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
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) 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')
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
threshhold 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
threshhold 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) element
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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