*2.3. Computational Efficiency*

Computational efficiency of an iterative method for solving *<sup>F</sup>*(*x*) = 0 is calculated by the efficiency index *E* = *<sup>p</sup>*1/*C*, (for detail see [21,22]), where *p* is the order of convergence and *C* is the total cost of computation. The cost of computation *C* is measured in terms of the total number of function evaluations per iteration and the number of operations (that means products and quotients) per iteration.

#### **3. The Method and Analysis of Convergence**

Let us begin with the following three-step scheme

$$\begin{aligned} y^{(k)} &= \mathbf{M}\_{2,1}(\mathbf{x}^{(k)}),\\ z^{(k)} &= y^{(k)} - [w^{(k)}, \mathbf{x}^{(k)}; F]^{-1} F(y^{(k)}),\\ x^{(k+1)} &= z^{(k)} - \left(I + \frac{1}{2} L\_F(y^{(k)})\right) F'(y^{(k)})^{-1} F(z^{(k)}),\end{aligned} \tag{13}$$

where *w*(*k*) = *x*(*k*) + *βF*(*x*(*k*)), *I* is *m* × *m* identity matrix and *LF*(*y*(*k*)) = *<sup>F</sup>* (*y*(*k*))−1*<sup>F</sup>* (*y*(*k*))*<sup>F</sup>* (*y*(*k*))−1*F*(*y<sup>k</sup>*).

Note that this is a scheme whose first two steps are that of third order Traub-Steffensen-type method (7) whereas third step is based on Chebyshev's method (3). The scheme requires first and second derivatives of *F* at *<sup>y</sup>*(*k*). To make this a derivative-free method, we describe an approach as follows:

Consider the Taylor expansion of *<sup>F</sup>*(*z*(*k*)) about *<sup>y</sup>*(*k*),

$$F(z^{(k)}) \approx F(y^{(k)}) + F'(y^{(k)})(z^{(k)} - y^{(k)}) + \frac{1}{2}F''(y^{(k)})(z^{(k)} - y^{(k)})^2. \tag{14}$$

Then, it follows that

$$\frac{1}{2}F''(y^{(k)})(z^{(k)}-y^{(k)})^2 \approx F(z^{(k)}) - F(y^{(k)}) - F'(y^{(k)})(z^{(k)}-y^{(k)}).\tag{15}$$

*Symmetry* **2019**, *11*, 891

> Using the fact that

$$F(z^{(k)}) - F(y^{(k)}) = [z^{(k)}, y^{(k)}; F](z^{(k)} - y^{(k)}),$$

(see, for example [4,5]), we can write (15) as

$$F''(y^{(k)})(z^{(k)} - y^{(k)}) \approx \mathcal{2}([z^{(k)}, y^{(k)}; F] - F'(y^{(k)})).\tag{16}$$

Then, using the second step of (13) in the above equation, it follows that

$$F^{\prime\prime}(y^{(k)})[w^{(k)}, \mathbf{x}^{(k)}; F]^{-1}F(y^{(k)}) \approx -2\left( [z^{(k)}, y^{(k)}; F] - F^{\prime}(y^{(k)}) \right). \tag{17}$$

Let us assume *<sup>F</sup>* (*y*(*k*)) ≈ [*w*(*k*), *x*(*k*) ; *<sup>F</sup>*], then (17) implies

$$F^{\prime\prime}(y^{(k)})[w^{(k)}, \mathbf{x}^{(k)}; F]^{-1}F(y^{(k)}) \approx -2\left( [z^{(k)}, y^{(k)}; F] - [w^{(k)}, \mathbf{x}^{(k)}; F] \right). \tag{18}$$

In addition, we have that

$$\begin{split} L\_F(y^{(k)}) &= F'(y^{(k)})^{-1} F''(y^{(k)}) F'(y^{(k)})^{-1} F(y^k) \\ &\approx [w^{(k)}, \mathbf{x}^{(k)}; F]^{-1} F''(y^{(k)}) [w^{(k)}, \mathbf{x}^{(k)}; F]^{-1} F(y^k). \end{split} \tag{19}$$

