*4.1. Approaches for Choosing a Nonlinear Function*

In the following, we consider two example functions that are suitable for selecting in the above algorithm.

#### 4.1.1. Nonlinear Function

Let *f*(*z*) has the following form

$$f(z) = e^z - z^2 - p\_\prime \tag{6}$$

where *p* is a real parameter such that *p* ≥ 1. For its first derivative,

$$f'(z) = e^z - 2z\_\prime$$

we conclude that *f* (*z*) <sup>&</sup>gt; 0 for all *<sup>z</sup>* <sup>∈</sup> <sup>R</sup>, i.e., the function *<sup>f</sup>*(*z*) is monotonically increasing for all *<sup>z</sup>* <sup>∈</sup> <sup>R</sup>. This and the limits

$$\lim\_{z \to -\infty} f(z) = -\infty \text{ and } \lim\_{z \to \infty} f(z) = \infty$$

show that the function *f*(*z*) has only one real root. From the second derivative of *f*

$$f''(z) = e^z - 2,$$

and because *f* (*z*) > 0 for ∀*z* > ln 2, it follows that the function *f*(*z*) is convex for *z* ∈ (ln 2, ∞).

Therefore, all these properties are also valid for the functions

$$g\_i(z) = f(z) - c\_{i\nu}$$

where *ci* is an integer value (the corresponding ASCII code). Then, for any *i*, the function *gi*(*z*) is monotonous, convex, and has a real root in the interval (ln 2, ∞). Indeed, it can be shown that each one function *gi*(*z*) has a real root in the finite interval (ln 2, 6).
