the first NB
   *>            columns, with the array TAUQ, represent the orthogonal
   *>            matrix Q as a product of elementary reflectors; and
   *>            elements above the diagonal in the first NB rows, with the
   *>            array TAUP, represent the orthogonal matrix P as a product
   *>            of elementary reflectors.
   *>          If m < n, elements below the diagonal in the first NB
   *>            columns, with the array TAUQ, represent the orthogonal
   *>            matrix Q as a product of elementary reflectors, and
   *>            elements on and above the diagonal in the first NB rows,
   *>            with the array TAUP, represent the orthogonal matrix P as
   *>            a product of elementary reflectors.
   *>          See Further Details.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,M).
   *> \endverbatim
   *>
   *> \param[out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension (NB)
   *>          The diagonal elements of the first NB rows and columns of
   *>          the reduced matrix.  D(i) = A(i,i).
   *> \endverbatim
   *>
   *> \param[out] E
   *> \verbatim
   *>          E is DOUBLE PRECISION array, dimension (NB)
   *>          The off-diagonal elements of the first NB rows and columns of
   *>          the reduced matrix.
   *> \endverbatim
   *>
   *> \param[out] TAUQ
   *> \verbatim
   *>          TAUQ is DOUBLE PRECISION array dimension (NB)
   *>          The scalar factors of the elementary reflectors which
   *>          represent the orthogonal matrix Q. See Further Details.
   *> \endverbatim
   *>
   *> \param[out] TAUP
   *> \verbatim
   *>          TAUP is DOUBLE PRECISION array, dimension (NB)
   *>          The scalar factors of the elementary reflectors which
   *>          represent the orthogonal matrix P. See Further Details.
   *> \endverbatim
   *>
   *> \param[out] X
   *> \verbatim
   *>          X is DOUBLE PRECISION array, dimension (LDX,NB)
   *>          The m-by-nb matrix X required to update the unreduced part
   *>          of A.
   *> \endverbatim
   *>
   *> \param[in] LDX
   *> \verbatim
   *>          LDX is INTEGER
   *>          The leading dimension of the array X. LDX >= max(1,M).
   *> \endverbatim
   *>
   *> \param[out] Y
   *> \verbatim
   *>          Y is DOUBLE PRECISION array, dimension (LDY,NB)
   *>          The n-by-nb matrix Y required to update the unreduced part
   *>          of A.
   *> \endverbatim
   *>
   *> \param[in] LDY
   *> \verbatim
   *>          LDY is INTEGER
   *>          The leading dimension of the array Y. LDY >= max(1,N).
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHERauxiliary
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The matrices Q and P are represented as products of elementary
   *>  reflectors:
   *>
   *>     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
   *>
   *>  Each H(i) and G(i) has the form:
   *>
   *>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
   *>
   *>  where tauq and taup are real scalars, and v and u are real vectors.
   *>
   *>  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
   *>  A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
   *>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
   *>
   *>  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
   *>  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
   *>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
   *>
   *>  The elements of the vectors v and u together form the m-by-nb matrix
   *>  V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
   *>  the transformation to the unreduced part of the matrix, using a block
   *>  update of the form:  A := A - V*Y**T - X*U**T.
   *>
   *>  The contents of A on exit are illustrated by the following examples
   *>  with nb = 2:
   *>
   *>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
   *>
   *>    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
   *>    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
   *>    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
   *>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
   *>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
   *>    (  v1  v2  a   a   a  )
   *>
   *>  where a denotes an element of the original matrix which is unchanged,
   *>  vi denotes an element of the vector defining H(i), and ui an element
   *>  of the vector defining G(i).
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,        SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
      $                   LDY )       $                   LDY )
 *  *
 *  -- LAPACK auxiliary routine (version 3.3.1) --  *  -- LAPACK auxiliary routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            LDA, LDX, LDY, M, N, NB        INTEGER            LDA, LDX, LDY, M, N, NB
Line 14 Line 223
      $                   TAUQ( * ), X( LDX, * ), Y( LDY, * )       $                   TAUQ( * ), X( LDX, * ), Y( LDY, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLABRD reduces the first NB rows and columns of a real general  
 *  m by n matrix A to upper or lower bidiagonal form by an orthogonal  
 *  transformation Q**T * A * P, and returns the matrices X and Y which  
 *  are needed to apply the transformation to the unreduced part of A.  