Using (18) in (19), we obtain that

$$\begin{split} \mathbb{L}\_F(y^{(k)}) &\approx [w^{(k)}, \mathbf{x}^{(k)}; F]^{-1} F'(y^{(k)}) [w^{(k)}, \mathbf{x}^{(k)}; F]^{-1} F(y^k) \\ &\approx -2 \{ [w^{(k)}, \mathbf{x}^{(k)}; F]^{-1} [\mathbf{z}^{(k)}, y^{(k)}; F] - I \}. \end{split} \tag{20}$$

Now, we can write the third-step of (13) in modified form as

$$\mathbf{x}^{(k+1)} = \mathbf{z}^{(k)} - \left(2I - [w^{(k)}, \mathbf{x}^{(k)}; F]^{-1}[\mathbf{z}^{(k)}, y^{(k)}; F]\right)[w^{(k)}, \mathbf{x}^{(k)}; F]^{-1}F(\mathbf{z}^{(k)}).\tag{21}$$

Thus, we define the following new method:

$$\begin{aligned} y^{(k)} &= \mathsf{M}\_{2,1}(\mathbf{x}^{(k)}),\\ z^{(k)} &= \mathsf{M}\_{3,1}(\mathbf{x}^{(k)}, y^{(k)}),\\ x^{(k+1)} &= z^{(k)} - H(\mathbf{x}^{(k)})[w^{(k)}, \mathbf{x}^{(k)}; F]^{-1}F(z^{(k)}),\end{aligned} \tag{22}$$

wherein *<sup>H</sup>*(*x*(*k*)) = 2*I* − [*w*(*k*), *x*(*k*) ; *<sup>F</sup>*]−<sup>1</sup>[*z*(*k*), *y*(*k*) ; *<sup>F</sup>*].

Since the scheme (22) is composed of Traub-Steffensen like steps, we call it the Traub-Steffensen-like method.

In order to explore the convergence properties of Traub-Steffensen-like method, we recall some important results from the theory of iteration functions. First, we state the following well-known result (see [3,23]):

**Lemma 1.** *Assume that M* : *D* ⊂ R*m* → R*m has a fixed point α* ∈ *int*(*D*) *and <sup>M</sup>*(*x*) *is Frechet* ´ *differentiable on α. If*

$$
\rho(M'(a)) = \sigma < 1,\tag{23}
$$

*then α is a point of attraction for the iteration x*(*k*+<sup>1</sup>) = *<sup>M</sup>*(*x*(*k*))*, where ρ is a spectral radius of <sup>M</sup>* (*α*)*.*

Next, we state a result which has been proven in [24] by Madhu et al. and that shows *α* is a point of attraction for a general iteration function of the form *<sup>M</sup>*(*x*) = *<sup>P</sup>*(*x*) − *Q*(*x*)*R*(*x*).

**Lemma 2.** *Let F* : *D* ⊂ R*m* → R*m be sufficiently Frechet* ´ *differentiable at each point of an open convex set D of α* ∈ *D, which is a solution of the nonlinear system <sup>F</sup>*(*x*) = 0*. Suppose that P*, *Q*, *R* : *D* ⊂ R*m* → R*m are sufficiently Frechet* ´ *differentiable functions (depending on F) at each point in the set D with the properties <sup>P</sup>*(*α*) = *α, Q*(*α*) = 0*, <sup>R</sup>*(*α*) = 0*. Then, there exists a ball*

$$S = S(\mathfrak{a}, \mathfrak{e}) = \{ \|\mathfrak{a} - \mathfrak{x}\| \le \mathfrak{e} \} \subset D, \mathfrak{e} > 0\_r$$

*on which the mapping*

$$\mathcal{M}: \mathcal{S} \to \mathbb{R}^m,\mathcal{M}(\mathfrak{x}) = P(\mathfrak{x}) - Q(\mathfrak{x})R(\mathfrak{x}), \forall \mathfrak{x} \in \mathcal{S}$$

