```      SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
\$                   CSR, SNR )
*
*  -- LAPACK auxiliary routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
INTEGER            LDA, LDB
DOUBLE PRECISION   CSL, CSR, SNL, SNR
*     ..
*     .. Array Arguments ..
DOUBLE PRECISION   A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ),
\$                   B( LDB, * ), BETA( 2 )
*     ..
*
*  Purpose
*  =======
*
*  DLAGV2 computes the Generalized Schur factorization of a real 2-by-2
*  matrix pencil (A,B) where B is upper triangular. This routine
*  computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
*  SNR such that
*
*  1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
*     types), then
*
*     [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
*     [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
*
*     [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
*     [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],
*
*  2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
*     then
*
*     [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
*     [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
*
*     [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
*     [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]
*
*     where b11 >= b22 > 0.
*
*
*  Arguments
*  =========
*
*  A       (input/output) DOUBLE PRECISION array, dimension (LDA, 2)
*          On entry, the 2 x 2 matrix A.
*          On exit, A is overwritten by the ``A-part'' of the
*          generalized Schur form.
*
*  LDA     (input) INTEGER
*          THe leading dimension of the array A.  LDA >= 2.
*
*  B       (input/output) DOUBLE PRECISION array, dimension (LDB, 2)
*          On entry, the upper triangular 2 x 2 matrix B.
*          On exit, B is overwritten by the ``B-part'' of the
*          generalized Schur form.
*
*  LDB     (input) INTEGER
*          THe leading dimension of the array B.  LDB >= 2.
*
*  ALPHAR  (output) DOUBLE PRECISION array, dimension (2)
*  ALPHAI  (output) DOUBLE PRECISION array, dimension (2)
*  BETA    (output) DOUBLE PRECISION array, dimension (2)
*          (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
*          pencil (A,B), k=1,2, i = sqrt(-1).  Note that BETA(k) may
*          be zero.
*
*  CSL     (output) DOUBLE PRECISION
*          The cosine of the left rotation matrix.
*
*  SNL     (output) DOUBLE PRECISION
*          The sine of the left rotation matrix.
*
*  CSR     (output) DOUBLE PRECISION
*          The cosine of the right rotation matrix.
*
*  SNR     (output) DOUBLE PRECISION
*          The sine of the right rotation matrix.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
*
*     .. Parameters ..
DOUBLE PRECISION   ZERO, ONE
PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
DOUBLE PRECISION   ANORM, ASCALE, BNORM, BSCALE, H1, H2, H3, QQ,
\$                   R, RR, SAFMIN, SCALE1, SCALE2, T, ULP, WI, WR1,
\$                   WR2
*     ..
*     .. External Subroutines ..
EXTERNAL           DLAG2, DLARTG, DLASV2, DROT
*     ..
*     .. External Functions ..
DOUBLE PRECISION   DLAMCH, DLAPY2
EXTERNAL           DLAMCH, DLAPY2
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
SAFMIN = DLAMCH( 'S' )
ULP = DLAMCH( 'P' )
*
*     Scale A
*
ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
\$        ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
ASCALE = ONE / ANORM
A( 1, 1 ) = ASCALE*A( 1, 1 )
A( 1, 2 ) = ASCALE*A( 1, 2 )