 *  
 *  If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower  
 *  bidiagonal form.  
 *  
 *  This is an auxiliary routine called by DGEBRD  
 *  
 *  Arguments  
 *  =========  
 *  
 *  M       (input) INTEGER  
 *          The number of rows in the matrix A.  
 *  
 *  N       (input) INTEGER  
 *          The number of columns in the matrix A.  
 *  
 *  NB      (input) INTEGER  
 *          The number of leading rows and columns of A to be reduced.  
 *  
 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)  
 *          On entry, the m by n general matrix to be reduced.  
 *          On exit, the first NB rows and columns of the matrix are  
 *          overwritten; the rest of the array is unchanged.  
 *          If m >= n, elements on and below the diagonal in the first NB  
 *            columns, with the array TAUQ, represent the orthogonal  
 *            matrix Q as a product of elementary reflectors; and  
 *            elements above the diagonal in the first NB rows, with the  
 *            array TAUP, represent the orthogonal matrix P as a product  
 *            of elementary reflectors.  
 *          If m < n, elements below the diagonal in the first NB  
 *            columns, with the array TAUQ, represent the orthogonal  
 *            matrix Q as a product of elementary reflectors, and  
 *            elements on and above the diagonal in the first NB rows,  
 *            with the array TAUP, represent the orthogonal matrix P as  
 *            a product of elementary reflectors.  
 *          See Further Details.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,M).  
 *  
 *  D       (output) DOUBLE PRECISION array, dimension (NB)  
 *          The diagonal elements of the first NB rows and columns of  
 *          the reduced matrix.  D(i) = A(i,i).  
 *  
 *  E       (output) DOUBLE PRECISION array, dimension (NB)  
 *          The off-diagonal elements of the first NB rows and columns of  
 *          the reduced matrix.  
 *  
 *  TAUQ    (output) DOUBLE PRECISION array dimension (NB)  
 *          The scalar factors of the elementary reflectors which  
 *          represent the orthogonal matrix Q. See Further Details.  
 *  
 *  TAUP    (output) DOUBLE PRECISION array, dimension (NB)  
 *          The scalar factors of the elementary reflectors which  
 *          represent the orthogonal matrix P. See Further Details.  
 *  
 *  X       (output) DOUBLE PRECISION array, dimension (LDX,NB)  
 *          The m-by-nb matrix X required to update the unreduced part  
 *          of A.  
 *  
 *  LDX     (input) INTEGER  
 *          The leading dimension of the array X. LDX >= max(1,M).  
 *  
 *  Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)  
 *          The n-by-nb matrix Y required to update the unreduced part  
 *          of A.  
 *  
 *  LDY     (input) INTEGER  
 *          The leading dimension of the array Y. LDY >= max(1,N).  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The matrices Q and P are represented as products of elementary  
 *  reflectors:  
 *  
 *     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)  
 *  
 *  Each H(i) and G(i) has the form:  
 *  
 *     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T  
 *  
 *  where tauq and taup are real scalars, and v and u are real vectors.  
 *  
 *  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in  
 *  A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in  
 *  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).  
 *  
 *  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in  
 *  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in  
 *  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).  
 *  
 *  The elements of the vectors v and u together form the m-by-nb matrix  
 *  V and the nb-by-n matrix U**T which are needed, with X and Y, to apply  
 *  the transformation to the unreduced part of the matrix, using a block  
 *  update of the form:  A := A - V*Y**T - X*U**T.  
 *  
 *  The contents of A on exit are illustrated by the following examples  
 *  with nb = 2:  
 *  
 *  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):  
 *  
 *    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )  
 *    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )  
 *    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )  
 *    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )  
 *    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )  
 *    (  v1  v2  a   a   a  )  
 *  
 *  where a denotes an element of the original matrix which is unchanged,  
 *  vi denotes an element of the vector defining H(i), and ui an element  
 *  of the vector defining G(i).  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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  Added in v.1.9

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