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incompatibility of the TLS problem
Ax
≈
b
, indicating just how closely
b
is
linearly related to
A
.Alarge
σ
p
+
1
means highly conflicting equations.
Remark 118 (Highly Conflicting Equations)
If
σ
p
+
1
→
σ
p
(
highly conflicting
equations
)
, the domain of convergence worsens because of the approach of the
saddle to the corresponding asymptote
(
good equations have faraway saddles
)
.
Furthermore, the motor gap of the saddles is smaller, thus worsening the conver-
gence speed of the iterative methods, and the maximum and the origin are much
farther from the saddle/solution.
Since
σ
p
+
1
=
σ
p
is the smallest singular value, it follows that
A
is nearly
rank-deficient, or else the set of equations is highly incompatible. The various
techniques devised to solve this problem are described in Section 1.7. In prac-
tice, close-to-nongeneric TLS problems are more common. In these cases the
generic TLS solution can still be computed, but it is unstable and becomes very
sensitive to data errors when
σ
p
approaches
σ
p
+
1
. Identifying the problem as non-
generic stabilizes the solution, making it insensitive to data errors. Unfortunately,
a numerical rank determination of a noisy matrix is required.
Remark 119 (No Observation Vector Identification)
If no matter which col-
umn is used as the right side, only an estimation of a linear relationship among
the columns of
[
A
0
, there is no need to solve the TLS
problem as nongeneric. It suffices to replace b by any column a
i
of A provided
that
;
b
]
is needed and
v
n
+
1,
n
+
1
=
v
i
,
n
+
1
=
0
.
5.4.1 Analysis of Convergence for
p
=
n
For
p
=
n
, the lowest critical point associated with
σ
n
+
1
goes to infinity in the
direction of
v
n
. Recalling (1.37), it holds that
±
v
n
0
b
⊥
u
n
v
n
+
1
=
and
(5.125)
The first property expresses the fact that the (
n
+
1)-dimensional vector
v
n
+
1
v
n
. From the second
is parallel to the TLS hyperplane, just in the direction of
property, it follows that
q
n
=
v
n
A
T
b
=
σ
n
u
n
b
=
0
(5.126)
which implies that for every hyperbola [eq. (5.104) with
ζ
=
0
.
5
0
q
n
λ
i
−
γ
nongeneric case
z
cn
=
=
(5.127)
≈
0
close-to-nongeneric case
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