version 1.1.1.1, 2010/01/26 15:22:45
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version 1.12, 2012/12/14 12:30:21
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*> \brief \b DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. |
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* |
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* =========== DOCUMENTATION =========== |
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* |
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* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download DLABRD + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlabrd.f"> |
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*> [TGZ]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlabrd.f"> |
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*> [ZIP]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlabrd.f"> |
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*> [TXT]</a> |
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*> \endhtmlonly |
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* |
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* Definition: |
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* =========== |
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* |
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* SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, |
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* LDY ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER LDA, LDX, LDY, M, N, NB |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), |
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* $ TAUQ( * ), X( LDX, * ), Y( LDY, * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> DLABRD reduces the first NB rows and columns of a real general |
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*> m by n matrix A to upper or lower bidiagonal form by an orthogonal |
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*> transformation Q**T * A * P, and returns the matrices X and Y which |
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*> are needed to apply the transformation to the unreduced part of A. |
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*> |
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*> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower |
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*> bidiagonal form. |
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*> |
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*> This is an auxiliary routine called by DGEBRD |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] M |
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*> \verbatim |
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*> M is INTEGER |
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*> The number of rows in the matrix A. |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The number of columns in the matrix A. |
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*> \endverbatim |
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*> |
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*> \param[in] NB |
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*> \verbatim |
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*> NB is INTEGER |
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*> The number of leading rows and columns of A to be reduced. |
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*> \endverbatim |
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*> |
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*> \param[in,out] A |
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*> \verbatim |
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*> A is DOUBLE PRECISION array, dimension (LDA,N) |
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*> On entry, the m by n general matrix to be reduced. |
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*> On exit, the first NB rows and columns of the matrix are |
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*> overwritten; the rest of the array is unchanged. |
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*> If m >= n, elements on and below the diagonal in the first NB |
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*> columns, with the array TAUQ, represent the orthogonal |
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*> matrix Q as a product of elementary reflectors; and |
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*> elements above the diagonal in the first NB rows, with the |
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*> array TAUP, represent the orthogonal matrix P as a product |
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*> of elementary reflectors. |
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*> If m < n, elements below the diagonal in the first NB |
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*> columns, with the array TAUQ, represent the orthogonal |
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*> matrix Q as a product of elementary reflectors, and |
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*> elements on and above the diagonal in the first NB rows, |
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*> with the array TAUP, represent the orthogonal matrix P as |
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*> a product of elementary reflectors. |
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*> See Further Details. |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of the array A. LDA >= max(1,M). |
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*> \endverbatim |
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*> |
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*> \param[out] D |
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*> \verbatim |
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*> D is DOUBLE PRECISION array, dimension (NB) |
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*> The diagonal elements of the first NB rows and columns of |
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*> the reduced matrix. D(i) = A(i,i). |
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*> \endverbatim |
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*> |
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*> \param[out] E |
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*> \verbatim |
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*> E is DOUBLE PRECISION array, dimension (NB) |
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*> The off-diagonal elements of the first NB rows and columns of |
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*> the reduced matrix. |
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*> \endverbatim |
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*> |
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*> \param[out] TAUQ |
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*> \verbatim |
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*> TAUQ is DOUBLE PRECISION array dimension (NB) |
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*> The scalar factors of the elementary reflectors which |
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*> represent the orthogonal matrix Q. See Further Details. |
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*> \endverbatim |
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*> |
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*> \param[out] TAUP |
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*> \verbatim |
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*> TAUP is DOUBLE PRECISION array, dimension (NB) |
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*> The scalar factors of the elementary reflectors which |
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*> represent the orthogonal matrix P. See Further Details. |
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*> \endverbatim |
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*> |
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*> \param[out] X |
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*> \verbatim |
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*> X is DOUBLE PRECISION array, dimension (LDX,NB) |
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*> The m-by-nb matrix X required to update the unreduced part |
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*> of A. |
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*> \endverbatim |
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*> |
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*> \param[in] LDX |
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*> \verbatim |
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*> LDX is INTEGER |
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*> The leading dimension of the array X. LDX >= max(1,M). |
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*> \endverbatim |
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*> |
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*> \param[out] Y |
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*> \verbatim |
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*> Y is DOUBLE PRECISION array, dimension (LDY,NB) |
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*> The n-by-nb matrix Y required to update the unreduced part |
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*> of A. |
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*> \endverbatim |
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*> |
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*> \param[in] LDY |
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*> \verbatim |
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*> LDY is INTEGER |
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*> The leading dimension of the array Y. LDY >= max(1,N). |
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*> \endverbatim |
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* |
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* Authors: |
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* ======== |
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* |
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*> \author Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \date September 2012 |
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* |
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*> \ingroup doubleOTHERauxiliary |
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* |
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*> \par Further Details: |
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* ===================== |
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*> |
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*> \verbatim |
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*> |
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*> The matrices Q and P are represented as products of elementary |
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*> reflectors: |
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*> |
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*> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) |
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*> |
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*> Each H(i) and G(i) has the form: |
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*> |
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*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T |
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*> |
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*> where tauq and taup are real scalars, and v and u are real vectors. |
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*> |
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*> If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in |
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*> A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in |
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*> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
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*> |
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*> If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in |
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*> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in |
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*> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
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*> |
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*> The elements of the vectors v and u together form the m-by-nb matrix |
