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330 lines
9.9 KiB
330 lines
9.9 KiB
2 years ago
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*> \brief \b SORBDB2
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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 SORBDB2 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sorbdb2.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/sorbdb2.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/sorbdb2.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 SORBDB2( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
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* TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
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* ..
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* .. Array Arguments ..
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* REAL PHI(*), THETA(*)
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* REAL TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
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* $ X11(LDX11,*), X21(LDX21,*)
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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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*> SORBDB2 simultaneously bidiagonalizes the blocks of a tall and skinny
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*> matrix X with orthonormal columns:
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*>
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*> [ B11 ]
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*> [ X11 ] [ P1 | ] [ 0 ]
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*> [-----] = [---------] [-----] Q1**T .
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*> [ X21 ] [ | P2 ] [ B21 ]
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*> [ 0 ]
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*>
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*> X11 is P-by-Q, and X21 is (M-P)-by-Q. P must be no larger than M-P,
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*> Q, or M-Q. Routines SORBDB1, SORBDB3, and SORBDB4 handle cases in
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*> which P is not the minimum dimension.
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*>
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*> The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
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*> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
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*> Householder vectors.
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*>
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*> B11 and B12 are P-by-P bidiagonal matrices represented implicitly by
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*> angles THETA, PHI.
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*>
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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 X11 plus the number of rows in X21.
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*> \endverbatim
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*>
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*> \param[in] P
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*> \verbatim
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*> P is INTEGER
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*> The number of rows in X11. 0 <= P <= min(M-P,Q,M-Q).
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*> \endverbatim
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*>
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*> \param[in] Q
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*> \verbatim
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*> Q is INTEGER
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*> The number of columns in X11 and X21. 0 <= Q <= M.
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*> \endverbatim
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*>
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*> \param[in,out] X11
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*> \verbatim
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*> X11 is REAL array, dimension (LDX11,Q)
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*> On entry, the top block of the matrix X to be reduced. On
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*> exit, the columns of tril(X11) specify reflectors for P1 and
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*> the rows of triu(X11,1) specify reflectors for Q1.
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*> \endverbatim
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*>
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*> \param[in] LDX11
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*> \verbatim
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*> LDX11 is INTEGER
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*> The leading dimension of X11. LDX11 >= P.
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*> \endverbatim
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*>
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*> \param[in,out] X21
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*> \verbatim
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*> X21 is REAL array, dimension (LDX21,Q)
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*> On entry, the bottom block of the matrix X to be reduced. On
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*> exit, the columns of tril(X21) specify reflectors for P2.
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*> \endverbatim
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*>
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*> \param[in] LDX21
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*> \verbatim
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*> LDX21 is INTEGER
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*> The leading dimension of X21. LDX21 >= M-P.
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*> \endverbatim
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*>
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*> \param[out] THETA
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*> \verbatim
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*> THETA is REAL array, dimension (Q)
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*> The entries of the bidiagonal blocks B11, B21 are defined by
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*> THETA and PHI. See Further Details.
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*> \endverbatim
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*>
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*> \param[out] PHI
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*> \verbatim
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*> PHI is REAL array, dimension (Q-1)
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*> The entries of the bidiagonal blocks B11, B21 are defined by
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*> THETA and PHI. See Further Details.
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*> \endverbatim
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*>
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*> \param[out] TAUP1
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*> \verbatim
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*> TAUP1 is REAL array, dimension (P-1)
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*> The scalar factors of the elementary reflectors that define
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*> P1.
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*> \endverbatim
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*>
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*> \param[out] TAUP2
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*> \verbatim
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*> TAUP2 is REAL array, dimension (Q)
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*> The scalar factors of the elementary reflectors that define
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*> P2.
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*> \endverbatim
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*>
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*> \param[out] TAUQ1
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*> \verbatim
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*> TAUQ1 is REAL array, dimension (Q)
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*> The scalar factors of the elementary reflectors that define
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*> Q1.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is REAL array, dimension (LWORK)
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= M-Q.
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*>
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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*> this value as the first entry of the WORK array, and no error
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*> message related to LWORK is issued by XERBLA.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit.
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*> < 0: if INFO = -i, the i-th argument had an illegal value.
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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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*> \ingroup realOTHERcomputational
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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 upper-bidiagonal blocks B11, B21 are represented implicitly by
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*> angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
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*> in each bidiagonal band is a product of a sine or cosine of a THETA
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*> with a sine or cosine of a PHI. See [1] or SORCSD for details.
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*>
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*> P1, P2, and Q1 are represented as products of elementary reflectors.
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*> See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
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*> and SORGLQ.
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*> \endverbatim
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*
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*> \par References:
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* ================
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*>
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*> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
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*> Algorithms, 50(1):33-65, 2009.
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*>
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* =====================================================================
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SUBROUTINE SORBDB2( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
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$ TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
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* ..
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* .. Array Arguments ..
