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.

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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= 2.

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.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= 2.

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.)

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 threshhold if the exact eigenvalue


          is sufficiently large.

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


          threshhold if the exact eigenvalue is sufficiently large.

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.

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.

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012