*Some Special Cases*

We can generate many iterative schemes as the special cases of the family in Equation (3) based on the forms of function *<sup>H</sup>*(*<sup>x</sup>*, *y*) that satisfy the conditions of Theorems 1, 2 and 6. However, we restrict ourselves to the choices of low-degree polynomials or simple rational functions. These choices should be such that the resulting methods may converge to the root with order four for *m* ≥ 1. Accordingly, the following simple forms are considered:

*Symmetry* **2019**, *11*, 1452

(1) Let us choose the function

$$H(\mathbf{x}\_k, y\_k) = \mathbf{x}\_k + m\mathbf{x}\_k^2 + (m-1)y\_k + m\mathbf{x}\_k y\_k$$

which satisfies the conditions of Theorems 1, 2 and 6. Then, the corresponding fourth-order iterative scheme is given by

$$t\_{k+1} = z\_k - \left(\mathbf{x}\_k + m\mathbf{x}\_k^2 + (m-1)y\_k + m\mathbf{x}\_k y\_k\right) \frac{f(t\_k)}{f[s\_{k'}t\_k]}.\tag{33}$$

(2) Next, consider the rational function

$$H(\mathbf{x}\_k, y\_k) = -\frac{\mathbf{x}\_k + m \, \mathbf{x}\_k^2 - (m - 1) y\_k (m \, y\_k - 1)}{m \, y\_k - 1},$$

satisfying the conditions of Theorems 1, 2 and 6. Then, corresponding fourth-order iterative scheme is given by

$$t\_{k+1} = z\_k + \frac{x\_k + m \, x\_k^2 - (m-1)y\_k(m \, y\_k - 1)}{m \, y\_k - 1} \frac{f(t\_k)}{f[s\_k, t\_k]}.\tag{34}$$

(3) Consider another rational function satisfying the conditions of Theorems 1, 2 and 6, which is given by

$$H(\mathbf{x}\_k, y\_k) = \frac{\mathbf{x}\_k - y\_k + m \, y\_k + 2m \, \mathbf{x}\_k \, y\_k - m^2 \, \mathbf{x}\_k \, y\_k}{1 - m \, \mathbf{x}\_k + \mathbf{x}\_k^2}.$$

The corresponding fourth-order iterative scheme is given by

$$t\_{k+1} = z\_k - \frac{x\_k - y\_k + m \, y\_k + 2m \, x\_k \, y\_k - m^2 \, x\_k \, y\_k}{1 - m \, x\_k + x\_k^2} \frac{f(t\_k)}{f[s\_k, t\_k]}.\tag{35}$$

For each of the above cases, *zk* = *tk* − *m f*(*tk* ) *f* [*sk*,*tk* ] . For future reference, the proposed methods in Equations (33)–(35) are denoted by NM1, NM2, and NM3, respectively.

## **4. Numerical Results**

To validate the theoretical results proven in previous sections, the special cases NM1, NM2, and NM3 of new family were tested numerically by implementing them on some nonlinear equations. Moreover, their comparison was also performed with some existing optimal fourth-order methods that use derivatives in the formulas. We considered, for example, the methods by Li et al. [7,8], Sharma and Sharma [9], Zhou et al. [10], Soleymani et al. [12], and Kansal et al. [14]. The methods are expressed as follows:

*Li–Liao–Cheng method* (LLC):

$$\begin{aligned} z\_k &= t\_k - \frac{2m}{m+2} \frac{f(t\_k)}{f'(t\_k)'} \\ t\_{k+1} &= t\_k - \frac{m(m-2) \left(\frac{m}{m+2}\right)^{-m} f'(z\_k) - m^2 f'(t\_k)}{f'(t\_k) - \left(\frac{m}{m+2}\right)^{-m} f'(z\_k)} \frac{f(t\_k)}{2f'(t\_k)}. \end{aligned}$$

*Symmetry* **2019**, *11*, 1452

> *Li–Cheng–Neta method* (LCN):

$$\begin{aligned} z\_k &= t\_k - \frac{2m}{m+2} \frac{f(t\_k)}{f'(t\_k)}, \\ t\_{k+1} &= t\_k - \alpha\_1 \frac{f(t\_k)}{f'(z\_k)} - \frac{f(t\_k)}{\alpha\_2 f'(t\_k) + \alpha\_3 f'(z\_k)} \end{aligned}$$

where

$$\begin{aligned} a\_1 &= -\frac{1}{2} \frac{\left(\frac{m}{m+2}\right)^m m (m^4 + 4m^3 - 16m - 16)}{m^3 - 4m + 8}, \\ a\_2 &= -\frac{(m^3 - 4m + 8)^2}{m (m^4 + 4m^3 - 4m^2 - 16m + 16)(m^2 + 2m - 4)}, \\ a\_3 &= \frac{m^2 (m^3 - 4m + 8)}{\left(\frac{m}{m+2}\right)^m (m^4 + 4m^3 - 4m^2 - 16m + 16)(m^2 + 2m - 4)}. \end{aligned}$$

