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1.     Introduction

We consider the following nonlinear system of equations:

                                                           𝐹(𝑥) = 0, 𝑥 = (𝑥₁,𝑥₂,⋯,𝑥_𝑛)^𝑇 ∈ 𝑅^𝑛,                                                                          (1)

© 2026 Zhanlav, T., Otgondorj, K., Ulziibayar, V., Enkhbayar, K.

                               This work is licensed under a Creative Commons “Attribution-NonCommercial 4.0 International” license.

where 𝐹 ∶ 𝐷 ⊆ 𝑅^𝑛 → 𝑅^𝑛 is a nonlinear and sufficiently Fréchet differentiable function in an open convex set 𝐷. Additionally, 𝐹^′(𝑥) is continuous and nonsingular at 𝛼, where 𝛼 is the simple and isolated solution of equation (1). Most physical systems are inherently nonlinear nature and described by nonlinear systems. The nonlinear systems (1) also appear in many fields of applied sciences and engineering[1–12]. Thesolutionofequation(1)cannotbecomputedexactlyandisoftenapproximated using iterative methods with different orders of convergence. A quite recently have been appeared some papers devoted to the constructing high efficient iterative methods containing vector and even scalar parameter coefficients [1, 3, 4, 6, 13–15]. For obtaining the numerical solution of the system (1) often used the following two-step and three-step iterative methods:

𝑦𝑘 = 𝑥𝑘 − 𝐹′(𝑥𝑘)−1𝐹(𝑥𝑘),

                                                                                       ′(𝑥𝑘)−1𝐹(𝑦𝑘),                                                                            (2)

𝑥𝑘+1 = 𝑦𝑘 − 𝜏̄𝑘𝐹

and

𝑦𝑘 = 𝑥𝑘 − 𝐹′(𝑥𝑘)−1𝐹(𝑥𝑘),

                                                               𝑧_𝑘 = 𝑦_𝑘 − 𝜏̄_𝑘𝐹^′(𝑥_𝑘)⁻¹𝐹(𝑦_𝑘),                                                                                    (3)

𝑥𝑘+1 = 𝑧𝑘 − 𝛼𝑘𝐹′(𝑥𝑘)−1𝐹(𝑧𝑘),

where 𝜏̄_𝑘 and 𝛼_𝑘 are iteration parameters to be determined properly.

The aim of this work is to develop derivative-free version of the iterations (2) and (3) with vector and scalar coefficients. In Section 2, we introduce new derivative-free two-step iterations of orders four and five. In Section 3, we present new derivative-free three-step iterations of order 𝜌 (𝜌 = 6,7,8) and an analysis of the efficiency of the proposed iterative methods. Section 4 devoted to analysis of efficiency of proposed methods compared with other methods. In Section 5, we present the results of our experiments and compare them with known methods of the same order. The article concludes

with some conclusions and references used in it.

2.       The construction of two-step derivative-free iterations

First, we employ 𝑅^𝑛 with point-wise multiplication and division of vectors. Let 𝑎 = (𝑎₁,𝑎₂,…,𝑎_𝑛)^𝑇 ∈ 𝑅^𝑛 and 𝑏 = (𝑏₁,𝑏₂,…,𝑏_𝑛)^𝑇 ∈ 𝑅^𝑛. The point-wise multiplication and division of two vectors are defined by

𝑎 ⋅ 𝑏 = (𝑎1𝑏1,𝑎2𝑏2,…,𝑎𝑛𝑏𝑛)𝑇 ∈ 𝑅𝑛,	(4a)
𝑎            𝑎1 𝑎2            𝑎𝑛   𝑇          𝑛.      = (       ,        ,…,         )     ∈ 𝑅	(4b)

                                                           𝑏           𝑏1 𝑏2             𝑏𝑛

The direct consequence of (4a) and (4b) is

𝑎² ∶ = (𝑎 ⋅ 𝑎) = (𝑎²₁,𝑎²₂,…,𝑎²_𝑛)^𝑇 ∈ 𝑅^𝑛,

𝑇

                                                            1 = (1,1,…,1)         ∈ 𝑅^𝑛.

