An Introduction
SIGLA/X group


Numbers and intervals 
Numerical results are exact only to a limited extent and they bear a certain amount of numerical error. Numerical results other than those carried by small integer numbers, are only estimated or approximated results of some measurement or computation Some times a single number seems enough to carry a determinate numerical information, because it is known beforehand what kind of measuring or computing process it comes from and what its corresponding spread of error is but, in general, single numbers are unable to usefully represent numerical information. For example, it is hardly acceptable to guess from the symbol 15 which of the information 15± 0.1 = [14.9,15.1] or 15± 5 = [10,20] is meant. The only way to indicate "the spread of a numerical result" is by pointing to a lower and an upper limit of its possible values, maybe through a notation like 15± 0.1 or 15± 5, or maybe thorough a more direct interval notation like [14.9,15.1] or [10,20]. Intervals, denoted generally by [a,b] with the condition a £ b, are the actual elementary items of numerical information. But when it comes to operate or doing some mathematics with numerical information, here is the departure point of interval mathematics from the usual way to handle numerical information through numbers. From an interval point of view the common alternative of computing with single numbers is able to provide only an indication of the true result that could be reached using interval calculus: only a single value pointing somewhere inside the complete interval result. Following this way systematically, interval mathematics uses trough the entire process of a computation, all the range of possible values that correspond to every item of the numerical information.
Real numbers generalise the concept of exact fraction to exact number
and set the geometrical model of the real line R, which is the ideal
ground conceptual and intuitive on which numerical models are conceived.
From a theoretical point of view, it is possible to deal with any real
number with a finite or infinite amount of digits, but real numbers are
not very suitable, as their name could indicate, because in practice it
is not possible to deal with numbers with an infinite number of digits.
Computer representation of a number contains only a limited amount of digits.
We could think that technology allows to work with a sufficient number
of digits, but this ignores the reality and would apply properties of R
to a set of numbers that, in fact, does not possess.
These outline facts are fully recognised and put to work by the Classical Interval Analysis's approach to numerical mathematics, when it decides systematically to keep the two nearest procedurally discernible digital bounds, a lower bound n1(DI), and a upper bound n2(DI), to represent any real value x(R) conceptually definite as something compatible enough with a definite measurement or actual computation. Thus every real number is between two consecutive digital numbers n1(DI) and n2(DI). The identification of the pair of digital numbers (n1,n2) bounding a real value to a settheoretical interval [n1,n2] makes the set of classic settheoretical intervals. In the Classical Interval Analysis approach to numerical computing the digital intervals
[n1(DI) , n2(DI)] = {xÎ DI ¦ n1(DI) £ x £ n2(DI)}
are the computational items. If I(R) denotes the set of intervals with real numbers as bounds
I(R) = {[x1,x2] ¦ x1Î R , x2Î R , x1 £ x2}
Interval mathematics identifies the interval [a,b] with the set of real numbers x which are between a and b
[a,b] = { x ¦ a £ x £ b}
What makes intervals non trivial for numerical computation and analysis
is that some fundamental regularities of real numbers are lost. This and
other shortcomings and defects of Classical Interval Analysis can lead
to the idea of giving it up, but actually it an effort is necessary to
complete its structure through an wider set.
Modal Intervals Analysis is a natural extension of the Classical Interval Analysis where the concept of interval is widened in the following way. Usings the notations
E(x,X) instead of ($ xÎ X) U(x,X) instead of (" xÎ X)
for the logical quantifiers and
[a,b]' := {xÎ R ¦ min(a,b) £ x £ max(a,b)} = [b,a]'
for the set of elements which belong to a classical interval (for example
[1,2]' = {xÎ R ¦ 1 £ x £ 2} = [2,1]')
a modal interval [a,b] is defined by
For example, [1,2] = ([1,2]',E) and [2,1] = ([2,1]',U). If a £ b we speak of an interval with the "proper" modality (or proper interval) and if a ³ b we speak of an interval with the "improper" modality (or improper interval). They are related by the "dual" operator
du([a,b]) = [b,a]
If the classical intervals set is
I(R) = {[a,b] ¦ aÎ R , bÎ R , a £ b}
the modal intervals set is
I*(R) = {[a,b] ¦ aÎ R , bÎ R}
and I*(R) is an extension of I(R), because I(R) Í I*(R).
Rational operations with modal intervals are natural extensions of the
rational operations between classical intervals. In I*(R)
questions as the following ones have an easy solution.
In the Classical Interval Analysis an equation as
[a,b] + [x,y] = [c,d]
has an interval solution holding the real relations a+x = c and b+y = d only when c+b £ a+d .
Example.
or
Even in this case, it fails to obtain its solution from any settheoretical interval operation between [a,b] and [c,d] because
In the Modal Interval Analysis it is easy and direct to get the true solution for the equations A+X = B, which is X = B du(A), and A*X = B, which is X = B/du(A).
Following the example, the solution is
because
which means
" (xÎ [1,3]) " (yÎ [1,4]) $ (zÎ [2,7]) (z = x+y)
And even the first equation has a solution
because
which means
" (xÎ
[1,3]) $ (yÎ
[2,3]) $ (zÎ
[4,5]) (z = x+y)
2) Meanings of the interval results
An interval result A of a function f evaluation in an interval X = (X1,...,Xn) in the Classical Interval Analysis has only the semantic interpretation
U(x1Î X1) ... U(xnÎ Xn) E(zÎ A) (z = f(x1,...,xn))
And this is the only accesible semantics. Because the semantics meaning of a function evaluation depends on the modality of intervals, it is possible in the Modal Interval Analysis to handle the modalities to get a desired semantics.