A( 2, 1 ) = ASCALE*A( 2, 1 )
A( 2, 2 ) = ASCALE*A( 2, 2 )
*
*     Scale B
*
BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 1, 2 ) )+ABS( B( 2, 2 ) ),
\$        SAFMIN )
BSCALE = ONE / BNORM
B( 1, 1 ) = BSCALE*B( 1, 1 )
B( 1, 2 ) = BSCALE*B( 1, 2 )
B( 2, 2 ) = BSCALE*B( 2, 2 )
*
*     Check if A can be deflated
*
IF( ABS( A( 2, 1 ) ).LE.ULP ) THEN
CSL = ONE
SNL = ZERO
CSR = ONE
SNR = ZERO
A( 2, 1 ) = ZERO
B( 2, 1 ) = ZERO
*
*     Check if B is singular
*
ELSE IF( ABS( B( 1, 1 ) ).LE.ULP ) THEN
CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
CSR = ONE
SNR = ZERO
CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
A( 2, 1 ) = ZERO
B( 1, 1 ) = ZERO
B( 2, 1 ) = ZERO
*
ELSE IF( ABS( B( 2, 2 ) ).LE.ULP ) THEN
CALL DLARTG( A( 2, 2 ), A( 2, 1 ), CSR, SNR, T )
SNR = -SNR
CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
CSL = ONE
SNL = ZERO
A( 2, 1 ) = ZERO
B( 2, 1 ) = ZERO
B( 2, 2 ) = ZERO
*
ELSE
*
*        B is nonsingular, first compute the eigenvalues of (A,B)
*
CALL DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2,
\$               WI )
*
IF( WI.EQ.ZERO ) THEN
*
*           two real eigenvalues, compute s*A-w*B
*
H1 = SCALE1*A( 1, 1 ) - WR1*B( 1, 1 )
H2 = SCALE1*A( 1, 2 ) - WR1*B( 1, 2 )
H3 = SCALE1*A( 2, 2 ) - WR1*B( 2, 2 )
*
RR = DLAPY2( H1, H2 )
QQ = DLAPY2( SCALE1*A( 2, 1 ), H3 )
*
IF( RR.GT.QQ ) THEN
*
*              find right rotation matrix to zero 1,1 element of
*              (sA - wB)
*
CALL DLARTG( H2, H1, CSR, SNR, T )
*
ELSE
*
*              find right rotation matrix to zero 2,1 element of
*              (sA - wB)
*
CALL DLARTG( H3, SCALE1*A( 2, 1 ), CSR, SNR, T )
*
END IF
*
SNR = -SNR
CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
*
*           compute inf norms of A and B
*
H1 = MAX( ABS( A( 1, 1 ) )+ABS( A( 1, 2 ) ),
\$           ABS( A( 2, 1 ) )+ABS( A( 2, 2 ) ) )
H2 = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
\$           ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
*
IF( ( SCALE1*H1 ).GE.ABS( WR1 )*H2 ) THEN
*
*              find left rotation matrix Q to zero out B(2,1)
*
CALL DLARTG( B( 1, 1 ), B( 2, 1 ), CSL, SNL, R )
*
ELSE
*
*              find left rotation matrix Q to zero out A(2,1)
*
CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
*
END IF
*
CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
*
A( 2, 1 ) = ZERO
B( 2, 1 ) = ZERO
*
ELSE
*
*           a pair of complex conjugate eigenvalues
*           first compute the SVD of the matrix B
*
CALL DLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR,
\$                   CSR, SNL, CSL )
*
*           Form (A,B) := Q(A,B)Z' where Q is left rotation matrix and
*           Z is right rotation matrix computed from DLASV2
*
CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
*
B( 2, 1 ) = ZERO
B( 1, 2 ) = ZERO
*
END IF
*
END IF
*
*     Unscaling
*
A( 1, 1 ) = ANORM*A( 1, 1 )
A( 2, 1 ) = ANORM*A( 2, 1 )
A( 1, 2 ) = ANORM*A( 1, 2 )
A( 2, 2 ) = ANORM*A( 2, 2 )
B( 1, 1 ) = BNORM*B( 1, 1 )
B( 2, 1 ) = BNORM*B( 2, 1 )
B( 1, 2 ) = BNORM*B( 1, 2 )
B( 2, 2 ) = BNORM*B( 2, 2 )
*
IF( WI.EQ.ZERO ) THEN
ALPHAR( 1 ) = A( 1, 1 )
ALPHAR( 2 ) = A( 2, 2 )
ALPHAI( 1 ) = ZERO
ALPHAI( 2 ) = ZERO
BETA( 1 ) = B( 1, 1 )
BETA( 2 ) = B( 2, 2 )
ELSE
ALPHAR( 1 ) = ANORM*WR1 / SCALE1 / BNORM
ALPHAI( 1 ) = ANORM*WI / SCALE1 / BNORM
ALPHAR( 2 ) = ALPHAR( 1 )
ALPHAI( 2 ) = -ALPHAI( 1 )
BETA( 1 ) = ONE
BETA( 2 ) = ONE
END IF
*
RETURN
*
*     End of DLAGV2
*
END

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