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SUBROUTINE CLAGS2( 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
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'*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 )
* where
*
* U = ( CSU SNU ), V = ( CSV SNV ),
* ( -CONJG(SNU) CSU ) ( -CONJG(SNV) CSV )
*
* Q = ( CSQ SNQ )
* ( -CONJG(SNQ) CSQ )
*
* Z' denotes the conjugate transpose of Z.
*
* 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'*A and V'*B,
* and (1,2) element of |U|'*|A| and |V|'*|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'*A and V'*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'*A and V'*B,
* and (2,2) element of |U|'*|A| and |V|'*|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'*A and V'*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'*A and V'*B,
* and (2,1) element of |U|'*|A| and |V|'*|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'*A and V'*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'*A and V'*B,
* and (1,1) element of |U|'*|A| and |V|'*|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'*A and V'*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
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