## The automatic computation of second-order slope tuples for some nonsmooth functions

Marco Schnurr

### Abstract

In this paper, we show how the automatic computation of second-order slope tuples can be performed. The algorithm allows for nonsmooth functions, such as $\varphi\left(x\right)=\left|u\left(x\right)\right|$ and $\varphi\left(x\right)=\max\left\{ u\left(x\right),v\left(x\right)\right\}$, to occur in the function expression of the underlying function. Furthermore, we allow the function expression to contain functions given by two or more branches. By using interval arithmetic, second-order slope tuples provide verified enclosures of the range of the underlying function. We give some examples comparing range enclosures given by a second-order slope tuple with enclosures from previous papers.

Full Text (PDF) [261 KB]

### Key words

slope tuple, interval analysis, automatic slope computation, range enclosure

65G20, 65G99

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