In [6] the following theorems were proven:

Theorem 6. [6] The two-step iteration (2) has a third, fourth and fifth-order convergence if and only if the parameter 𝜏̄_𝑘 satisfies

𝜏̄_𝑘 = 1 + 𝑂(ℎ),

                                             𝜏̄_𝑘 = 1 + 2𝛩_𝑘 + 𝑂(ℎ²),                                                                                                                  (6a)

𝜏̄_𝑘𝐹^′(𝑥_𝑘)⁻¹𝐹(𝑦_𝑘) = (1 + 𝛩_𝑘²)𝐹^′(𝑦_𝑘)⁻¹𝐹(𝑦_𝑘) + 𝑂(ℎ³),

Theorem 7. [6] The three-step iteration (3) have order of convergence 𝜌 + 1, 𝜌 + 2, 𝜌 + 3 if and only if the parameter 𝛼_𝑘 satisfies

𝛼_𝑘 = 1 + 𝑂(ℎ),

                                                                  𝛼_𝑘 = 1 + 2𝛩_𝑘 + 𝑂(ℎ²),                                                                                    (7a)

𝛼_𝑘𝐹^′(𝑥_𝑘)⁻¹𝐹(𝑧_𝑘) = (1 + 2𝛩_𝑘²)𝐹^′(𝑦_𝑘)⁻¹𝐹(𝑧_𝑘) + 𝑂(ℎ³),

where

                                                       𝛩_𝑘 = 𝐹(𝑦^𝑘),   𝛩),

𝐹(𝑥_𝑘)

and 𝜌 is the order of convergence of iteration (2).

We note that the conditions (6a) and (7a) can be replaced by

1 + 𝑎𝛩^𝑘

                                                𝜏̄_𝑘 = 𝛼_𝑘 = ,          𝑎,𝑏,𝑑 ∈ 𝑅,

1 + (𝑎 − 2)𝛩_𝑘

and in this case the convergence order maintained. We now proceed with the construction of a derivative-free analog of (2) as follows:

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

(8)

₋₁

                                                           𝑥_𝑘+1 = 𝑦_𝑘 − 𝑇_𝑘 [𝑤_𝑘,𝑠_𝑘;𝐹]           𝐹(𝑦_𝑘),

where [𝑤_𝑘,𝑠_𝑘;𝐹] is first order divided difference with

                                      𝑤_𝑘 = 𝑥_𝑘 + 𝛾₁𝐹(𝑥_𝑘),            𝑠_𝑘 = 𝑥_𝑘 − 𝛾₁𝐹(𝑥_𝑘),         𝛾₁ ≠ 0,       𝛾₁ ∈ 𝑅.

It is easy to show that

                                                                 𝑇_𝑘 = 𝜏̄_𝑘𝐹^′(𝑥_𝑘)⁻¹ [𝑤_𝑘,𝑠_𝑘;𝐹]^,                                                                                 (9)

or

                                                                𝜏̄𝑘 = 𝑇𝑘 [𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹′(𝑥𝑘).                                                                              (10)

The passing of (2) to (8) is realized by (9). The converse is realized by (10). It is easy to show that

where

𝐹′(𝑥𝑘)−1 [𝑤𝑘,𝑠𝑘;𝐹] = 𝐼 + 𝐵𝑘 + 𝑂(ℎ4),

(11)

                                                       𝐵_𝑘 = 𝐹^′(𝑥_𝑘)⁻¹𝐹^‴(𝑥_𝑘)𝛾₁²𝐹(𝑥_𝑘)² = 𝑂(ℎ²).                                                                  (12)

If we take (12) into account, then from (11) it follows that
𝐹′(𝑥𝑘)−1 = [𝑤𝑘,𝑠𝑘;𝐹]−1 + 𝑂(ℎ2). Analogously, using (11) and the Taylor expansion of 𝐹(𝑦𝑘) at point 𝑥𝑘, we easily obtain	(13)
𝐹(𝑦𝑘) = 𝑂(ℎ2),             𝐹′(𝑦𝑘) = [𝑢𝑘,𝜛𝑘;𝐹] + 𝑂(ℎ4),	(14)

where

                                    𝑢_𝑘 = 𝑦_𝑘 + 𝛽₁𝐹(𝑥_𝑘),            𝜛_𝑘 = 𝑦_𝑘 − 𝛽₁𝐹(𝑥_𝑘),          𝛽₁ ≠ 0,        𝛽₁ ∈ 𝑅.