*is well defined. Moreover, <sup>M</sup>*(*x*) *is Frechet differentiable at* ´ *α, thus*

$$M'(a) = P'(a) - Q(a)R'(a).$$

Let us also recall the definition (10) of divided difference operator. Then, expanding *<sup>F</sup>* (*x* + *th*) in (10) by Taylor series at the point *x* and thereafter integrating, we have that

$$\left[\mathbf{x} + \boldsymbol{h}, \mathbf{x}; F\right] = \int\_0^1 F'(\mathbf{x} + t\boldsymbol{h}) \, dt = F'(\mathbf{x}) + \frac{1}{2} F''(\mathbf{x}) \boldsymbol{h} + \frac{1}{6} F'''(\mathbf{x}) \boldsymbol{h}^2 + \frac{1}{24} F^{(iv)}(\mathbf{x}) \boldsymbol{h}^3 + O(h^4),\tag{24}$$

where *hi* = (*h*, *h*, *i* · · ··, *h*), *h* ∈ R*<sup>m</sup>*. Let *e*(*k*) = *x*(*k*) − *α*. Assuming that Γ = *<sup>F</sup>* (*α*)−<sup>1</sup> exists, then expanding *<sup>F</sup>*(*x*(*k*)) and its first three derivatives in a neighborhood of *α* by Taylor's series, we have that

$$F(\mathbf{x}^{(k)}) = F'(\mathbf{a}) \left( \varepsilon^{(k)} + A\_2(\varepsilon^{(k)})^2 + A\_3(\varepsilon^{(k)})^3 + A\_4(\varepsilon^{(k)})^4 + A\_5(\varepsilon^{(k)})^5 + O((\varepsilon^{(k)})^6) \right), \tag{25}$$

$$F'(\mathbf{x}^{(k)}) = F'(\mathbf{a}) \left( I + 2A\_2 \mathbf{e}^{(k)} + 3A\_3 (\mathbf{e}^{(k)})^2 + 4A\_4 (\mathbf{e}^{(k)})^3 + 5A\_5 (\mathbf{e}^{(k)})^4 + O((\mathbf{e}^{(k)})^5) \right), \tag{26}$$

$$F^{\prime\prime}(\mathbf{x}^{(k)}) = F^{\prime}(\mathbf{a}) \left( 2A\_2 + 6A\_3 \mathbf{e}^{(k)} + 12A\_4 (\mathbf{e}^{(k)})^2 + 20A\_5 (\mathbf{e}^{(k)})^3 + O((\mathbf{e}^{(k)})^4) \right) \tag{27}$$

and

$$F^{\prime\prime\prime}(\mathbf{x}^{(k)}) = F^{\prime}(\mathbf{a}) \left( 6A\_3 + 24A\_4 e^{(k)} + 60A\_5 (e^{(k)})^2 + O((e^{(k)})^3) \right),\tag{28}$$

where *Ai* = 1*i*! <sup>Γ</sup>*F*(*i*)(*α*) ∈ *Li*(R*<sup>m</sup>*, R*m*) and (*e*(*k*))*<sup>i</sup>* = (*e*(*k*),*<sup>e</sup>*(*k*), *i*−*times* · · ·· ,*<sup>e</sup>*(*k*)), *e*(*k*) ∈ R*<sup>m</sup>*.

We are in a situation to analyze the behavior of Traub-Steffensen-like method. Thus, the following theorem is proved:

**Theorem 1.** *Let F* : *D* ⊂ R*m* → R*m be sufficiently Frechet* ´ *differentiable at each point of an open convex set D of α* ∈ R*m, which is a solution of <sup>F</sup>*(*x*) = 0*. Assume that x* ∈ *S* = *<sup>S</sup>*¯(*<sup>α</sup>*, ) *and <sup>F</sup>* (*x*) *is continuous and nonsingular at α, and x*(0) *close to α. Then, α is a point of attraction of the sequence* {*x*(*k*)} *generated by the Traub-Steffensen-like method (22). Furthermore, the sequence so developed converges locally to α with order at least 5.*

**Proof.** First we show that *α* is a point of attraction of Traub-Steffensen-like iteration. In this case, we have that

$$P(\mathbf{x}) = z(\mathbf{x}), \ Q(\mathbf{x}) = H(\mathbf{x})[w, \mathbf{x}; F]^{-1}, \ R(\mathbf{x}) = F(z(\mathbf{x})).$$

