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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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