*> \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. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DLAGS2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, * SNV, CSQ, SNQ ) * * .. Scalar Arguments .. * LOGICAL UPPER * DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ, * \$ SNU, SNV * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such *> that if ( UPPER ) then *> *> U**T *A*Q = U**T *( A1 A2 )*Q = ( x 0 ) *> ( 0 A3 ) ( x x ) *> and *> V**T*B*Q = V**T *( B1 B2 )*Q = ( x 0 ) *> ( 0 B3 ) ( x x ) *> *> or if ( .NOT.UPPER ) then *> *> U**T *A*Q = U**T *( A1 0 )*Q = ( x x ) *> ( A2 A3 ) ( 0 x ) *> and *> V**T*B*Q = V**T*( B1 0 )*Q = ( x x ) *> ( B2 B3 ) ( 0 x ) *> *> The rows of the transformed A and B are parallel, where *> *> U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ ) *> ( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ ) *> *> Z**T denotes the transpose of Z. *> *> \endverbatim * * Arguments: * ========== * *> \param[in] UPPER *> \verbatim *> UPPER is LOGICAL *> = .TRUE.: the input matrices A and B are upper triangular. *> = .FALSE.: the input matrices A and B are lower triangular. *> \endverbatim *> *> \param[in] A1 *> \verbatim *> A1 is DOUBLE PRECISION *> \endverbatim *> *> \param[in] A2 *> \verbatim *> A2 is DOUBLE PRECISION *> \endverbatim *> *> \param[in] A3 *> \verbatim *> A3 is DOUBLE PRECISION *> On entry, A1, A2 and A3 are elements of the input 2-by-2 *> upper (lower) triangular matrix A. *> \endverbatim *> *> \param[in] B1 *> \verbatim *> B1 is DOUBLE PRECISION *> \endverbatim *> *> \param[in] B2 *> \verbatim *> B2 is DOUBLE PRECISION *> \endverbatim *> *> \param[in] B3 *> \verbatim *> B3 is DOUBLE PRECISION *> On entry, B1, B2 and B3 are elements of the input 2-by-2 *> upper (lower) triangular matrix B. *> \endverbatim *> *> \param[out] CSU *> \verbatim *> CSU is DOUBLE PRECISION *> \endverbatim *> *> \param[out] SNU *> \verbatim *> SNU is DOUBLE PRECISION *> The desired orthogonal matrix U. *> \endverbatim *> *> \param[out] CSV *> \verbatim *> CSV is DOUBLE PRECISION *> \endverbatim *> *> \param[out] SNV *> \verbatim *> SNV is DOUBLE PRECISION *> The desired orthogonal matrix V. *> \endverbatim *> *> \param[out] CSQ *> \verbatim *> CSQ is DOUBLE PRECISION *> \endverbatim *> *> \param[out] SNQ *> \verbatim *> SNQ is DOUBLE PRECISION *> The desired orthogonal matrix Q. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup doubleOTHERauxiliary * * ===================================================================== SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, \$ SNV, CSQ, SNQ ) * * -- LAPACK auxiliary routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. LOGICAL UPPER DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ, \$ SNU, SNV * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) * .. * .. Local Scalars .. DOUBLE PRECISION A, AUA11, AUA12, AUA21, AUA22, AVB11, AVB12, \$ AVB21, AVB22, B, C, CSL, CSR, D, R, S1, S2, \$ SNL, SNR, UA11, UA11R, UA12, UA21, UA22, UA22R, \$ VB11, VB11R, VB12, VB21, VB22, VB22R * .. * .. External Subroutines .. EXTERNAL DLARTG, DLASV2 * .. * .. Intrinsic Functions .. INTRINSIC ABS * .. * .. Executable Statements .. * IF( UPPER ) THEN * * Input matrices A and B are upper triangular matrices * * Form matrix C = A*adj(B) = ( a b ) * ( 0 d ) * A = A1*B3 D = A3*B1 B = A2*B1 - A1*B2 * * The SVD of real 2-by-2 triangular C * * ( CSL -SNL )*( A B )*( CSR SNR ) = ( R 