slag2.f

          SUBROUTINE SLAG2( 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
          REAL               SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
    *     ..
    *     .. Array Arguments ..
          REAL               A( LDA, * ), B( LDB, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SLAG2 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) REAL 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) REAL 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) REAL
    *          The smallest positive number s.t. 1/SAFMIN does not
    *          overflow.  (This should always be SLAMCH('S') -- it is an
    *          argument in order to avoid having to call SLAMCH frequently.)
    *
    *  SCALE1  (output) REAL
    *          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) REAL
    *          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) REAL
    *          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) REAL
    *          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) REAL
    *          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 ..
          REAL               ZERO, ONE, TWO
          PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 )
          REAL               HALF
          PARAMETER          ( HALF = ONE / TWO )
          REAL               FUZZY1
          PARAMETER          ( FUZZY1 = ONE+1.0E-5 )
    *     ..
    *     .. Local Scalars ..
          REAL               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 SLAG2
    *
          RETURN
          END