Using (9), (11), (13) and (14) it is easy to show that the 𝜌-order conditions (6) can be rewritten in term of 𝑇_𝑘 as:

𝑇_𝑘 = 1 + 𝑂(ℎ),

1 + 2𝛩

2

                                            𝑇𝑘 = 1 + 2𝛩𝑘 + 𝑂(ℎ ) =                       𝑘 +2𝑏𝛩𝑘2 + 𝑂(ℎ2),                                                     (15a)

1 + 𝑑𝛩𝑘

                                               𝑇_𝑘 [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑦_𝑘) = (1 + 𝛩_𝑘²)[𝑢_𝑘,𝜛_𝑘;𝐹]⁻¹ 𝐹(𝑦_𝑘).                                                    (15b)

Using (15a) in (8) we obtain the following family of fourth order iterations (𝑀₁⁴)

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

                                            1     1                               −1                                                                                  1 + 𝑏𝛩𝑘2)𝐹(𝑦𝑘(16)

                         𝑥𝑘+1 = 𝑦𝑘 −                     2 [𝑤𝑘,𝑠𝑘;𝐹]        [(

+ 𝑑𝛩𝑘

Analogously, using (15b) in (8) we obtain the following fifth order iteration (𝑀₂⁵)

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

(17)

1 + 𝛩_𝑘²)[𝑢_𝑘,𝜛_𝑘;𝐹]₋₁ 𝐹(𝑦_𝑘). 𝑥_𝑘+1 = 𝑦_𝑘 − (

If 𝛾₁ → 0 and 𝛽₁ → 0 then (16) and (17) lead to the iteration with derivative, considered in [6] and in

[4]. The scalar coefficients versions of (16) and (17) are [1]

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

                                                  1                              ₋₁                                                                                                                                                                      (18)

                          𝑥_𝑘+1 = 𝑦_𝑘 −      [𝑤_𝑘,𝑠_𝑘;𝐹]             [(1 + 𝑏𝑣_𝑘)𝐹(𝑦_𝑘) + 2𝑣_𝑘𝐹(𝑥_𝑘)], 𝑑,𝑏 ∈ 𝑅,

1 + 𝑑𝑣_𝑘

‖𝐹(𝑦_𝑘)‖² 𝑣𝑘 = ‖𝐹(𝑥_𝑘)‖2 ,

and

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

(19)

₋₁ 𝑥_𝑘+1 = 𝑦_𝑘 − (1 + 𝑣_𝑘)[𝑢_𝑘,𝜛_𝑘;𝐹]                𝐹(𝑦_𝑘),

with convergence order 4 and 5 respectively. The iteration (18) completely coincides with scheme given in [13], while (19) can be considered as new scheme with fifth order of convergence. Let’s denote the methods (18) and (19) as (𝑀₃⁴) and (𝑀₄⁵), respectively.

To analyze the convergence behavior of the proposed method, we first present a lemma that will be used to develop the Taylor expansion of vector functions (see [16]).

Lemma 1. Let 𝐹 ∶ 𝐷 ⊆ 𝑅^𝑛 → 𝑅^𝑛 be 𝑝-times Fréchet differentiable in a open convex set 𝐷 ⊆ 𝑅^𝑛, then for any 𝑥,ℎ ∈ 𝐷̂                the following expression holds:

𝐹(𝑥 + ℎ) = 𝐹(̂  𝑥) + 𝐹^′           ̂       ̂                    𝐹^(𝑝−1)(𝑥)ℎ^𝑝−1̂                    + 𝑅_𝑝, where                _𝑝

                                                  1                ^(𝑝)(𝑥 + 𝑡ℎ)̂ ‖‖ℎ‖̂ ^𝑝 𝑎𝑛𝑑 ℎ^𝑝̂ = (^{⏞⎴⎴⏞⎴⎴⏞}ℎ,̂ ℎ,⋯,̂   ℎ)̂ ,

                                     ‖𝑅_𝑝‖ ⩽           sup ‖𝐹

𝑝! 0<𝑡<1

and ‖ ⋅ ‖ denotes any norm in 𝑅^𝑛, or a corresponding operator norm.

Definition 1. Let 𝑒_𝑘 = 𝑥_𝑘 − 𝛼 be the error in the 𝑘-th iteration, we call the relation

𝑒𝑘+1 = 𝐿(𝑒𝑘)𝑝 + 𝑂((𝑒𝑘)𝑝+1),

as the error equation.                 Here,          𝑝 is the order of convergence,                𝐿 is a 𝑝-linear function,           i.e.

