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+      SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
+     $                  WR2, WI )
+*
+*  -- 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   SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
+*     ..
+*     .. Array Arguments ..
+      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
+*     ..
+*
+*  Purpose
+*  =======
+*
+*  DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
+*  problem  A - w B, with scaling as necessary to avoid over-/underflow.
+*
+*  The scaling factor "s" results in a modified eigenvalue equation
+*
+*      s A - w B
+*
+*  where  s  is a non-negative scaling factor chosen so that  w,  w B,
+*  and  s A  do not overflow and, if possible, do not underflow, either.
+*
+*  Arguments
+*  =========
+*
+*  A       (input) DOUBLE PRECISION array, dimension (LDA, 2)
+*          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm
+*          is less than 1/SAFMIN.  Entries less than
+*          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
+*
+*  LDA     (input) INTEGER
+*          The leading dimension of the array A.  LDA >= 2.
+*
+*  B       (input) DOUBLE PRECISION array, dimension (LDB, 2)
+*          On entry, the 2 x 2 upper triangular matrix B.  It is
+*          assumed that the one-norm of B is less than 1/SAFMIN.  The
+*          diagonals should be at least sqrt(SAFMIN) times the largest
+*          element of B (in absolute value); if a diagonal is smaller
+*          than that, then  +/- sqrt(SAFMIN) will be used instead of
+*          that diagonal.
+*
+*  LDB     (input) INTEGER
+*          The leading dimension of the array B.  LDB >= 2.
+*
+*  SAFMIN  (input) DOUBLE PRECISION
+*          The smallest positive number s.t. 1/SAFMIN does not
+*          overflow.  (This should always be DLAMCH('S') -- it is an
+*          argument in order to avoid having to call DLAMCH frequently.)
+*
+*  SCALE1  (output) DOUBLE PRECISION
+*          A scaling factor used to avoid over-/underflow in the
+*          eigenvalue equation which defines the first eigenvalue.  If
+*          the eigenvalues are complex, then the eigenvalues are
+*          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
+*          exponent range of the machine), SCALE1=SCALE2, and SCALE1
+*          will always be positive.  If the eigenvalues are real, then
+*          the first (real) eigenvalue is  WR1 / SCALE1 , but this may
+*          overflow or underflow, and in fact, SCALE1 may be zero or
+*          less than the underflow threshhold if the exact eigenvalue
+*          is sufficiently large.
+*
+*  SCALE2  (output) DOUBLE PRECISION
+*          A scaling factor used to avoid over-/underflow in the
+*          eigenvalue equation which defines the second eigenvalue.  If
+*          the eigenvalues are complex, then SCALE2=SCALE1.  If the
+*          eigenvalues are real, then the second (real) eigenvalue is
+*          WR2 / SCALE2 , but this may overflow or underflow, and in
+*          fact, SCALE2 may be zero or less than the underflow
+*          threshhold if the exact eigenvalue is sufficiently large.
+*
+*  WR1     (output) DOUBLE PRECISION
+*          If the eigenvalue is real, then WR1 is SCALE1 times the
+*          eigenvalue closest to the (2,2) element of A B**(-1).  If the
+*          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
+*          part of the eigenvalues.
+*
+*  WR2     (output) DOUBLE PRECISION
+*          If the eigenvalue is real, then WR2 is SCALE2 times the
+*          other eigenvalue.  If the eigenvalue is complex, then
+*          WR1=WR2 is SCALE1 times the real part of the eigenvalues.
+*
+*  WI      (output) DOUBLE PRECISION
+*          If the eigenvalue is real, then WI is zero.  If the
+*          eigenvalue is complex, then WI is SCALE1 times the imaginary
+*          part of the eigenvalues.  WI will always be non-negative.
+*
+*  =====================================================================
+*
+*     .. Parameters ..
+      DOUBLE PRECISION   ZERO, ONE, TWO
+      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
+      DOUBLE PRECISION   HALF
+      PARAMETER          ( HALF = ONE / TWO )
+      DOUBLE PRECISION   FUZZY1
+      PARAMETER          ( FUZZY1 = ONE+1.0D-5 )
+*     ..
+*     .. Local Scalars ..
+      DOUBLE PRECISION   A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
+     $                   AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
+     $                   BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
+     $                   DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
+     $                   SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
+     $                   WSCALE, WSIZE, WSMALL
+*     ..
+*     .. Intrinsic Functions ..
+      INTRINSIC          ABS, MAX, MIN, SIGN, SQRT
+*     ..
+*     .. Executable Statements ..
