Graphics Programs Reference
In-Depth Information
which says that
v
is an eigenvector for
S
witheigenvalue 1
+
(
α/λ
)
e
α
T
. Since
the eigenvalues of
S
are the
N
throots of unity, whichare evenly spaced
around the unit circle in the complex plane, and closely spaced together for
large
N
, there is potential instability whenever 1
+
(
α/λ
)
e
α
T
has absolute
value 1 for some
α
with positive real part: that is, whenever (
α
T
/λ
T
)
e
α
T
can
be of the form
e
i
θ
−
1 for some
α
T
with positive real part. Whether instability
occurs or not depends on the value of the product
λ
T
. We can see this by
plotting values of
z
exp(
z
) for
z
=
α
T
=
iy
a complex number on the critical
line Re
z
=
0, and comparing withplots of
λ
T
(
e
i
θ
−
1) for various values of
the parameter
λ
T
.
syms y; expand(i*y*(cos(y) + i*sin(y)))
ans =
i*y*cos(y)-y*sin(y)
ezplot(-y*sin(y), y*cos(y), [-2*pi, 2*pi]); hold on
theta = 0:0.05*pi:2*pi;
plot((1/2)*(cos(theta) - 1), (1/2)*sin(theta), '-');
plot(cos(theta) - 1, sin(theta), ':')
plot(2*(cos(theta) - 1), 2*sin(theta), '--');
title('iyeˆ{iy} and circles \lambda T(eˆ{i\theta}-1)');
hold off
iye
iy
and circles
T(e
i
θ
-1)
λ
6
4
2
0
-2
-4
-6
-6
-4
-2
0
2
4
6
8
x
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