0 ) * ( SNL CSL ) ( 0 D ) ( -SNR CSR ) ( 0 T ) * CALL DLASV2( A, B, D, S1, S2, SNR, CSR, SNL, CSL ) * IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) ) \$ THEN * * Compute the (1,1) and (1,2) elements of U**T *A and V**T *B, * and (1,2) element of |U|**T *|A| and |V|**T *|B|. * UA11R = CSL*A1 UA12 = CSL*A2 + SNL*A3 * VB11R = CSR*B1 VB12 = CSR*B2 + SNR*B3 * AUA12 = ABS( CSL )*ABS( A2 ) + ABS( SNL )*ABS( A3 ) AVB12 = ABS( CSR )*ABS( B2 ) + ABS( SNR )*ABS( B3 ) * * zero (1,2) elements of U**T *A and V**T *B * IF( ( ABS( UA11R )+ABS( UA12 ) ).NE.ZERO ) THEN IF( AUA12 / ( ABS( UA11R )+ABS( UA12 ) ).LE.AVB12 / \$ ( ABS( VB11R )+ABS( VB12 ) ) ) THEN CALL DLARTG( -UA11R, UA12, CSQ, SNQ, R ) ELSE CALL DLARTG( -VB11R, VB12, CSQ, SNQ, R ) END IF ELSE CALL DLARTG( -VB11R, VB12, CSQ, SNQ, R ) END IF * CSU = CSL SNU = -SNL CSV = CSR SNV = -SNR * ELSE * * Compute the (2,1) and (2,2) elements of U**T *A and V**T *B, * and (2,2) element of |U|**T *|A| and |V|**T *|B|. * UA21 = -SNL*A1 UA22 = -SNL*A2 + CSL*A3 * VB21 = -SNR*B1 VB22 = -SNR*B2 + CSR*B3 * AUA22 = ABS( SNL )*ABS( A2 ) + ABS( CSL )*ABS( A3 ) AVB22 = ABS( SNR )*ABS( B2 ) + ABS( CSR )*ABS( B3 ) * * zero (2,2) elements of U**T*A and V**T*B, and then swap. * IF( ( ABS( UA21 )+ABS( UA22 ) ).NE.ZERO ) THEN IF( AUA22 / ( ABS( UA21 )+ABS( UA22 ) ).LE.AVB22 / \$ ( ABS( VB21 )+ABS( VB22 ) ) ) THEN CALL DLARTG( -UA21, UA22, CSQ, SNQ, R ) ELSE CALL DLARTG( -VB21, VB22, CSQ, SNQ, R ) END IF ELSE CALL DLARTG( -VB21, VB22, CSQ, SNQ, R ) END IF * CSU = SNL SNU = CSL CSV = SNR SNV = CSR * END IF * ELSE * * Input matrices A and B are lower triangular matrices * * Form matrix C = A*adj(B) = ( a 0 ) * ( c d ) * A = A1*B3 D = A3*B1 C = A2*B3 - A3*B2 * * The SVD of real 2-by-2 triangular C * * ( CSL -SNL )*( A 0 )*( CSR SNR ) = ( R 0 ) * ( SNL CSL ) ( C D ) ( -SNR CSR ) ( 0 T ) * CALL DLASV2( A, C, D, S1, S2, SNR, CSR, SNL, CSL ) * IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) ) \$ THEN * * Compute the (2,1) and (2,2) elements of U**T *A and V**T *B, * and (2,1) element of |U|**T *|A| and |V|**T *|B|. * UA21 = -SNR*A1 + CSR*A2 UA22R = CSR*A3 * VB21 = -SNL*B1 + CSL*B2 VB22R = CSL*B3 * AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS( A2 ) AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS( B2 ) * * zero (2,1) elements of U**T *A and V**T *B. * IF( ( ABS( UA21 )+ABS( UA22R ) ).NE.ZERO ) THEN IF( AUA21 / ( ABS( UA21 )+ABS( UA22R ) ).LE.AVB21 / \$ ( ABS( VB21 )+ABS( VB22R ) ) ) THEN CALL DLARTG( UA22R, UA21, CSQ, SNQ, R ) ELSE CALL DLARTG( VB22R, VB21, CSQ, SNQ, R ) END IF ELSE CALL DLARTG( VB22R, VB21, CSQ, SNQ, R ) END IF * CSU = CSR SNU = -SNR CSV = CSL SNV = -SNL * ELSE * * Compute the (1,1) and (1,2) elements of U**T *A and V**T *B, * and (1,1) element of |U|**T *|A| and |V|**T *|B|. * UA11 = CSR*A1 + SNR*A2 UA12 = SNR*A3 * VB11 = CSL*B1 + SNL*B2 VB12 = SNL*B3 * AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS( A2 ) AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS( B2 ) * * zero (1,1) elements of U**T*A and V**T*B, and then swap. * IF( ( ABS( UA11 )+ABS( UA12 ) ).NE.ZERO ) THEN IF( AUA11 / ( ABS( UA11 )+ABS( UA12 ) ).LE.AVB11 / \$ ( ABS( VB11 )+ABS( VB12 ) ) ) THEN CALL DLARTG( UA12, UA11, CSQ, SNQ, R ) ELSE CALL DLARTG( VB12, VB11, CSQ, SNQ, R ) END IF ELSE CALL DLARTG( VB12, VB11, CSQ, SNQ, R ) END IF * CSU = SNR SNU = CSR CSV = SNL SNV = CSL * END IF * END IF * RETURN * * End of DLAGS2 * END