+*
+      RTMIN = SQRT( SAFMIN )
+      RTMAX = ONE / RTMIN
+      SAFMAX = ONE / SAFMIN
+*
+*     Scale A
+*
+      ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
+     $        ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
+      ASCALE = ONE / ANORM
+      A11 = ASCALE*A( 1, 1 )
+      A21 = ASCALE*A( 2, 1 )
+      A12 = ASCALE*A( 1, 2 )
+      A22 = ASCALE*A( 2, 2 )
+*
+*     Perturb B if necessary to insure non-singularity
+*
+      B11 = B( 1, 1 )
+      B12 = B( 1, 2 )
+      B22 = B( 2, 2 )
+      BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
+      IF( ABS( B11 ).LT.BMIN )
+     $   B11 = SIGN( BMIN, B11 )
+      IF( ABS( B22 ).LT.BMIN )
+     $   B22 = SIGN( BMIN, B22 )
+*
+*     Scale B
+*
+      BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
+      BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
+      BSCALE = ONE / BSIZE
+      B11 = B11*BSCALE
+      B12 = B12*BSCALE
+      B22 = B22*BSCALE
+*
+*     Compute larger eigenvalue by method described by C. van Loan
+*
+*     ( AS is A shifted by -SHIFT*B )
+*
+      BINV11 = ONE / B11
+      BINV22 = ONE / B22
+      S1 = A11*BINV11
+      S2 = A22*BINV22
+      IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
+         AS12 = A12 - S1*B12
+         AS22 = A22 - S1*B22
+         SS = A21*( BINV11*BINV22 )
+         ABI22 = AS22*BINV22 - SS*B12
+         PP = HALF*ABI22
+         SHIFT = S1
+      ELSE
+         AS12 = A12 - S2*B12
+         AS11 = A11 - S2*B11
+         SS = A21*( BINV11*BINV22 )
+         ABI22 = -SS*B12
+         PP = HALF*( AS11*BINV11+ABI22 )
+         SHIFT = S2
+      END IF
+      QQ = SS*AS12
+      IF( ABS( PP*RTMIN ).GE.ONE ) THEN
+         DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
+         R = SQRT( ABS( DISCR ) )*RTMAX
+      ELSE
+         IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
+            DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
+            R = SQRT( ABS( DISCR ) )*RTMIN
+         ELSE
+            DISCR = PP**2 + QQ
+            R = SQRT( ABS( DISCR ) )
+         END IF
+      END IF
+*
+*     Note: the test of R in the following IF is to cover the case when
+*           DISCR is small and negative and is flushed to zero during
+*           the calculation of R.  On machines which have a consistent
+*           flush-to-zero threshhold and handle numbers above that
+*           threshhold correctly, it would not be necessary.
+*
+      IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
+         SUM = PP + SIGN( R, PP )
+         DIFF = PP - SIGN( R, PP )
+         WBIG = SHIFT + SUM
+*
+*        Compute smaller eigenvalue
+*
+         WSMALL = SHIFT + DIFF
+         IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
+            WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
+            WSMALL = WDET / WBIG
+         END IF
+*
+*        Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
+*        for WR1.
+*
+         IF( PP.GT.ABI22 ) THEN
+            WR1 = MIN( WBIG, WSMALL )
+            WR2 = MAX( WBIG, WSMALL )
+         ELSE
+            WR1 = MAX( WBIG, WSMALL )
+            WR2 = MIN( WBIG, WSMALL )
+         END IF
+         WI = ZERO
+      ELSE
+*
+*        Complex eigenvalues
+*
+         WR1 = SHIFT + PP
+         WR2 = WR1
+         WI = R
+      END IF
+*
+*     Further scaling to avoid underflow and overflow in computing
+*     SCALE1 and overflow in computing w*B.
+*
+*     This scale factor (WSCALE) is bounded from above using C1 and C2,
+*     and from below using C3 and C4.
+*        C1 implements the condition  s A  must never overflow.
+*        C2 implements the condition  w B  must never overflow.
+*        C3, with C2,
+*           implement the condition that s A - w B must never overflow.
+*        C4 implements the condition  s    should not underflow.
+*        C5 implements the condition  max(s,|w|) should be at least 2.
+*
+      C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
+      C2 = SAFMIN*MAX( ONE, BNORM )
+      C3 = BSIZE*SAFMIN
+      IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
+         C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
+      ELSE
+         C4 = ONE
+      END IF
+      IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
+         C5 = MIN( ONE, ASCALE*BSIZE )
+      ELSE
+         C5 = ONE
+      END IF
+*
+*     Scale first eigenvalue
+*
+      WABS = ABS( WR1 ) + ABS( WI )
+      WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
+     $        MIN( C4, HALF*MAX( WABS, C5 ) ) )
+      IF( WSIZE.NE.ONE ) THEN
+         WSCALE = ONE / WSIZE
+         IF( WSIZE.GT.ONE ) THEN
+            SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
+     $               MIN( ASCALE, BSIZE )
+         ELSE
+            SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
+     $               MAX( ASCALE, BSIZE )
+         END IF
+         WR1 = WR1*WSCALE
+         IF( WI.NE.ZERO ) THEN
+            WI = WI*WSCALE
+            WR2 = WR1
+            SCALE2 = SCALE1
+         END IF
+      ELSE
+         SCALE1 = ASCALE*BSIZE
+         SCALE2 = SCALE1
+      END IF
+*
+*     Scale second eigenvalue (if real)
+*
+      IF( WI.EQ.ZERO ) THEN
+         WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
+     $           MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
+         IF( WSIZE.NE.ONE ) THEN
+            WSCALE = ONE / WSIZE
+            IF( WSIZE.GT.ONE ) THEN
+               SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
+     $                  MIN( ASCALE, BSIZE )
+            ELSE
+               SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
+     $                  MAX( ASCALE, BSIZE )
+            END IF
+            WR2 = WR2*WSCALE
+         ELSE
+            SCALE2 = ASCALE*BSIZE
+         END IF
+      END IF
+*
+*     End of DLAG2
+*
+      RETURN
+      END