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REAL PHI(*), THETA(*)
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REAL TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
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$ X11(LDX11,*), X21(LDX21,*)
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* ..
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*
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* ====================================================================
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*
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* .. Parameters ..
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REAL NEGONE, ONE
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PARAMETER ( NEGONE = -1.0E0, ONE = 1.0E0 )
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* ..
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* .. Local Scalars ..
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REAL C, S
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INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
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$ LWORKMIN, LWORKOPT
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LOGICAL LQUERY
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* ..
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* .. External Subroutines ..
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EXTERNAL SLARF, SLARFGP, SORBDB5, SROT, SSCAL, XERBLA
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* ..
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* .. External Functions ..
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REAL SNRM2
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EXTERNAL SNRM2
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* ..
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* .. Intrinsic Function ..
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INTRINSIC ATAN2, COS, MAX, SIN, SQRT
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* ..
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* .. Executable Statements ..
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*
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* Test input arguments
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*
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INFO = 0
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LQUERY = LWORK .EQ. -1
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*
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IF( M .LT. 0 ) THEN
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INFO = -1
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ELSE IF( P .LT. 0 .OR. P .GT. M-P ) THEN
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INFO = -2
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ELSE IF( Q .LT. 0 .OR. Q .LT. P .OR. M-Q .LT. P ) THEN
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INFO = -3
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ELSE IF( LDX11 .LT. MAX( 1, P ) ) THEN
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INFO = -5
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ELSE IF( LDX21 .LT. MAX( 1, M-P ) ) THEN
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INFO = -7
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END IF
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*
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* Compute workspace
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*
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IF( INFO .EQ. 0 ) THEN
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ILARF = 2
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LLARF = MAX( P-1, M-P, Q-1 )
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IORBDB5 = 2
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LORBDB5 = Q-1
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LWORKOPT = MAX( ILARF+LLARF-1, IORBDB5+LORBDB5-1 )
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LWORKMIN = LWORKOPT
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WORK(1) = LWORKOPT
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IF( LWORK .LT. LWORKMIN .AND. .NOT.LQUERY ) THEN
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INFO = -14
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END IF
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END IF
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IF( INFO .NE. 0 ) THEN
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CALL XERBLA( 'SORBDB2', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* Reduce rows 1, ..., P of X11 and X21
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*
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DO I = 1, P
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*
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IF( I .GT. 1 ) THEN
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CALL SROT( Q-I+1, X11(I,I), LDX11, X21(I-1,I), LDX21, C, S )
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END IF
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CALL SLARFGP( Q-I+1, X11(I,I), X11(I,I+1), LDX11, TAUQ1(I) )
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C = X11(I,I)
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X11(I,I) = ONE
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CALL SLARF( 'R', P-I, Q-I+1, X11(I,I), LDX11, TAUQ1(I),
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$ X11(I+1,I), LDX11, WORK(ILARF) )
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CALL SLARF( 'R', M-P-I+1, Q-I+1, X11(I,I), LDX11, TAUQ1(I),
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$ X21(I,I), LDX21, WORK(ILARF) )
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S = SQRT( SNRM2( P-I, X11(I+1,I), 1 )**2
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$ + SNRM2( M-P-I+1, X21(I,I), 1 )**2 )
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THETA(I) = ATAN2( S, C )
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*
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CALL SORBDB5( P-I, M-P-I+1, Q-I, X11(I+1,I), 1, X21(I,I), 1,
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$ X11(I+1,I+1), LDX11, X21(I,I+1), LDX21,
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$ WORK(IORBDB5), LORBDB5, CHILDINFO )
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CALL SSCAL( P-I, NEGONE, X11(I+1,I), 1 )
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CALL SLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1, TAUP2(I) )
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IF( I .LT. P ) THEN
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CALL SLARFGP( P-I, X11(I+1,I), X11(I+2,I), 1, TAUP1(I) )
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PHI(I) = ATAN2( X11(I+1,I), X21(I,I) )
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C = COS( PHI(I) )
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S = SIN( PHI(I) )
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X11(I+1,I) = ONE
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CALL SLARF( 'L', P-I, Q-I, X11(I+1,I), 1, TAUP1(I),
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$ X11(I+1,I+1), LDX11, WORK(ILARF) )
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END IF
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X21(I,I) = ONE
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CALL SLARF( 'L', M-P-I+1, Q-I, X21(I,I), 1, TAUP2(I),
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$ X21(I,I+1), LDX21, WORK(ILARF) )
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*
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END DO
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*
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* Reduce the bottom-right portion of X21 to the identity matrix
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*
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DO I = P + 1, Q
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CALL SLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1, TAUP2(I) )
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X21(I,I) = ONE
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CALL SLARF( 'L', M-P-I+1, Q-I, X21(I,I), 1, TAUP2(I),
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$ X21(I,I+1), LDX21, WORK(ILARF) )
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END DO
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*
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RETURN
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*
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* End of SORBDB2
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*
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END
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