version 1.1, 2010/01/26 15:22:46
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version 1.15, 2016/08/27 15:34:27
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*> \brief \b DLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel. |
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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 DLAGS2 + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlags2.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/dlags2.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/dlags2.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 DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, |
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* SNV, CSQ, SNQ ) |
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* |
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* .. Scalar Arguments .. |
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* LOGICAL UPPER |
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* DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ, |
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* $ SNU, SNV |
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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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*> DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such |
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*> that if ( UPPER ) then |
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*> |
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*> U**T *A*Q = U**T *( A1 A2 )*Q = ( x 0 ) |
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*> ( 0 A3 ) ( x x ) |
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*> and |
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*> V**T*B*Q = V**T *( B1 B2 )*Q = ( x 0 ) |
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*> ( 0 B3 ) ( x x ) |
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*> |
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*> or if ( .NOT.UPPER ) then |
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*> |
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*> U**T *A*Q = U**T *( A1 0 )*Q = ( x x ) |
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*> ( A2 A3 ) ( 0 x ) |
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*> and |
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*> V**T*B*Q = V**T*( B1 0 )*Q = ( x x ) |
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*> ( B2 B3 ) ( 0 x ) |
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*> |
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*> The rows of the transformed A and B are parallel, where |
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*> |
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*> U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ ) |
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*> ( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ ) |
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*> |
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*> Z**T denotes the transpose of Z. |
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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] UPPER |
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*> \verbatim |
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*> UPPER is LOGICAL |
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*> = .TRUE.: the input matrices A and B are upper triangular. |
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*> = .FALSE.: the input matrices A and B are lower triangular. |
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*> \endverbatim |
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*> |
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*> \param[in] A1 |
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*> \verbatim |
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*> A1 is DOUBLE PRECISION |
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*> \endverbatim |
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*> |
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*> \param[in] A2 |
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*> \verbatim |
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*> A2 is DOUBLE PRECISION |
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*> \endverbatim |
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*> |
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*> \param[in] A3 |
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*> \verbatim |
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*> A3 is DOUBLE PRECISION |
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*> On entry, A1, A2 and A3 are elements of the input 2-by-2 |
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*> upper (lower) triangular matrix A. |
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*> \endverbatim |
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*> |
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*> \param[in] B1 |
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*> \verbatim |
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*> B1 is DOUBLE PRECISION |
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*> \endverbatim |
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*> |
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*> \param[in] B2 |
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*> \verbatim |
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*> B2 is DOUBLE PRECISION |
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*> \endverbatim |
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*> |
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*> \param[in] B3 |
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*> \verbatim |
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*> B3 is DOUBLE PRECISION |