Now, since *<sup>F</sup>*(*α*) = 0, [*α*, *α* ; *F*] = *<sup>F</sup>* (*α*) = *O*, we have

$$y(\mathfrak{a}) = \mathfrak{a} - [\mathfrak{a}, \mathfrak{a}; F]^{-1} F(\mathfrak{a}) = \mathfrak{a} - F'(\mathfrak{a})^{-1} F(\mathfrak{a}) = \mathfrak{a}\_{\prime\prime}$$

$$z(\mathfrak{a}) = \mathfrak{a} - [\mathfrak{a}, \mathfrak{a}; F]^{-1} F(\mathfrak{a}) - [\mathfrak{a}, \mathfrak{a}; F]^{-1} F(\mathfrak{a}) = \mathfrak{a} - F'(\mathfrak{a})^{-1} F(\mathfrak{a}) - F'(\mathfrak{a})^{-1} F(\mathfrak{a}) = \mathfrak{a},$$

$$H(\mathfrak{a}) = 2I - [\mathfrak{a}, \mathfrak{a}; F]^{-1} [\mathfrak{a}, \mathfrak{a}; F] = I,$$

$$P(\mathfrak{a}) = z(\mathfrak{a}), P'(\mathfrak{a}) = z'(\mathfrak{a}),$$

$$Q(\alpha) = H(\alpha)[\alpha, \alpha; F]^{-1} = I\left[\alpha, \alpha; F\right]^{-1} = [\alpha, \alpha; F]^{-1} = F'(\alpha)^{-1} \neq 0,$$

$$R(\alpha) = F(z(\alpha)) = F(\alpha) = 0,$$

$$R'(\alpha) = F'(z(\alpha))z'(\alpha) = F'(\alpha)z'(\alpha),$$

$$M'(\alpha) = P'(\alpha) - Q(\alpha)R'(\alpha) = z'(\alpha) - F'(\alpha)^{-1}F'(\alpha)z'(\alpha) = O,$$

so that *ρ*(*<sup>M</sup>* (*α*)) = 0 < 1 and by Lemma 1, *α* is a point of attraction of (22).

Let *e*(*k*) *w* = *w*(*k*) − *α* = *x*(*k*) + *βF*(*x*(*k*)) − *α* = *e*(*k*) + *βF*(*x*(*k*)). Then using (25), it follows that

$$\varepsilon\_{w}^{(k)} = \left(I + \beta F'(a)\right)\varepsilon^{(k)} + \beta F'(a)\left((A\_2(\varepsilon^{(k)})^2 + A\_3(\varepsilon^{(k)})^3) + O((\varepsilon^{(k)})^4)\right). \tag{29}$$

Setting *x* + *h* = *<sup>w</sup>*(*k*), *x* = *<sup>x</sup>*(*k*), *h* = *e*(*k*) *w* − *e*(*k*) in Equation (24) and then using (26)–(29), we can write

$$\begin{aligned} [w^{(k)}, \mathbf{x}^{(k)}; F] &= F^t(\mathbf{a}) \left( I + X\_1 A\_2 e^{(k)} + (\lambda A\_2^2 + X\_2 A\_3) (e^{(k)})^2 + X\_1 (2\lambda A\_2 A\_3) \\ &+ X\_3 A\_4 ) (e^{(k)})^3 + O((e^{(k)})^4) \right), \end{aligned} \tag{30}$$

where *λ* = *β<sup>F</sup>* (*α*), *X*1 = *λ* + 2, *X*2 = *λ*<sup>2</sup> + 3*λ* + 3 and *X*3 = *λ*<sup>2</sup> + 2*λ* + 2.

