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SUBROUTINE CLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
$ SNV, CSQ, SNQ ) * * -- LAPACK auxiliary routine (version 3.3.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2011 -- * * .. Scalar Arguments .. LOGICAL UPPER REAL A1, A3, B1, B3, CSQ, CSU, CSV COMPLEX A2, B2, SNQ, SNU, SNV * .. * * Purpose * ======= * * CLAGS2 computes 2-by-2 unitary matrices U, V and Q, such * that if ( UPPER ) then * * U**H *A*Q = U**H *( A1 A2 )*Q = ( x 0 ) * ( 0 A3 ) ( x x ) * and * V**H*B*Q = V**H *( B1 B2 )*Q = ( x 0 ) * ( 0 B3 ) ( x x ) * * or if ( .NOT.UPPER ) then * * U**H *A*Q = U**H *( A1 0 )*Q = ( x x ) * ( A2 A3 ) ( 0 x ) * and * V**H *B*Q = V**H *( B1 0 )*Q = ( x x ) * ( B2 B3 ) ( 0 x ) * where * * U = ( CSU SNU ), V = ( CSV SNV ), * ( -SNU**H CSU ) ( -SNV**H CSV ) * * Q = ( CSQ SNQ ) * ( -SNQ**H CSQ ) * * The rows of the transformed A and B are parallel. Moreover, if the * input 2-by-2 matrix A is not zero, then the transformed (1,1) entry * of A is not zero. If the input matrices A and B are both not zero, * then the transformed (2,2) element of B is not zero, except when the * first rows of input A and B are parallel and the second rows are * zero. * * 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) REAL * A2 (input) COMPLEX * A3 (input) REAL * On entry, A1, A2 and A3 are elements of the input 2-by-2 * upper (lower) triangular matrix A. * * B1 (input) REAL * B2 (input) COMPLEX * B3 (input) REAL * On entry, B1, B2 and B3 are elements of the input 2-by-2 * upper (lower) triangular matrix B. * * CSU (output) REAL * SNU (output) COMPLEX * The desired unitary matrix U. * * CSV (output) REAL * SNV (output) COMPLEX * The desired unitary matrix V. * * CSQ (output) REAL * SNQ (output) COMPLEX * The desired unitary matrix Q. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. REAL A, AUA11, AUA12, AUA21, AUA22, AVB11, AVB12, $ AVB21, AVB22, CSL, CSR, D, FB, FC, S1, S2, SNL, $ SNR, UA11R, UA22R, VB11R, VB22R COMPLEX B, C, D1, R, T, UA11, UA12, UA21, UA22, VB11, $ VB12, VB21, VB22 * .. * .. External Subroutines .. EXTERNAL CLARTG, SLASV2 * .. * .. Intrinsic Functions .. INTRINSIC ABS, AIMAG, CMPLX, CONJG, REAL * .. * .. Statement Functions .. REAL ABS1 * .. * .. Statement Function definitions .. ABS1( T ) = ABS( REAL( T ) ) + ABS( AIMAG( T ) ) * .. * .. 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 FB = ABS( B ) * * Transform complex 2-by-2 matrix C to real matrix by unitary * diagonal matrix diag(1,D1). * D1 = ONE IF( FB.NE.ZERO ) $ D1 = B / FB * * 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 SLASV2( A, FB, 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**H *A and V**H *B, * and (1,2) element of |U|**H *|A| and |V|**H *|B|. * UA11R = CSL*A1 UA12 = CSL*A2 + D1*SNL*A3 * VB11R = CSR*B1 VB12 = CSR*B2 + D1*SNR*B3 * AUA12 = ABS( CSL )*ABS1( A2 ) + ABS( SNL )*ABS( A3 ) AVB12 = ABS( CSR )*ABS1( B2 ) + ABS( SNR )*ABS( B3 ) * * zero (1,2) elements of U**H *A and V**H *B * IF( ( ABS( UA11R )+ABS1( UA12 ) ).EQ.ZERO ) THEN CALL CLARTG( -CMPLX( VB11R ), CONJG( VB12 ), CSQ, SNQ, $ R ) ELSE IF( ( ABS( VB11R )+ABS1( VB12 ) ).EQ.ZERO ) THEN CALL CLARTG( -CMPLX( UA11R ), CONJG( UA12 ), CSQ, SNQ, $ R ) ELSE IF( AUA12 / ( ABS( UA11R )+ABS1( UA12 ) ).LE.AVB12 / $ ( ABS( VB11R )+ABS1( VB12 ) ) ) THEN CALL CLARTG( -CMPLX( UA11R ), CONJG( UA12 ), CSQ, SNQ, $ R ) ELSE CALL CLARTG( -CMPLX( VB11R ), CONJG( VB12 ), CSQ, SNQ, $ R ) END IF * CSU = CSL SNU = -D1*SNL CSV = CSR SNV = -D1*SNR * ELSE * * Compute the (2,1) and (2,2) elements of U**H *A and V**H *B, * and (2,2) element of |U|**H *|A| and |V|**H *|B|. * UA21 = -CONJG( D1 )*SNL*A1 UA22 = -CONJG( D1 )*SNL*A2 + CSL*A3 * VB21 = -CONJG( D1 )*SNR*B1 VB22 = -CONJG( D1 )*SNR*B2 + CSR*B3 * AUA22 = ABS( SNL )*ABS1( A2 ) + ABS( CSL )*ABS( A3 ) AVB22 = ABS( SNR )*ABS1( B2 ) + ABS( CSR )*ABS( B3 ) * * zero (2,2) elements of U**H *A and V**H *B, and then swap. * IF( ( ABS1( UA21 )+ABS1( UA22 ) ).EQ.ZERO ) THEN CALL CLARTG( -CONJG( VB21 ), CONJG( VB22 ), CSQ, SNQ, R ) ELSE IF( ( ABS1( VB21 )+ABS( VB22 ) ).EQ.ZERO ) THEN CALL CLARTG( -CONJG( UA21 ), CONJG( UA22 ), CSQ, SNQ, R ) ELSE IF( AUA22 / ( ABS1( UA21 )+ABS1( UA22 ) ).LE.AVB22 / $ ( ABS1( VB21 )+ABS1( VB22 ) ) ) THEN CALL CLARTG( -CONJG( UA21 ), CONJG( UA22 ), CSQ, SNQ, R ) ELSE CALL CLARTG( -CONJG( VB21 ), CONJG( VB22 ), CSQ, SNQ, R ) END IF * CSU = SNL SNU = D1*CSL CSV = SNR SNV = D1*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 FC = ABS( C ) * * Transform complex 2-by-2 matrix C to real matrix by unitary * diagonal matrix diag(d1,1). * D1 = ONE IF( FC.NE.ZERO ) $ D1 = C / FC * * 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 SLASV2( A, FC, 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**H *A and V**H *B, * and (2,1) element of |U|**H *|A| and |V|**H *|B|. * UA21 = -D1*SNR*A1 + CSR*A2 UA22R = CSR*A3 * VB21 = -D1*SNL*B1 + CSL*B2 VB22R = CSL*B3 * AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS1( A2 ) AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS1( B2 ) * * zero (2,1) elements of U**H *A and V**H *B. * IF( ( ABS1( UA21 )+ABS( UA22R ) ).EQ.ZERO ) THEN CALL CLARTG( CMPLX( VB22R ), VB21, CSQ, SNQ, R ) ELSE IF( ( ABS1( VB21 )+ABS( VB22R ) ).EQ.ZERO ) THEN CALL CLARTG( CMPLX( UA22R ), UA21, CSQ, SNQ, R ) ELSE IF( AUA21 / ( ABS1( UA21 )+ABS( UA22R ) ).LE.AVB21 / $ ( ABS1( VB21 )+ABS( VB22R ) ) ) THEN CALL CLARTG( CMPLX( UA22R ), UA21, CSQ, SNQ, R ) ELSE CALL CLARTG( CMPLX( VB22R ), VB21, CSQ, SNQ, R ) END IF * CSU = CSR SNU = -CONJG( D1 )*SNR CSV = CSL SNV = -CONJG( D1 )*SNL * ELSE * * Compute the (1,1) and (1,2) elements of U**H *A and V**H *B, * and (1,1) element of |U|**H *|A| and |V|**H *|B|. * UA11 = CSR*A1 + CONJG( D1 )*SNR*A2 UA12 = CONJG( D1 )*SNR*A3 * VB11 = CSL*B1 + CONJG( D1 )*SNL*B2 VB12 = CONJG( D1 )*SNL*B3 * AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS1( A2 ) AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS1( B2 ) * * zero (1,1) elements of U**H *A and V**H *B, and then swap. * IF( ( ABS1( UA11 )+ABS1( UA12 ) ).EQ.ZERO ) THEN CALL CLARTG( VB12, VB11, CSQ, SNQ, R ) ELSE IF( ( ABS1( VB11 )+ABS1( VB12 ) ).EQ.ZERO ) THEN CALL CLARTG( UA12, UA11, CSQ, SNQ, R ) ELSE IF( AUA11 / ( ABS1( UA11 )+ABS1( UA12 ) ).LE.AVB11 / $ ( ABS1( VB11 )+ABS1( VB12 ) ) ) THEN CALL CLARTG( UA12, UA11, CSQ, SNQ, R ) ELSE CALL CLARTG( VB12, VB11, CSQ, SNQ, R ) END IF * CSU = SNR SNU = CONJG( D1 )*CSR CSV = SNL SNV = CONJG( D1 )*CSL * END IF * END IF * RETURN * * End of CLAGS2 * END |