slag2(3F)

NAME
     SLAG2 - compute the eigenvalues of a 2 x 2 generalized eigenvalue problem
     A - w B, with scaling as necessary to avoid over-/underflow

SYNOPSIS
     SUBROUTINE SLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI )

         INTEGER       LDA, LDB

         REAL          SAFMIN, SCALE1, SCALE2, WI, WR1, WR2

         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.