**3. Numerical Examples**

We consider some of the weight functions to solve a variety of univariate problems that are depicted in Examples 1–3.

Tables 1–3 display the minimum number of iterations necessary to obtain the required accuracy for the zeros of the functions <sup>Ω</sup>(*x*) in Examples 1–3. Moreover, we include also the initial guess, the radius of convergence of the corresponding function, and the theoretical order of convergence. Additionally, we calculate the *COC* approximated by means of (37) and (38).

All computations used the package *Mathematica* 9 with multiple precision arithmetic, adopting = 10−<sup>50</sup> as a tolerance error and the stopping criteria:

$$(i) \ |\delta\_{s+1} - \delta\_{\mathfrak{s}}| < \epsilon \text{ and } (ii) \ |\Omega(\delta\_{\mathfrak{s}})| < \epsilon.$$

**Example 1.** *Let* S = R, P = [−*π*, *<sup>π</sup>*], *x*<sup>∗</sup> = 0*. Let us define function* Ω *on* P *by:*

$$
\Omega(\mathbf{x}) = \cos \mathbf{x} - \mathbf{x} - \mathbf{1}.\tag{39}
$$

*Consequently, it results α* = 1 + |*β*|+*M<sup>k</sup>* |Ω(*p*∗)|*<sup>k</sup>*−<sup>1</sup> *γk*−<sup>1</sup> , *γ* = 12 , *b* = |Ω(*p*∗)| = 1 *and M* = 2*. We obtain a different radius of convergence when using distinct types of weight functions (for details, please see [3]), COC (ξ) and s presented in Table 1.*

**Cases Different Values of the Parameters that Satisfy Theorem 1** *β k <sup>H</sup>***(***<sup>υ</sup>***,** *η***)** *R***1** *Rq R λ***0** *s ξ* 1. −1 1 1+*υ* 1−*η* 0.10526 0.27008 0.02535 0.024 4 4 2. 3 2 1 + 2*υ* 0.00250 0.03749 0.00082 0.0007 3 4 3. −33 1 + 2*η* 0.00020 0.01013 0.00004 0.0003 3 4 4. 0.1 4 1 1−2*η*0.00962 0.07090 0.00160 0.0005 3 4

**Table 1.** Radii of convergence according to the adopted weight function.

**Example 2.** *Let* S = R, P = [−1, 1], *x*<sup>∗</sup> = 0.714806 *(approximated root), and let us assume function* Ω *on* P *by*

$$
\Omega(\mathbf{x}) = \mathbf{e}^{\mathbf{x}} - 4\mathbf{x}^2. \tag{40}
$$

*As a consequence, we get α* = 1 + |*β*|+*M<sup>k</sup>* |Ω(*p*∗)|*<sup>k</sup>*−<sup>1</sup> *γk*−<sup>1</sup> , *γ* = 2, *b* = |Ω(*p*∗)| = |*ex*<sup>∗</sup> − <sup>8</sup>*p*∗| ≈ 3.67466 *and M* = 2*. We have the distinct radius of convergence when using several weight functions (for details, please see [3]), COC (ξ) and s listed in Table 2.*


**Table 2.** Radii of convergence according to the adopted weight function.

**Example 3.** *Using the example of the introduction, we have α* = 1 + |*β*|+*M<sup>k</sup>* |Ω(*p*∗)|*<sup>k</sup>*−<sup>1</sup> *γk*−<sup>1</sup> , *γ* = 2, *b* = |Ω(*p*∗)| = 2*π*+1 *π*<sup>3</sup> ≈ 0.23489, *M* = 2*, and the required zero is p*∗ = 1*π* ≈ 0.318309886*. We have different radii of convergence by adopting distinct types of weight functions (for details, please see [3]), COC (ξ) and s in Table 3.*

**Table 3.** Radii of convergence according to the adopted weight function.