Example. For the simple function z = x+y we have (as well in the classical intervals as in the modal intervals context)
[5,11] = [1,3] + [4,8] Þ U(x,[1,3]') U(y,[4,8]') E(z,[5,11]') (z = x+y )
If we want the semantics
U(x,[1,3]') U(z, ? ) E(y,[4,8]') (z = x+y )
the calculations have to be
[9,7] = [1,3] + [8,4] Þ U(x,[1,3]') U(z,[9,7]') E(y,[8,4]') (z = x+y )
Similarly, we can get other semantics handling the modalities of the operands
[7,9] = [3,1] + [4,8] Þ U(y,[4,8]') E(x,[3,1]') E(z,[7,9]') (z = x+y )
[11,5] = [3,1] + [8,4] Þ U(z,[11,5]')
E(x,[3,1]') E(y,[8,4]') (z = x+y )
In the Classical Interval Analysis, for a real function f continuous in a box
X = (X1,...,Xn) Í Rn
the united extension Rf is the range of f in the box X, defined by
For a rational function f(x1,...,xn) a rational interval extension fR(X1,...,Xn) is defined through the sintactic tree of f, with their numerical arguments x1,...,xn replaced by the interval arguments X1,...,Xn and with the numerical operations of f replaced by their corresponding interval operations, which in case of approximate computations must be externally rounded. It is well known that both extensions Rf are related by
Rf(X1,...,Xn) Í fR(X1,...,Xn)
Extension Rf is not computable in general but fR is and the loss of information which represent fR respect to Rf is, in most cases, very big.
Example . For f(x,y) = (x–y)/(y+x) and xÎ [1,2], yÎ [3,6] it is possible to obtain the united extension with differential calculus tools
and the rational interval extension through the interval arithmetics
both verifying Rf([1,2],[3,6]) Í fR([1,2],[3,6]).
But in the Modal Interval Analysis it is very easy, in this case, to get Rf through intervals calculations.
where the "intervals" [2,1] and [6,3] are the dual of the intervals [1,2] and [3,6]. The semantic meaning of this result is
U(x, [1,2]') U(y, [3,6]') E(z, [
0.625,  0.25]') (z = (x–y)/(y+x))
4) Simulations with mathematical models
Many of the physical and technological processes are difficult to describe in terms of their behaviour. The cause and effect relationships are not easily discernibles because of the many important variables involved and their critical interactions. That is why many physical processes, whose dynamic characterization need to be well understood, are quite complex so that subsequently a formal mathematical approach is required. Defining the physical variables, making assumptions about the relations and factors in characterizing the responses of the physical process and the use of standard physical theories lead to the mathematical equations which define the mathematical state model for the physical systems. Considerations as the uncomplete knowledge of the system, the use of simplified models due to the intrinsecal complexity, the uncertainty and variation with time of the parameters and the unpredictible or unmodeled phenomenons lead to interval models.
Example . The water of a tank, of volume v, is
heated by means of a hydraulic system formed by a primary circuit, where
warm water is pumped by the pump B1 at a flow rate of q1,
a secondary circuit, where water is pumped by the pump B2 at a flow
rate of q2, and a heat exchanger, from which the water of both circuits
goes out at the same temperature.
The mathematical model for the energetic balance is
where
t4 is the temperature of the water in the point 4 and in the tank, mesured by TT
t3 is the temperature of the water in the primary circuit at point 3
ta is the temperature of the air surrounding the system
k is the disipation constant
d is the density of water
c is the heat capacity of the water
Simulation performed with an finite differences scheme
for
k = 7000 W/K, d = 1000 kg/m3, c = 4180 J/K× kg, ta = 300 K, t3 = 340 K,
q1 = 0.005 m3/s, q2 = 0.025 m3/s, v = 1 m3 and t4(0) = 301 K and D k = 1
gives the results shown in Figure 1
Fig 1
Let us consider the following intervals of variation: Q1 for the flow rate q1, Q2 for the flow rate q2, T3 for the temperature t3, T4 for the temperature t4, Ta for the temperature ta, V for the volume v (let us consider k, d and c as constants) and discretizing through finite differences, the physical system can be represented by the following discretetime interval model
Simulation performed for
k = 7000 W/K, d = 1000 kg/m3, c = 4180 J/K× kg, Ta = [299,301] K,
T3 = [339,341] K, Q1 = [0.005,0.006] m3/s, Q2 = [0.024,0.026] m3/s,
V = [0.9,1.1] m3, T4(0) = [301,303]
gives the results shown in Figure 2.
Fig 2
But for the following interval model
the results obtained are shown in Figure 3.
Fig 3
That means, for example at the instant n = 1000
U(ta,[299,301]') U(t3,[339,341]') U(q1,[0.005,0.006]') U(q2,[0.024,0.026]')
U(v,[0.9,1.1]') U(t(0),[301,303]') E(t(1000),[325.6,330.7]')
(t(1000) = f(ta,t3,q1,q2,v,t(0))
Solutions to problems as interval equations, evaluation of functions and the possibility to get results with different meanings, depending on the quantification of the variables, make the Model Interval Analysis a powerful tool in all the fields where intervals are a suitable procedure: simulation and control of physical systems, optimization, boundary problems,..,etc.
People interested in possible applications of this new theory can contact
with the group which works on its development (email
sainz@ima.udg.edu)