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This derivative is an odd function and is always positive. It means that the growth
of the distance with respect to
ε
is the same on both the left and right of
ζ
=
0
.
5.
In particular, for
ζ
=
0
.
5 (TLS),
TLS
=
2
σ
d
ε
d
2
i
(6.77)
6.5.2 Analysis of the Maximum
α
1
The analysis of the maximum
α
1
is the same as in the case of saddles. However,
some crossovers miss.
For
ζ
≤
0
.
5, the following
strictest
inequalities derived from the previous
theory are true:
0
OLS
1
−
ζ
b
T
b
2
(
1
−
ζ )
l
3
2
1
2
(
1
−
ζ )
l
2
2
1
2
ζ
u
1
ζ
−
≤
ζ
≤
0
.
5
γ
σ
σ
=
≤
≤
α
1
≤
(6.78)
σ
1
2
2
1
0
≤
ζ
≤
ζ
−
σ
≤
α
i
≤
ζ
l
4
ζ
u
1
2
ζ
→
0 (OLS) the lower bound
l
4 and the upper bound
u
1tendtoinfinity;
as seen before,
For
α
1
→∞
ζ
≥
.
.For
0
5, there is no right crossover and the
strictest
inequality is given by
2
1
2
ζ
l
3
2
1
2
(
1
−
ζ )
u
2
σ
σ
≤
α
1
≤
(6.79)
However, from the inequality (6.40) it follows that
0
OLS
1
−
ζ
γ
≤
α
(6.80)
1
l
ζ
→
For
1 (DLS), this lower bound tends to infinity, as seen previously. The
same is true for the upper bound
u
2. Indeed, the maximum eigenvalue for DLS
is infinite.
6.5.3 Analysis of the Minimum
α
n
+1
Also in the analysis of the minimum
α
n
+
1
, previous considerations for the max-
imum can be repeated. For
ζ
≤
0
.
5, there is no left crossover and the
strictest
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