slag2()

subroutine slag2	(	real, dimension( lda, * )	a,
		integer	lda,
		real, dimension( ldb, * )	b,
		integer	ldb,
		real	safmin,
		real	scale1,
		real	scale2,
		real	wr1,
		real	wr2,
		real	wi
	)

SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.

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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.

Parameters

[in]	A	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.
[in]	LDA	LDA is INTEGER The leading dimension of the array A. LDA >= 2.
[in]	B	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.
[in]	LDB	LDB is INTEGER The leading dimension of the array B. LDB >= 2.
[in]	SAFMIN	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.)
[out]	SCALE1	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.
[out]	SCALE2	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.
[out]	WR1	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.
[out]	WR2	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.
[out]	WI	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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 154 of file slag2.f.

*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. 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

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