dlags2.f

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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.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
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*
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*
*  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.
*
*> \ingroup lags2
*
*  =====================================================================
      SUBROUTINE dlags2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
     $                   SNV, CSQ, SNQ )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. 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