𝑝

𝐿 ∈ ℒ(⏞⎴⎴⎴⏞⎴⎴⎴⏞𝑅^𝑛 × ⋯ × 𝑅^𝑛,𝑅^𝑛).

In the following result, we establish the convergence of the family of methods given by (16) under the conditions stated in Lemma 1.

Theorem 8. Let the function 𝐹 ∶ 𝐷 ⊆ 𝑅^𝑛 → 𝑅^𝑛 be sufficiently differentiable in a convex set 𝐷 containing a zero 𝛼 of 𝐹(𝑥). Further, assume that 𝐹^′(𝑥) is continuous and non-singular at 𝛼 and the initial guess 𝑥₀ is sufficiently close to the solution. Then, the sequence generated by method (16) converges to the solution 𝛼

with order four, for any nonzero value of parameter 𝛾₁ and for any values of 𝑏 and 𝑑.

Proof. By applying the Taylor expansion of 𝐹(𝑥_𝑘) around 𝛼, we obtain

                                               𝐹(𝑥_𝑘) = 𝐹^′(𝛼)(𝑒_𝑘 + 𝐴₂𝑒²_𝑘 + 𝐴₃𝑒³_𝑘 + 𝐴₄𝑒⁴_𝑘) + 𝑂(𝑒⁵_𝑘)^,                                                       (20)

𝐹^′(𝑥_𝑘) = 𝐹^′(𝛼)(𝐼 + 2𝐴₂𝑒_𝑘 + 3𝐴₃𝑒²_𝑘 + 4𝐴₄𝑒³_𝑘) + 𝑂(𝑒⁴_𝑘),

𝐹^″(𝑥_𝑘) = 𝐹^′(𝛼)(2𝐴₂ + 6𝐴₃𝑒_𝑘 + 12𝐴₄𝑒²_𝑘) + 𝑂(𝑒³_𝑘),

                                                       𝐹^‴(𝑥_𝑘) = 𝐹^′(𝛼)(6𝐴₃ + 24𝐴₄𝑒_𝑘) + 𝑂(𝑒²_𝑘),                                                                  (21)

where

1 ^′(𝛼)]⁻¹𝐹^(𝑖)(𝛼), 𝑖 = 2,3,…. 𝐴_𝑖 = [𝐹

𝑖!

Using the Genocchi–Hermite formula [17] and (20)–(21), we obtain

[𝑤_𝑘,𝑠_𝑘;𝐹] = 𝐹^′(𝑥_𝑘) + 𝐹^‴(𝑥_𝑘)(𝛾₁𝐹(𝑥_𝑘))² + 𝑂((𝛾₁𝐹(𝑥_𝑘))³) =

= 𝐹^′(𝛼)(𝐼 + 2𝐴₂𝑒_𝑘 + 3𝐴₃𝑒²_𝑘) + 𝐹^′(𝛼)6𝐴₃𝛾₁²𝐹^′(𝛼)²(𝑒_𝑘)² + 𝑂(𝑒³_𝑘) =

= 𝐹^′(𝛼)(𝐼 + 2𝐴₂𝑒_𝑘 + 𝐴₃(3𝐼 + 𝛾₁²𝐹^′(𝛼)²)𝑒²_𝑘) + 𝑂(𝑒³_𝑘).

Inversion of [𝑤_𝑘,𝑠_𝑘;𝐹] yields

                                                 [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ = (𝐼 + 𝐶₁𝑒_𝑘 + 𝐶₂𝑒²_𝑘)𝐹^′(𝛼)⁻¹ + 𝑂(𝑒³_𝑘),                                                          (22)

where 𝐶₁ = −2𝐴₂, 𝐶₂ = 4𝐴²₂ − 𝐴₃(3𝐼 + 𝛾₁²𝐹^′(𝛼)²).