*Sharma–Sharma method* (SS):

$$\begin{aligned} z\_k &= t\_k - \frac{2m}{m+2} \frac{f(t\_k)}{f'(t\_k)}, \\ t\_{k+1} &= t\_k - \frac{m}{8} \left[ (m^3 - 4m + 8) - (m+2)^2 \left( \frac{m}{m+2} \right)^m \frac{f'(t\_k)}{f'(z\_k)} \right. \\ &\times \left( 2(m-1) - (m+2) \left( \frac{m}{m+2} \right)^m \frac{f'(t\_k)}{f'(z\_k)} \right) \Big] \frac{f(t\_k)}{f'(t\_k)}. \end{aligned}$$

*Zhou–Chen–Song method* (ZCS):

$$\begin{split} z\_{k} &= t\_{k} - \frac{2m}{m+2} \frac{f(t\_{k})}{f'(t\_{k})}, \\ t\_{k+1} &= t\_{k} - \frac{m}{8} \left[ m^{3} \left( \frac{m+2}{m} \right)^{2m} \left( \frac{f'(z\_{k})}{f'(t\_{k})} \right)^{2} - 2m^{2} (m+3) \left( \frac{m+2}{m} \right)^{m} \frac{f'(z\_{k})}{f'(t\_{k})} \right] \\ &\quad + (m^{3} + 6m^{2} + 8m + 8) \left] \frac{f(t\_{k})}{f'(t\_{k})}. \end{split}$$

*Soleymani–Babajee–Lotfi method* (SBL):

$$\begin{aligned} z\_k &= t\_k - \frac{2m}{m+2} \frac{f(t\_k)}{f'(t\_k)}, \\ t\_{k+1} &= t\_k - \frac{f'(z\_k)f(t\_k)}{q\_1(f'(z\_k))^2 + q\_2f'(z\_k)f'(t\_k) + q\_3(f'(t\_k))^2}, \end{aligned}$$

where

$$\begin{aligned} q\_1 &= \frac{1}{16} m^{3-m} (2+m)^m, \\ q\_2 &= \frac{8-m(2+m)(-2+m^2)}{8m}, \\ q\_3 &= \frac{1}{16} (-2+m) m^{-1+m} (2+m)^{3-m}. \end{aligned}$$

*Kansal–Kanwar–Bhatia method* (KKB):

$$\begin{aligned} z\_k &= t\_k - \frac{2m}{m+2} \frac{f(t\_k)}{f'(t\_k)},\\ t\_{k+1} &= t\_k - \frac{m}{4} f(t\_k) \left( 1 + \frac{m^4 p^{-2m} \left( -\frac{f'(z\_k)}{f'(t\_k)} + p^{-1+m} \right)^2 (-1 + p^m)}{8(2p^m + m(-1 + p^m))} \right) \\ &\times \left( \frac{4 - 2m + m^2 (-1 + p^{-m})}{f'(t\_k)} - \frac{p^{-m} (2p^m + m(-1 + p^m))^2}{f'(t\_k) - f'(z\_k)} \right). \end{aligned}$$

where *p* = *m m*+2 .

Computational work was compiled in the programming package of Mathematica software [18] in a PC with Intel(R) Pentium(R) CPU B960 @ 2.20 GHz, 2.20 GHz (32-bit Operating System) Microsoft Windows 7 Professional and 4 GB RAM. Performance of the new methods was tested by choosing value of the parameter *β* = 0.01. The tabulated results obtained by the methods for each problem include: (a) the number of iterations (*k*) required to obtain the solution using the stopping criterion |*tk*+<sup>1</sup> − *tk*| + | *f*(*tk*)| < <sup>10</sup>−100; (b) the estimated error |*tk*+<sup>1</sup> − *tk*| in the first three iterations; (c) the calculated convergence order (CCO); and (d) the elapsed time (CPU time in seconds) in execution of a program, which was measured by the command "TimeUsed[ ]". The calculated convergence order (CCO) to confirm the theoretical convergence order was calculated by the formula (see [19])

$$\text{CCO} = \frac{\log \left| (t\_{k+2} - a) / (t\_{k+1} - a) \right|}{\log \left| (t\_{k+1} - a) / (t\_k - a) \right|}, \text{ for each } k = 1, 2, \dots \tag{36}$$

The following numerical examples were chosen for experimentation:

**Example 1.** *Planck law of radiation to calculate the energy density in an isothermal black body [20] is stated as*

$$\phi(\lambda) = \frac{8\pi ch\lambda^{-5}}{\epsilon^{ch/\lambda kT} - 1}.\tag{37}$$

