Error detected in the first version of the book
page 21, line 4 of (2.54), replace ]-oo,1/ybar] by ]-oo,1/yunderlined].
page 60, (Thank you Philippe Planchon).
Step 4 of imageSP, replace “[y] \subset”
by “y \in”.
page 61, formula (3.31) Replace x by x1 and y by x2.
page 66, Table 4.1, line 5. The f(x) should be in bold
font.
page 74, replace [10,10] by [-10,10] : three times.
page 86, (Thank you Stefan Ratschan).
The first implication of (4.92) is false. Indeed, we apply a multi-variate mean-value theorem, using
the Jacobian at a certain point \xi. However, in that
case, this \xi might be different for each component of the function f (see the
counterexample below).
So, instead of one \xi, we need n of them!
However, the resulting algorithm is correct, since by overestimating the Jacobi
matrix over the whole interval, the wrong information is not used anyway.
Counter-example :
> let [x]= [0,1] x [0, 0], let x_0 = (0, 0)^T, let x = (1, 0)^T
> let f1 be s.t. f1((0,0)^T)=0,
f1(((1,0)^T)=0, let f2 be s.t.
f2((0,0)^T)=0, f2(((1,0)^T)=0
> and such that the zero of df1/dx in [0, 1]x[0,0] is different from the
zero of df2/dx in [0, 1]x[0,0]
> then f(x)-f(x_0)= (0, 0)^T,
> but Jf(xi)(x-x0)= Jf(xi)(1,
0)^T= [df1/dx, df2/dx]^T
> so both df1/dx and df2/dx have to be zero at a certain xi in [x], which is
a contradiction
page 87, Table 4.10. In the arguments of the algorithm C_N, replace
“inout : [p]” by “inout
[x]”.
page 87, The parallel linearisation
has first been proposed by L. Kolev
in the paper “A new method for global solution of nonlinear equations. Reliable
Computing, 4 , № 2, 1998, pp.
125-146”. Thank you Lubomir
for this rectification.
page 92, (Thank you Arnaud Lallouet).
two lines after (4.106), C(Coo([x])=Coo([x]) is false
and thus the Theorem 4.3 is false. One should add the assumption that the sets
(or constraints) involved are all closed. A counterexample of the theorem is
the following.
If S1={x1,x2|x1+x2=0} and if S2={x1,x2| 2x1+x2=0 and
(x1.x2)^2>0}. It is clear that the intersection between S1 and S2 is empty.
If C1 and C2 are the optimal contractors by S1 and S2 and if C=C1oC2. For a
given box [x], then Coo([x])={0}, whereas
C(Coo([x]))=\emptyset. In such a case, Coo([x]) is not the largest subbox
which is a fixed point of [x]. .
page 95, line 4. Replace ‘remain’ by
remains.
page 98, formula (4.116) replace \cup by \sqcup, twice. Formula (4.117), replace the left \cup
by \sqcup.
page 126, formula (5.6), replace the first min by max
and the first max by min
page 185, figure 6.25, for k=1, one subfigure is
missing.
page 218, For the theorem 7.7
we have cited Malan et al., “Robust analysis and design of control
systems using interval arithmetics”, Automatica, 1997, v.33, No 7, pp. 1363-1372. Now, the
theorem 4.13 of the book “L. Kolev. "Interval Methods for Circuit Analysis". World
page 222, formula (7.115) add “max c
\in [c]” just after the sign’=’.
page 245, Figure 8.11 (a). Change the sense
of the arrows (a-s2 , a-s3) and (a-s1 , a-s6).
Error detected in the second version of the book
(These errors are also in the first version of the book)
Page 42. Formula (2.162).
First line replace the last two \cap (intersection) by \wedge
(minimum).
Second line replace the last two \cup (union) by \vee
(maximum).
Page 119,
line 4 of the algorithm. Replace cbar:=GoDown(mid([p]),c(.)) by cbar:=min(cbar, GoDown(mid([p]),c(.))). (Thank you Marteen Van Emden for detecting the error).
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