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*> V and the nb-by-n matrix U**T which are needed, with X and Y, to apply |
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*> the transformation to the unreduced part of the matrix, using a block |
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*> update of the form: A := A - V*Y**T - X*U**T. |
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*> |
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*> The contents of A on exit are illustrated by the following examples |
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*> with nb = 2: |
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*> |
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*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): |
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*> |
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*> ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) |
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*> ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) |
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*> ( v1 v2 a a a ) ( v1 1 a a a a ) |
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*> ( v1 v2 a a a ) ( v1 v2 a a a a ) |
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*> ( v1 v2 a a a ) ( v1 v2 a a a a ) |
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*> ( v1 v2 a a a ) |
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*> |
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*> where a denotes an element of the original matrix which is unchanged, |
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*> vi denotes an element of the vector defining H(i), and ui an element |
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*> of the vector defining G(i). |
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*> \endverbatim |
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*> |
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* ===================================================================== |
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 3.2) -- |
* -- LAPACK auxiliary routine (version 3.4.2) -- |
* -- 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..-- |
* November 2006 |
* September 2012 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER LDA, LDX, LDY, M, N, NB |
INTEGER LDA, LDX, LDY, M, N, NB |
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$ TAUQ( * ), X( LDX, * ), Y( LDY, * ) |
$ TAUQ( * ), X( LDX, * ), Y( LDY, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DLABRD reduces the first NB rows and columns of a real general |
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* m by n matrix A to upper or lower bidiagonal form by an orthogonal |
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* transformation Q' * A * P, and returns the matrices X and Y which |
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* are needed to apply the transformation to the unreduced part of A. |
|
* |
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* If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower |
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* bidiagonal form. |
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* |
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* This is an auxiliary routine called by DGEBRD |
|
* |
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* Arguments |
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* ========= |
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* |
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* M (input) INTEGER |
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* The number of rows in the matrix A. |
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* |
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* N (input) INTEGER |
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* The number of columns in the matrix A. |
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* |
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* NB (input) INTEGER |
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* The number of leading rows and columns of A to be reduced. |
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* |
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* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
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* On entry, the m by n general matrix to be reduced. |
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* On exit, the first NB rows and columns of the matrix are |
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* overwritten; the rest of the array is unchanged. |
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* If m >= n, elements on and below the diagonal in the first NB |
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* columns, with the array TAUQ, represent the orthogonal |
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* matrix Q as a product of elementary reflectors; and |
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* elements above the diagonal in the first NB rows, with the |
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* array TAUP, represent the orthogonal matrix P as a product |
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* of elementary reflectors. |
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* If m < n, elements below the diagonal in the first NB |
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* columns, with the array TAUQ, represent the orthogonal |
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* matrix Q as a product of elementary reflectors, and |
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* elements on and above the diagonal in the first NB rows, |
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* with the array TAUP, represent the orthogonal matrix P as |
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* a product of elementary reflectors. |
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* See Further Details. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= max(1,M). |
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* |
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* D (output) DOUBLE PRECISION array, dimension (NB) |
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* The diagonal elements of the first NB rows and columns of |
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* the reduced matrix. D(i) = A(i,i). |
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* |
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* E (output) DOUBLE PRECISION array, dimension (NB) |
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* The off-diagonal elements of the first NB rows and columns of |
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* the reduced matrix. |
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* |
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* TAUQ (output) DOUBLE PRECISION array dimension (NB) |
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* The scalar factors of the elementary reflectors which |
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* represent the orthogonal matrix Q. See Further Details. |
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* |
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* TAUP (output) DOUBLE PRECISION array, dimension (NB) |
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* The scalar factors of the elementary reflectors which |
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* represent the orthogonal matrix P. See Further Details. |
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* |
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* X (output) DOUBLE PRECISION array, dimension (LDX,NB) |
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* The m-by-nb matrix X required to update the unreduced part |
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* of A. |
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* |
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* LDX (input) INTEGER |
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* The leading dimension of the array X. LDX >= M. |
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* |
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* Y (output) DOUBLE PRECISION array, dimension (LDY,NB) |
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* The n-by-nb matrix Y required to update the unreduced part |
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* of A. |
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* |
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* LDY (input) INTEGER |
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* The leading dimension of the array Y. LDY >= N. |
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* |
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* Further Details |
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* =============== |
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* |
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* The matrices Q and P are represented as products of elementary |
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* reflectors: |
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* |
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* Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) |
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* |
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* Each H(i) and G(i) has the form: |
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* |
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* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' |
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* |
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* where tauq and taup are real scalars, and v and u are real vectors. |
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* |
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* If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in |
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* A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in |
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* A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
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* |
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* If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in |
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* A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in |
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* A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
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* |
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* The elements of the vectors v and u together form the m-by-nb matrix |
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* V and the nb-by-n matrix U' which are needed, with X and Y, to apply |
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* the transformation to the unreduced part of the matrix, using a block |
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* update of the form: A := A - V*Y' - X*U'. |
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* |
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* The contents of A on exit are illustrated by the following examples |
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* with nb = 2: |
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* |
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* m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): |
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* |
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* ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) |
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* ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) |
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* ( v1 v2 a a a ) ( v1 1 a a a a ) |
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* ( v1 v2 a a a ) ( v1 v2 a a a a ) |
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* ( v1 v2 a a a ) ( v1 v2 a a a a ) |
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* ( v1 v2 a a a ) |
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* |
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* where a denotes an element of the original matrix which is unchanged, |
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* vi denotes an element of the vector defining H(i), and ui an element |
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* of the vector defining G(i). |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |