*> \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
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*> [TXT]</a>
*> \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