DLAG2(3)                   MathKeisan LAPACK routine                  DLAG2(3)



NAME
       DLAG2 - the eigenvalues of a 2 x 2 generalized eigenvalue problem A - w
       B, with scaling as necessary to avoid over-/underflow

SYNOPSIS
       SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2,  WI
                         )

           INTEGER       LDA, LDB

           DOUBLE        PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2

           DOUBLE        PRECISION A( LDA, * ), B( LDB, * )

PURPOSE
       DLAG2  computes the eigenvalues of a 2 x 2 generalized eigenvalue prob-
       lem  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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
               The  smallest  positive number s.t. 1/SAFMIN does not overflow.
               (This should always be DLAMCH('S') --  it  is  an  argument  in
               order to avoid having to call DLAMCH frequently.)

       SCALE1  (output) DOUBLE PRECISION
               A  scaling  factor  used to avoid over-/underflow in the eigen-
               value 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 posi-
               tive.  If the eigenvalues  are  real,  then  the  first  (real)
               eigenvalue  is   WR1 / SCALE1 , but this may overflow or under-
               flow, and in fact, SCALE1 may be zero or less than  the  under-
               flow  threshhold if the exact eigenvalue is sufficiently large.

       SCALE2  (output) DOUBLE PRECISION
               A scaling factor used to avoid over-/underflow  in  the  eigen-
               value  equation  which  defines  the second eigenvalue.  If the
               eigenvalues are complex, then SCALE2=SCALE1.  If the  eigenval-
               ues 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) DOUBLE PRECISION
               If the eigenvalue is real, then WR1 is SCALE1 times the  eigen-
               value closest to the (2,2) element of A B**(-1).  If the eigen-
               value is complex, then WR1=WR2 is SCALE1 times the real part of
               the eigenvalues.

       WR2     (output) DOUBLE PRECISION
               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) DOUBLE PRECISION
               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.



 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                        DLAG2(3)