SLAG2 (3lapack)

compute the eigenvalues of a 2 x 2 generalized eigenvalue problem A  (3lapack)
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.



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