d standard functions . The following inclusion monotone holds.
for (5)
where,f([x]) is an interval also and which stands for an interval arithmetic evaluation of f over .As x∈[x] , the relation (6) can be obtained.
(6)
whence
(7)
Where R(f,[x]) denotes the range of f over .
(2) A new approach to evaluate interval functions
Overestimation is a major drawback in interval computation. Based on the inclusion monotone relation (7) and the physical/real means expressed by the interval function, a new approach to evaluate interval functions was proposed in this work.
Relation (7) is the fundamental property on which nearly all applications of interval arithmetic are based. It shows that it is possible to compute lower and upper bounds for the range over an interval by using only the bounds of the given interval without any further assumption.
Obviously, the true value of is existing and unique. means the range of over . One of the original idea to introduce the interval function is to evaluate the range of the value of the function when the variable changes in the range of in a statement of interval way. However, because only some of the algebraic laws which valid for real numbers hold only in a weaker form for interval numbers, the computational results of depends on the calculating order severely, and they are often larger than the value of . A number of literatur英语论文网 【http://www.51lunwen.org】es took efforts on finding the skills to obtain the better results of . And some valuable rules were found. For example, (1) If each variable , , occurs at most once in , then ; (2) To make the most of the subdistributivity, i.e., to execute the addition and subtraction operation first, to execute multiplication and division operations then. For instance, the better result of the polynomial can be obtained through computing its reformed form . However, the similar results to improve the results of a rational function have not been found.
In fact, the best result of can be obtained through calculate directly. The bounds of can be obtained through solving the following two optimization problems.
(8)
(9)
It is clear that the optima indicated in optimization problem (8) and optimization problem (9) refer to the global optima of in . When is monotone in , there are only one local maximum and one local minimum of , they are the global maximum and the global minimum of in respectively. Many optimization algorithms (for example, Newton algorithm, Nowton-Raphson algorithm, Gauss Newton algorithm, etc) that only can locate the local optima of the problems could be used. However, when the expression is not monotonic in , or the monotone property is unknown, the global optimization method that has good capability to locate the global optima of in is needed.
In this work, a real-code genetic algor
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