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*> On entry, B1, B2 and B3 are elements of the input 2-by-2 |
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*> upper (lower) triangular matrix B. |
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*> \endverbatim |
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*> |
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*> \param[out] CSU |
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*> \verbatim |
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*> CSU is DOUBLE PRECISION |
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*> \endverbatim |
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*> |
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*> \param[out] SNU |
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*> \verbatim |
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*> SNU is DOUBLE PRECISION |
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*> The desired orthogonal matrix U. |
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*> \endverbatim |
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*> |
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*> \param[out] CSV |
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*> \verbatim |
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*> CSV is DOUBLE PRECISION |
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*> \endverbatim |
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*> |
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*> \param[out] SNV |
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*> \verbatim |
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*> SNV is DOUBLE PRECISION |
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*> The desired orthogonal matrix V. |
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*> \endverbatim |
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*> |
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*> \param[out] CSQ |
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*> \verbatim |
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*> CSQ is DOUBLE PRECISION |
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*> \endverbatim |
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*> |
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*> \param[out] SNQ |
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*> \verbatim |
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*> SNQ is DOUBLE PRECISION |
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*> The desired orthogonal matrix Q. |
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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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* ===================================================================== |
SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, |
SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, |
$ SNV, CSQ, SNQ ) |
$ SNV, CSQ, SNQ ) |
* |
* |
* -- 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 .. |
LOGICAL UPPER |
LOGICAL UPPER |
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$ SNU, SNV |
$ SNU, SNV |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such |
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* that if ( UPPER ) then |
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* |
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* U'*A*Q = U'*( A1 A2 )*Q = ( x 0 ) |
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* ( 0 A3 ) ( x x ) |
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* and |
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* V'*B*Q = V'*( B1 B2 )*Q = ( x 0 ) |
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* ( 0 B3 ) ( x x ) |
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* |
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* or if ( .NOT.UPPER ) then |
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* |
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* U'*A*Q = U'*( A1 0 )*Q = ( x x ) |
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* ( A2 A3 ) ( 0 x ) |
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* and |
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* V'*B*Q = V'*( B1 0 )*Q = ( x x ) |
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* ( B2 B3 ) ( 0 x ) |
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* |
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* The rows of the transformed A and B are parallel, where |
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* |
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* U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ ) |
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* ( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ ) |
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* |
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* Z' denotes the transpose of Z. |
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* |
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* |
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* Arguments |
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* ========= |
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* |
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* UPPER (input) LOGICAL |
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* = .TRUE.: the input matrices A and B are upper triangular. |
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* = .FALSE.: the input matrices A and B are lower triangular. |
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* |
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* A1 (input) DOUBLE PRECISION |
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* A2 (input) DOUBLE PRECISION |
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* A3 (input) DOUBLE PRECISION |
