SLAG2(1) LAPACK auxiliary routine (version 3.2) SLAG2(1)NAME
SLAG2 - computes the eigenvalues of a 2 x 2 generalized eigenvalue
problem A - 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 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) 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 eigenval‐
ue equation which defines the first eigenvalue. If the eigen‐
values 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 eigenval‐
ue equation which defines the second eigenvalue. If the eigen‐
values 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 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) 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.
LAPACK auxiliary routine (versioNovember 2008 SLAG2(1)