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# dlag2

```
NAME
DLAG2 - compute 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 eigen-
value 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) 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 entry of B (in abso-
lute 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 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.

SCALE2  (output) DOUBLE PRECISION
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.

WR1     (output) DOUBLE PRECISION
If the eigenvalue is real, then WR1 is SCALE1 times
the eigenvalue closest to the (2,2) entry of A
B**(-1).  If the eigenvalue 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.
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