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this function over an interval vector [x] is given by:
f ([x]) = { f (a) | a ∈ [x]} (4)
Then, one calls an inclusion function denoted by [ f ]
for the real function f an interval application that sat-
isfies the following inclusion
∀[x] ∈ IR
n
, f ([x]) ⊂ [ f ]([x]) (5)
In practice, the simplest manner to obtain an inclusion
function [ f ] for real function f consists in replacing
each occurrence of a real variable by the correspond-
ing interval and each standard function by its inter-
val counterpart. The resulting function is called the
natural inclusion function and the tightness of the en-
closure provided by [ f ] depends on the formal expres-
sion of f . In fact, it is well known if the same variable
x
i
has many occurrences in the mathematical expres-
sion of f , the dependence effect (Moore, 1966) (Jaulin
et al., 2001) will induce pessimism while computing
an enclosure of the range of the real function. Hence,
formal pre-processing of the function expression is
advisable in order to minimize the number of variable
occurrences. In the next paragraph, we will show how
thanks to interval computation one can solve easily
some constraint satisfaction problems (CSP) for con-
tinuous domains (Jaulin et al., 2001).
3.2 Constraint Satisfaction Problem
Historically the constraint satisfaction problems were
firstly formulated and studied in the context of dis-
crete domains, viz the variables belonged to finite sets
(Mackworth, 1977). Then, they have been extended
to continuous domains where the variables are part of
subsets of R or intervals (Sam-Haroud and Faltings,
1996). In continuous setting one defines a CSP as fol-
lows
H : (g(x) = y, x ∈ [x],y ∈ [y]) (6)
where the vectors x and y belong respectively to R
n
and R
m
with n not necessarily equal to m. Further-
more, the set of constraints can be assimilated to a
subset S of R
n
defined by
S = {x ∈ [x] | g(x) ∈ [y]} (7)
In practice, thanks to special algorithms (contractors)
(Jaulin et al., 2001) based on interval analysis and
consistency techniques, one can get with a polyno-
mial complexity in time and space, an outer approx-
imation S of the solution set S ⊆ S. Moreover, the
outer approximation set S is said global and guaran-
teed in the sense that it encloses all the possible so-
lutions of the CSP H . Thus, contracting an interval
vector (box) [x] under the set of constraints H con-
sists in computing a smaller box [x
0
], which contains
in guaranteed way the set resulting from the follow-
ing intersection S ∩ [x] ⊂ [x
0
]. That means, one at-
tempts to reduce the width of the initial box [x] ac-
cording to the set of constraints H . In the literature
several types of contractors are developed in order to
deal with specific CSP. For instance, we can cite those
inspired from point algorithms such as Gauss elimina-
tion, Gauss-Seidel algorithm, Krawczyk method, lin-
ear programming and Newton algorithm (Jaulin et al.,
2001). Hereafter, let C be a contractor and note that
all contractors have the following properties:
∀[x], C ([x]) ⊂ [x] (contractance),
∀[x], [x] ∩ S ⊂ C ([x]) (correctness),
[x] ⊂ [y] ⇒ C ([x]) ⊂ C ([y]) (monotonicity)
(8)
and a fixed point of a contractor C is a box [x] such
that C ([x]) = [x]. To close this section, we present
in the following a contractor based on the forward-
backward constraint propagation.
3.3 Forward-backward Contractor
In this paragraph, we illustrate the core idea of con-
tracting a box [x] according to a set defined by H
using the constraints propagation on intervals (Ben-
hamou et al., 1999). Note that, for the kind of
contractor the dimension of the vector function g
is no longer necessarily equal to the dimension of
the variable vector x. Assume that the mathemati-
cal expression g(x) = y is decomposable into a se-
quence of primitive constraints. Roughly speak-
ing, a primitive constraint is a constraint involving
only a single operator {+,−,×,÷} or single func-
tion {sin,cos,exp,...}. The forward-backward con-
tractor, here denoted by C
f b
, incorporates two proce-
dures called forward propagation and backward prop-
agation. Using interval computation, a direct evalua-
tion of all the primitive constraints are achieved in the
forward propagation procedure in order to obtain in
a sequential way an inclusion function for g. Then,
using consistency techniques (Sam-Haroud and Falt-
ings, 1996) the width of the computed inclusion func-
tion is reduced in accordance with the given box [y].
After this correction a backward propagation is per-
formed from the corrected inclusion function to its
elementary arguments gathered in the interval vector
[x]. Now, let us explain how works this kind of con-
tractor through a simple example. Consider the non-
linear constraint
g(x
1
,x
2
) = x
1
exp(x
2
) = y, (x
1
,x
2
) ∈ [x
1
]×[x
2
], y ∈ [Y ]
(9)
which can be decomposed into sequence of two prim-
itive constraints
a
1
= exp(x
2
)
y = x
1
a
1
(10)
UsingForward-backwardContractorstoIdentifyParasiticParametersofElectricalCircuitsWorkinginHighFrequency
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