SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, $ SNV, CSQ, SNQ ) * * -- LAPACK auxiliary routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver 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