Expansion of the inverse of preceding divided difference operator is given as

$$\begin{aligned} \left[w^{(k)}, \mathbf{x}^{(k)}; \mathcal{F}\right]^{-1} &= \left(I - X\_1 A\_2(\boldsymbol{\epsilon}^{(k)}) + ((1 + X\_2)A\_2^2 - X\_2 A\_3)(\boldsymbol{\epsilon}^{(k)})^2 - X\_1((2 + X\_3)A\_2^3 - X\_2 A\_3)\right) \\ &- 2(1 + X\_3)A\_2 A\_3 + X\_3 A\_4)(\boldsymbol{\epsilon}^{(k)})^3 + O((\boldsymbol{\epsilon}^{(k)})^3))\Gamma. \end{aligned} \tag{31}$$

By using (25) and (31) in the first step of method (22), we ge<sup>t</sup>

$$e\_y^{(k)} = y^{(k)} - \mathfrak{a} = (-1 + X\_1)A\_2(e^{(k)})^2 - (X\_3A\_2^2 + (1 - X\_2)A\_3)(e^{(k)})^3 + O((e^{(k)})^4). \tag{32}$$

Taylor expansion of *<sup>F</sup>*(*y<sup>k</sup>*) about *α* yields,

$$F(y^{(k)}) = F'(a) \left( e\_y^{(k)} + A\_2(e\_y^{(k)})^2 + O((e\_y^{(k)})^3) \right). \tag{33}$$

From the second step of (22), on using (31) and (33), it follows that

$$\begin{split} e\_2^{(k)} &= z^{(k)} - u \\ &= X\_1 A\_2(e^{(k)}) e\_y^{(k)} - A\_2(e\_y^{(k)})^2 - ((1 + X\_2)A\_2^2 - X\_2 A\_3)(e^{(k)})^2 e\_y^{(k)} + O((e^{(k)})^5). \end{split} \tag{34}$$

By Taylor expansion of *<sup>F</sup>*(*z<sup>k</sup>*) about *α*,

$$F(z^{(k)}) = F'(a) \left( e\_z^{(k)} + A\_2(e\_z^{(k)})^2 + O((e\_z^{(k)})^3) \right). \tag{35}$$

Equation (24), for *x* + *h* = *<sup>z</sup>*(*k*), *x* = *y*(*k*) and *h* = *e*(*k*) *z* − *e*(*k*) *y* , yields

$$\begin{split} \left[ z^{(k)}, y^{(k)} ; F \right] = \mathbf{F}'(\boldsymbol{\alpha}) \left( I + A\_2 (\boldsymbol{e}\_z^{(k)} + \boldsymbol{e}\_y^{(k)}) + O((\boldsymbol{e}^{(k)})^3) \right) \\ = \mathbf{F}'(\boldsymbol{\alpha}) \left( I + (\boldsymbol{\lambda} + 1) A\_2^2 (\boldsymbol{e}^{(k)})^2 + O((\boldsymbol{e}^{(k)})^3) \right). \end{split} \tag{36}$$

From (31) and (36), we have

$$\begin{split} H(\mathbf{x}^{(k)}) &= 2I - [w^{(k)}, \mathbf{x}^{(k)}; \mathbf{F}]^{-1} [\mathbf{z}^{(k)}, y^{(k)}; \mathbf{F}] \\ &= I + X\_1 A\_2 e^{(k)} + \left( X\_2 A\_3 - (X\_1 + X\_2) A\_2^2 \right) (e^{(k)})^2 + O((e^{(k)})^3). \end{split} \tag{37}$$

Equations (31) and (37) yield

$$H(\mathbf{x}^{(k)})[w^{(k)},\mathbf{x}^{(k)};\mathcal{F}]^{-1} = \left(I - (\lambda^2 + 5\lambda + 5)A\_2^2(\mathbf{e}^{(k)})^2 + O((\mathbf{e}^{(k)})^3)\right)\Gamma. \tag{38}$$

Applying Equations (34), (35) and (38) in the last step of method (22) and then simplifying, we ge<sup>t</sup> the error equation

$$
\varepsilon^{(k+1)} = (\lambda + 1)(\lambda + 2)(\lambda^2 + 5\lambda + 5)A\_2^4(e^{(k)})^5 + O((e^{(k)})^6). \tag{39}
$$

This completes the proof of Theorem 1.

Thus, the Traub-Steffensen-like method (22) defines a one-parameter (*β*) family of derivative-free fifth order methods. Now onwards we denote it by M5,1. In terms of computational cost M5,1 utilizes four functions, two divided difference and one matrix inversion per each step. In the next section we will compare the computational efficiency of the new method with the existing derivative-free methods.
