slag2.f

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*> \brief \b SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
*                         WR2, WI )
* 
*       .. Scalar Arguments ..
*       INTEGER            LDA, LDB
*       REAL               SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), B( LDB, * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] A
*> \verbatim
*>          A is 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= 2.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is 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.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= 2.
*> \endverbatim
*>
*> \param[in] SAFMIN
*> \verbatim
*>          SAFMIN is 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.)
*> \endverbatim
*>
*> \param[out] SCALE1
*> \verbatim
*>          SCALE1 is 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 threshold if the exact eigenvalue
*>          is sufficiently large.
*> \endverbatim
*>
*> \param[out] SCALE2
*> \verbatim
*>          SCALE2 is 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
*>          threshold if the exact eigenvalue is sufficiently large.
*> \endverbatim
*>
*> \param[out] WR1
*> \verbatim
*>          WR1 is 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.
*> \endverbatim
*>
*> \param[out] WR2
*> \verbatim
*>          WR2 is 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.
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*>          WI is 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.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date June 2016
*
*> \ingroup realOTHERauxiliary
*
*  =====================================================================
      SUBROUTINE slag2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
     $                  wr2, wi )
*
*  -- LAPACK auxiliary routine (version 3.6.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
*     .. Scalar Arguments ..
      INTEGER            LDA, LDB
      REAL               SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
*     ..
*     .. Array Arguments ..
      REAL               A( lda, * ), B( ldb, * )
*     ..
*
*  =====================================================================
*
*     .. 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 threshold and handle numbers above that
*           threshold 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