dlags2.f

          SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
         $                   SNV, CSQ, SNQ )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          LOGICAL            UPPER
          DOUBLE PRECISION   A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ,
         $                   SNU, SNV
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
    *  that if ( UPPER ) then
    *
    *            U'*A*Q = U'*( A1 A2 )*Q = ( x  0  )
    *                        ( 0  A3 )     ( x  x  )
    *  and
    *            V'*B*Q = V'*( B1 B2 )*Q = ( x  0  )
    *                        ( 0  B3 )     ( x  x  )
    *
    *  or if ( .NOT.UPPER ) then
    *
    *            U'*A*Q = U'*( A1 0  )*Q = ( x  x  )
    *                        ( A2 A3 )     ( 0  x  )
    *  and
    *            V'*B*Q = V'*( 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' denotes the transpose of Z.
    *
    *
    *  Arguments
    *  =========
    *
    *  UPPER   (input) LOGICAL
    *          = .TRUE.: the input matrices A and B are upper triangular.
    *          = .FALSE.: the input matrices A and B are lower triangular.
    *
    *  A1      (input) DOUBLE PRECISION
    *  A2      (input) DOUBLE PRECISION
    *  A3      (input) DOUBLE PRECISION
    *          On entry, A1, A2 and A3 are elements of the input 2-by-2
    *          upper (lower) triangular matrix A.
    *
    *  B1      (input) DOUBLE PRECISION
    *  B2      (input) DOUBLE PRECISION
    *  B3      (input) DOUBLE PRECISION
    *          On entry, B1, B2 and B3 are elements of the input 2-by-2
    *          upper (lower) triangular matrix B.
    *
    *  CSU     (output) DOUBLE PRECISION
    *  SNU     (output) DOUBLE PRECISION
    *          The desired orthogonal matrix U.
    *
    *  CSV     (output) DOUBLE PRECISION
    *  SNV     (output) DOUBLE PRECISION
    *          The desired orthogonal matrix V.
    *
    *  CSQ     (output) DOUBLE PRECISION
    *  SNQ     (output) DOUBLE PRECISION
    *          The desired orthogonal matrix Q.
    *
    *  =====================================================================
    *
    *     .. 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'*A and V'*B,
    *           and (1,2) element of |U|'*|A| and |V|'*|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'*A and V'*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'*A and V'*B,
    *           and (2,2) element of |U|'*|A| and |V|'*|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'*A and V'*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'*A and V'*B,
    *           and (2,1) element of |U|'*|A| and |V|'*|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'*A and V'*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'*A and V'*B,
    *           and (1,1) element of |U|'*|A| and |V|'*|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'*A and V'*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