Let us denote 𝑒̄_𝑘 = 𝑦_𝑘 − 𝛼. From (20) and (22), we get

𝑒̄_𝑘 = 𝑥_𝑘 − 𝛼 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘) = 𝐵₁𝑒²_𝑘 + 𝐵₂𝑒³_𝑘 + 𝑂(𝑒⁴_𝑘),

where 𝐵₁ = 𝐴₂, 𝐵₂ = −2𝐴²₂ + 𝐴₃(2𝐼 + 𝛾₁²𝐹^′(𝛼)²). We then obtain

                             𝐹(𝑦_𝑘) = 𝐹^′(𝛼)(𝑒̄_𝑘 + 𝐴₂𝑒̄²_𝑘 + 𝐴₃𝑒̄³_𝑘) + 𝑂(𝑒⁴_𝑘) = 𝐹^′(𝛼)(𝐵₁𝑒²_𝑘 + 𝐵₂𝑒³_𝑘) + 𝑂(𝑒⁴_𝑘).                                (23)

Next, we expand the term 𝛩_𝑘 = ^{𝐹(𝑦𝑘)}, which appears in the second step of (16). From (20) and (23),

𝐹(𝑥_𝑘) we obtain

                                       𝛩𝑘2 = 𝐴22𝑒2𝑘 + (−6𝐴32 + 2𝐴2𝐴3(2𝐼 + 𝛾12𝐹′(𝛼)2))𝑒3𝑘 + 𝑂(𝑒4𝑘).                                             (24)

Then, from (24), we can get

^{1 + 𝑏𝛩2}

            𝑃_𝑘 = 𝐼 1 + 𝑑𝛩^𝑘_𝑘2 = 𝐼 + 𝐴²₂(𝑏 − 𝑑)𝑒²_𝑘 − 2(𝐴₂(𝑏 − 𝑑)(3𝐴²₂ − 𝐴₃(2𝐼 + 𝛾₁²𝐹^′(𝛼)²)))𝑒³_𝑘 + 𝑂(𝑒⁴_𝑘),        (25)

and

𝑄𝑘 = 2𝛩^𝑘2 2 ²𝑒²_𝑘 + 4𝐴₂(−3𝐴²₂ + 𝐴₃(2𝐼 + 𝛾₁²𝐹^′(𝛼)²))𝑒³_𝑘 + 𝑂(𝑒⁴_𝑘). (26) = 2𝐴₂

1 + 𝑑𝛩𝑘

From (25) and (26), it follows that

𝑃𝑘𝐹(𝑦𝑘) + 𝑄𝑘𝐹(𝑥𝑘) = 𝐴22𝑒2𝑘 + 𝐴3(2𝐼 + 𝛾12𝐹′(𝛼)2))𝑒3𝑘 +

                                                                        + 𝐴³₂(−10 + 𝑏 − 𝑑) + 4𝐴₂𝐴₃(2𝐼 + 𝛾₁²𝐹^′(𝛼)²)𝑒⁴_𝑘 + 𝑂(𝑒⁵_𝑘).           (27)

Then, using (22), and (27), the second step of the method (16) gives the error equation as

𝑒𝑘+1 = 𝑥𝑘+1 − 𝛼 = 10𝐴32 − 𝑏𝐴32 − 8𝐴2𝐴3 + 𝐴32𝑑 − 4𝐴2𝐴3𝛾12𝐹′(𝛼)2 +

+ 2𝐴₂𝐴₃(2𝐼 + 𝛾₁²𝐹^′(𝛼)²) − 𝐴₂(4𝐴²₂ − 𝐴₃(3𝐼 + 𝛾₁²𝐹^′(𝛼)²))𝑒⁴_𝑘 + 𝑂(𝑒⁵_𝑘) =

= −𝐴₂(𝐴₃ + 𝐴²₂(−6 + 𝑏 − 𝑑) + 𝐴₃𝛾₁²𝐹^′(𝛼)²)𝑒⁴_𝑘 + 𝑂(𝑒⁵_𝑘).

This shows the fourth order convergence of the proposed family (16).                                                                               

The convergence analysis of the other proposed methods follows a similar approach to the proof of Theorem 8. Therefore, we omit it here.

3.       The construction of three-step derivative-free iterations

The derivative-free analogy of iteration (3) obtained as:

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

𝑧_𝑘 = 𝑦_𝑘 − 𝑇_𝑘 [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑦_𝑘),

𝑥𝑘+1 = 𝑧𝑘 − 𝐻𝑘 [𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑧𝑘),

where 𝑇_𝑘 is given by (15) and 𝐻_𝑘 determined as:

𝐻_𝑘 = 𝛼_𝑘𝐹(𝑥_𝑘)⁻¹ [𝑤_𝑘,𝑠_𝑘;𝐹].