*where λ is wavelength of the radiation, c is speed of light, T is absolute temperature of the black body, k is Boltzmann's constant, and h is Planck's constant. The problem is to determine the wavelength λ corresponding to maximum energy density φ*(*λ*)*. Thus, Equation* (37) *leads to*

$$\phi'(\lambda) = \left(\frac{8\pi ch\lambda^{-6}}{e^{ch/\lambda kT} - 1}\right)\left(\frac{(ch/\lambda kT)e^{ch/\lambda kT}}{e^{ch/\lambda kT} - 1} - 5\right) = A.B. \text{ (say)}$$

*Note that a maxima for φ will occur when B* = 0*, that is when*

$$\frac{(ch/\lambda kT)e^{ch/\lambda kT}}{e^{ch/\lambda kT}-1} = 5.$$

*Setting t* = *ch*/*λkT, the above equation assumes the form*

$$1 - \frac{t}{5} = e^{-t}.\tag{38}$$

*Define*

$$f\_1(t) = e^{-t} - 1 + \frac{t}{5}.\tag{39}$$

*The root t* = 0 *is trivial and thus is not taken for discussion. Observe that for t* = 5 *the left-hand side of Equation* (38) *is zero and the right-hand side is e*<sup>−</sup><sup>5</sup> ≈ 6.74 × 10−3*. Thus, we guess that another root might occur*

*somewhere near to t* = 5*. In fact, the expected root of Equation* (39) *is given by α* ≈ 4.96511423174427630369 *with t*0 = 5.5*. Then, the wavelength of radiation ( λ) corresponding to maximum energy density is*

$$
\lambda \approx \frac{ch}{4.96511423174427630369(kT)}.
$$

*The results so obtained are shown in Table 1.*

**Example 2.** *Consider the van der Waals equation (see [15])*

$$\left(P + \frac{a\_1 n^2}{V^2}\right)(V - n a\_2) = nRT\_{\prime \prime}$$

*that explains the nature of a real gas by adding two parameters a*1 *and a*2 *in the ideal gas equation. To find the volume V in terms of rest of the parameters, one requires solving the equation*

$$PV^3 - (na\_2P + nRT)V^2 + a\_1n^2V - a\_1a\_2n^2 = 0.$$

*One can find values of n, P, and T, for a given a set of values of a*1 *and a*2 *of a particular gas, so that the equation has three roots. Using a particular set of values, we have the function*

$$f\_2(t) = t^3 - 5.22t^2 + 9.0825t - 5.2675\_r$$

*that has three roots from which one is simple zero α* = 1.72 *and other one is a multiple zero α* = 1.75 *of multiplicity two. However, our desired zero is α* = 1.75*. The methods are tested for initial guess t*0 = 2.5*. Computed results are given in Table 2.*

**Example 3.** *Next, we assume a standard nonlinear test function which is defined as*

$$f\_3(t) = \left[\tan^{-1}\left(\frac{\sqrt{5}}{2}\right) - \tan^{-1}(\sqrt{t^2 - 1}) + \sqrt{6}\left(\tan^{-1}\left(\sqrt{\frac{t^2 - 1}{6}}\right) - \tan^{-1}\left(\frac{1}{2}\sqrt{\frac{5}{6}}\right)\right) - \frac{11}{63}\right]^3.$$

*The function f*3 *has multiple zero at α* = 1.8411027704926161 ... *of multiplicity three. We select initial approximation t*0 = 1.6 *to obtain zero of this function. Numerical results are exhibited in Table 3.*

**Example 4.** *Lastly, we consider another standard test function, which is defined as*

$$f\_4(t) = t(t^2 + 1)(2\epsilon^{t^2 + 1} + t^2 - 1)\cosh^2\left(\frac{\pi t}{2}\right).$$

*The function f*4 *has multiple zero at i of multiplicity four. We choose the initial approximation x*0 = 1.2*i for obtaining the zero of the function. Numerical results are displayed in Table 4.*


**Table 1.** Numerical results for Example 1.


**Table 2.** Numerical results for Example 2.

**Table 3.** Numerical results for Example 3.


**Table 4.** Numerical results for Example 4.


From the computed results shown in Tables 1–4, we can observe a good convergence behavior of the proposed methods similar to those of existing methods. The reason for good convergence is the increase in accuracy of the successive approximations per iteration, as is evident from numerical results. This also points to the stable nature of methods. It is also clear that the approximations to the solutions by the new methods have accuracies greater than or equal to those computed by existing methods. We display the value 0 of |*tk*+<sup>1</sup> − *tk*| at the stage when stopping criterion |*tk*+<sup>1</sup> − *tk*| + | *f*(*tk*)| < 10−<sup>100</sup> has been satisfied. From the calculation of computational order of convergence shown in the penultimate column in each table, we verify the theoretical fourth-order of convergence.

The efficient nature of presented methods can be observed by the fact that the amount of CPU time consumed by the methods is less than the time taken by existing methods (result confirmed by similar numerical experiments on many other different problems). The methods requiring repeated evaluations of the roots (such as the ones tackled in [21–24]), also may benefit greatly from the use of proposed methods (NM1–NM3, Equations (33)–(35)).