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* On entry, A1, A2 and A3 are elements of the input 2-by-2 |
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* upper (lower) triangular matrix A. |
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* |
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* B1 (input) DOUBLE PRECISION |
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* B2 (input) DOUBLE PRECISION |
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* B3 (input) DOUBLE PRECISION |
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* On entry, B1, B2 and B3 are elements of the input 2-by-2 |
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* upper (lower) triangular matrix B. |
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* |
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* CSU (output) DOUBLE PRECISION |
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* SNU (output) DOUBLE PRECISION |
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* The desired orthogonal matrix U. |
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* |
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* CSV (output) DOUBLE PRECISION |
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* SNV (output) DOUBLE PRECISION |
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* The desired orthogonal matrix V. |
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* |
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* CSQ (output) DOUBLE PRECISION |
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* SNQ (output) DOUBLE PRECISION |
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* The desired orthogonal matrix Q. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) ) |
IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) ) |
$ THEN |
$ THEN |
* |
* |
* Compute the (1,1) and (1,2) elements of U'*A and V'*B, |
* Compute the (1,1) and (1,2) elements of U**T *A and V**T *B, |
* and (1,2) element of |U|'*|A| and |V|'*|B|. |
* and (1,2) element of |U|**T *|A| and |V|**T *|B|. |
* |
* |
UA11R = CSL*A1 |
UA11R = CSL*A1 |
UA12 = CSL*A2 + SNL*A3 |
UA12 = CSL*A2 + SNL*A3 |
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AUA12 = ABS( CSL )*ABS( A2 ) + ABS( SNL )*ABS( A3 ) |
AUA12 = ABS( CSL )*ABS( A2 ) + ABS( SNL )*ABS( A3 ) |
AVB12 = ABS( CSR )*ABS( B2 ) + ABS( SNR )*ABS( B3 ) |
AVB12 = ABS( CSR )*ABS( B2 ) + ABS( SNR )*ABS( B3 ) |
* |
* |
* zero (1,2) elements of U'*A and V'*B |
* zero (1,2) elements of U**T *A and V**T *B |
* |
* |
IF( ( ABS( UA11R )+ABS( UA12 ) ).NE.ZERO ) THEN |
IF( ( ABS( UA11R )+ABS( UA12 ) ).NE.ZERO ) THEN |
IF( AUA12 / ( ABS( UA11R )+ABS( UA12 ) ).LE.AVB12 / |
IF( AUA12 / ( ABS( UA11R )+ABS( UA12 ) ).LE.AVB12 / |
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* |
* |
ELSE |
ELSE |
* |
* |
* Compute the (2,1) and (2,2) elements of U'*A and V'*B, |
* Compute the (2,1) and (2,2) elements of U**T *A and V**T *B, |
* and (2,2) element of |U|'*|A| and |V|'*|B|. |
* and (2,2) element of |U|**T *|A| and |V|**T *|B|. |
* |
* |
UA21 = -SNL*A1 |
UA21 = -SNL*A1 |
UA22 = -SNL*A2 + CSL*A3 |
UA22 = -SNL*A2 + CSL*A3 |
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AUA22 = ABS( SNL )*ABS( A2 ) + ABS( CSL )*ABS( A3 ) |
AUA22 = ABS( SNL )*ABS( A2 ) + ABS( CSL )*ABS( A3 ) |
AVB22 = ABS( SNR )*ABS( B2 ) + ABS( CSR )*ABS( B3 ) |
AVB22 = ABS( SNR )*ABS( B2 ) + ABS( CSR )*ABS( B3 ) |
* |
* |
* zero (2,2) elements of U'*A and V'*B, and then swap. |
* zero (2,2) elements of U**T*A and V**T*B, and then swap. |
* |
* |
IF( ( ABS( UA21 )+ABS( UA22 ) ).NE.ZERO ) THEN |
IF( ( ABS( UA21 )+ABS( UA22 ) ).NE.ZERO ) THEN |
IF( AUA22 / ( ABS( UA21 )+ABS( UA22 ) ).LE.AVB22 / |
IF( AUA22 / ( ABS( UA21 )+ABS( UA22 ) ).LE.AVB22 / |
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IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) ) |
IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) ) |
$ THEN |
$ THEN |
* |
* |
* Compute the (2,1) and (2,2) elements of U'*A and V'*B, |
* Compute the (2,1) and (2,2) elements of U**T *A and V**T *B, |
* and (2,1) element of |U|'*|A| and |V|'*|B|. |
* and (2,1) element of |U|**T *|A| and |V|**T *|B|. |
* |
* |
UA21 = -SNR*A1 + CSR*A2 |
UA21 = -SNR*A1 + CSR*A2 |
UA22R = CSR*A3 |
UA22R = CSR*A3 |
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AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS( A2 ) |
AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS( A2 ) |
AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS( B2 ) |
AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS( B2 ) |
* |
* |
* zero (2,1) elements of U'*A and V'*B. |
* zero (2,1) elements of U**T *A and V**T *B. |
* |
* |
IF( ( ABS( UA21 )+ABS( UA22R ) ).NE.ZERO ) THEN |
IF( ( ABS( UA21 )+ABS( UA22R ) ).NE.ZERO ) THEN |
IF( AUA21 / ( ABS( UA21 )+ABS( UA22R ) ).LE.AVB21 / |
IF( AUA21 / ( ABS( UA21 )+ABS( UA22R ) ).LE.AVB21 / |
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* |
* |
ELSE |
ELSE |
* |
* |
* Compute the (1,1) and (1,2) elements of U'*A and V'*B, |
* Compute the (1,1) and (1,2) elements of U**T *A and V**T *B, |
* and (1,1) element of |U|'*|A| and |V|'*|B|. |
* and (1,1) element of |U|**T *|A| and |V|**T *|B|. |
* |
* |
UA11 = CSR*A1 + SNR*A2 |
UA11 = CSR*A1 + SNR*A2 |
UA12 = SNR*A3 |
UA12 = SNR*A3 |
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AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS( A2 ) |
AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS( A2 ) |
AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS( B2 ) |
AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS( B2 ) |
* |
* |
* zero (1,1) elements of U'*A and V'*B, and then swap. |
* zero (1,1) elements of U**T*A and V**T*B, and then swap. |
* |
* |
IF( ( ABS( UA11 )+ABS( UA12 ) ).NE.ZERO ) THEN |
IF( ( ABS( UA11 )+ABS( UA12 ) ).NE.ZERO ) THEN |
IF( AUA11 / ( ABS( UA11 )+ABS( UA12 ) ).LE.AVB11 / |
IF( AUA11 / ( ABS( UA11 )+ABS( UA12 ) ).LE.AVB11 / |