As before, the condition (7) can be rewritten in term of 𝐻_𝑘 as:

𝐻_𝑘 = 1 + 𝑂(ℎ),

2

                                    𝐻_𝑘 =                                           

1 + 𝑑𝛩

𝐻_𝑘 [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑧_𝑘[𝑢_𝑘,𝜛_𝑘;𝐹]⁻¹ 𝐹(𝑧_𝑘) + 𝑂(ℎ³).

Theorem 7 and the combination of choices (15) and (28) yields different derivative-free three-step iterations and we list some of these methods below.

Sixth-order iterations:

𝑦𝑘 = 𝑥𝑘 − [𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑥𝑘),
1                                   𝑧𝑘 = 𝑦𝑘 −  + 𝑑𝛩𝑘2 [𝑤𝑘,𝑠𝑘;𝐹]−1 [(1 + 𝑏𝛩𝑘)𝐹(𝑦𝑘) + 2𝛩𝑘2𝐹(𝑥𝑘)], 1 𝑥𝑘+1 = 𝑧𝑘 − (1 + 2𝛩𝑘)[𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑧𝑘), and 𝑦𝑘 = 𝑥𝑘 − [𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑥𝑘), 𝑧𝑘 = 𝑦𝑘 − (1 + 𝛩𝑘2)[𝑢𝑘,𝜛𝑘;𝐹]−1 𝐹(𝑦𝑘), 𝑥𝑘+1 = 𝑧𝑘 − [𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑧𝑘), and 𝑦𝑘 = 𝑥𝑘 − [𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑥𝑘), 𝑧𝑘 = 𝑦𝑘 − [𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑦𝑘), 𝑥𝑘+1 = 𝑧𝑘 − (1 + 2𝛩𝑘2)[𝑢𝑘,𝜛𝑘;𝐹]−1 𝐹(𝑧𝑘). Seventh-order iterations 𝑦𝑘 = 𝑥𝑘 − [𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑥𝑘), 1 𝑧𝑘 = 𝑦𝑘 − 2 [𝑤𝑘,𝑠𝑘;𝐹]−1 [(1 + 𝑏𝛩𝑘)𝐹(𝑦𝑘) + 2𝛩𝑘2𝐹(𝑥𝑘)], 1 + 𝑑𝛩𝑘 𝑥𝑘+1 = 𝑧𝑘 − (1 + 2𝛩𝑘2)[𝑢𝑘,𝜛𝑘;𝐹]−1 𝐹(𝑧𝑘), and 𝑦𝑘 = 𝑥𝑘 − [𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑥𝑘), 𝑧𝑘 = 𝑦𝑘 − (1 + 𝛩𝑘2)[𝑢𝑘,𝜛𝑘;𝐹]−1 𝐹(𝑦𝑘), 𝑥𝑘+1 = 𝑧𝑘 − (1 + 2𝛩𝑘)[𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑧𝑘). Eighth-order iterations 𝑦𝑘 = 𝑥𝑘 − [𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑥𝑘),	(29)
𝑧𝑘 = 𝑦𝑘 − (1 + 𝛩𝑘2)[𝑢𝑘,𝜛𝑘;𝐹]−1 𝐹(𝑦𝑘), 𝑥𝑘+1 = 𝑧𝑘 − (1 + 2𝛩𝑘2)[𝑢𝑘,𝜛𝑘;𝐹]−1 𝐹(𝑧𝑘).	(30)

In the remainder of the paper, the methods (29)–(30) will be denoted by 𝑀₅⁶, 𝑀₆⁶, 𝑀₇⁶, 𝑀₈⁷, 𝑀₉⁷ and 𝑀₁₀⁸ , respectively. If we take the transition rule that established in [14] into account, then we easily obtain from (29)–(30) its scalar coefficients variants: Sixth-order iterations

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

                                                        1                              ⁻¹

                               𝑧_𝑘 = 𝑦_𝑘 − 1 + 𝑑𝑣_𝑘 [𝑤_𝑘,𝑠_𝑘;𝐹]       [(1 + 𝑏𝑣_𝑘)𝐹(𝑦_𝑘) + 2𝑣_𝑘𝐹(𝑥_𝑘)],                                         (31)

