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# Interval FAQ Derivative of an interval function

From Claudio M. Rocco S. (crocco@reacciun.ve):

## Derivative of an interval function?

My first question is related to the derivative of an interval function. For example for the Newton method, you need the derivative of the function.

Usually if you have a real function f(x) then you define F(X) as an interval extension with the property f(x)=F(x), and the same for F'(X): as an interval extension of f '(x).

On the other side: Given the interval function F(X) (X interval), how you define F'(X)? (for example if F(X) = [1,2]X^2 )

Claudio

### Response by R. Baker Kearfott:

(rbk@usl.edu), University of Southwestern Louisiana:

How do we want to use the derivative? Also, how is the interval [1,2] to be interpreted? If it is to be interpreted as a single point value that is known only to lie in the interval [1,2], then the logical derivative would be 2*[1,2]*X. This is because f(x) = a x^2 for a in [1,2], so f'(x) = 2ax, for the SAME a in [1,2]. When I speak of "interval derivative" or "interval Jacobi matrix," this is what I mean.

If, however, say, F(1) is meant to be the entire set [1,2], then I'm not sure how F'(1) should be defined. Perhaps the well-developed theory of subgradients may provide an answer.

A number of people have considered function strips.

### Claudio continues:

In the book Computer Methods for the Range of Functions, page 75, Ratscheck and Rokne present the mean-value and Taylor-forms. For the case of one real function f you need to expand the function f around a point c in X. In this case, you need to evaluate f'(x). They mention: "Moore now assumed that f' had an inclusion F' , that is f'(x) belongs to F'(X) for X in I and all x in X. We note that F' is not the derivative of F". What they mean ?

### Baker responds:

I believe that they meant that the relation of F' to F is different from that of a point derivative to its original point function, and not all deductions or manipulations can be carried out. However, when I say "interval derivative", I mean "inclusion for the derivative," precisely as they describe.

### Claudio continues:

The other question is related to integration. What is the definition of the integral of an interval function?

### Baker responds:

Moore, in "Methods and Applications of Interval Analysis," SIAM, 1979, has an interesting definition, derived from first principles, on pp. 50-59.

R. Baker Kearfott

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