𝑥𝑘+1 = 𝑧𝑘 − [𝑢𝑘,𝜛𝑘;𝐹]−1 𝐹(𝑧𝑘),

where

‖𝐹(𝑦_𝑘)‖² 𝑣𝑘 = ‖𝐹(𝑥_𝑘)‖2 ,

and

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

𝑧_𝑘 = 𝑦_𝑘 − (1 + 𝑣_𝑘)[𝑢_𝑘,𝜛_𝑘;𝐹]⁻¹ 𝐹(𝑦_𝑘),

𝑥𝑘+1 = 𝑧𝑘 − [𝑤𝑘,𝑠𝑘;𝐹]−1 𝐹(𝑧𝑘).

and

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

𝑧_𝑘 = 𝑦_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑦_𝑘),

𝑥_𝑘+1 = 𝑧_𝑘 − (1 + 2𝑣_𝑘)[𝑢_𝑘,𝜛_𝑘;𝐹]⁻¹ 𝐹(𝑧_𝑘).

Seventh-order methods:

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

                                                        1                              ⁻¹

𝑧_𝑘 = 𝑦_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹] [(1 + 𝑏𝑣_𝑘)𝐹(𝑦_𝑘) + 2𝑣_𝑘𝐹(𝑥_𝑘)], 1 + 𝑑𝑣_𝑘

𝑥_𝑘+1 = 𝑧_𝑘 − (1 + 2𝑣_𝑘)[𝑢_𝑘,𝜛_𝑘;𝐹]⁻¹ 𝐹(𝑧_𝑘),

and

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

𝑧_𝑘 = 𝑦_𝑘 − (1 + 𝑣_𝑘)[𝑢_𝑘,𝜛_𝑘;𝐹]⁻¹ 𝐹(𝑦_𝑘),

                                                                       −1                                                             ‖𝐹(𝑧_𝑘)‖²

                                    𝑥𝑘+1 = 𝑧𝑘 − [𝑢𝑘,𝜛𝑘;𝐹]            (𝐹(𝑧𝑘) − 𝛽𝑘𝐹(𝑥𝑘)), 𝛽𝑘 = ‖𝐹(𝑦_𝑘)‖2 .

Eight-order method:

𝑦_𝑘 = 𝑥_𝑘 − [𝑤_𝑘,𝑠_𝑘;𝐹]⁻¹ 𝐹(𝑥_𝑘),

                                                       𝑧_𝑘 = 𝑦_𝑘 − (1 + 𝑣_𝑘)[𝑢_𝑘,𝜛_𝑘;𝐹]⁻¹ 𝐹(𝑦_𝑘),                                                                        (32)

𝑥_𝑘+1 = 𝑧_𝑘 − (1 + 2𝑣_𝑘)[𝑢_𝑘,𝜛_𝑘;𝐹]⁻¹ 𝐹(𝑧_𝑘). In the rest of the paper, the methods (31)–(32) will be denoted by 𝑀₁₁⁶ , 𝑀₁₂⁶ , 𝑀₁₃⁶ , 𝑀₁₄⁷ , 𝑀₁₅⁷ and 𝑀₁₆⁸ , respectively.

4.      Computational efficiency

The computational efficiency index of an iterative method for solving a nonlinear system is defined

1 by 𝐶𝐼 = 𝜌𝐶 , where 𝜌 is the order of convergence and 𝐶 is the computational cost of each method. We will study the computational efficiency of the presented methods and compare it with that of other methods presented in the literature, namely 𝑀₃⁴ [13], 𝑀_6,2 [18], NM7 [19] and 𝑃𝑀1 [20]. To compute 𝐹 in any iterative method we evaluate 𝑛 scalar functions, whereas the number of scalar evaluations is

𝑛(𝑛 − 1) scalar functions for any divided difference [⋅,⋅;𝐹]. In addition, we must include the number of operations shown in Table 1.

As we can see in Figs. 1, 2 and in Table 2, in terms of computational efficiency the proposed method 𝑀₅⁶ is significantly superior to other considered methods. Additionally, fourth-order 𝑀₃⁴ and eighth-order 𝑀₁₆⁸ also have high computational efficiency.

Table 1 Computational cost of different operations

Computational cost

LU decomposition	(𝑛3 − 𝑛)
Solution of two triangular systems	𝑛2
Quotients in divided difference operator	𝑛2
Matrix-vector multiplication	𝑛2
Scalar-vector multiplication	𝑛
Component-wise multiplication (division) of vectors	𝑛

Table 2

5.       Numerical results and discussion

To evaluate the effectiveness of the new method and provide a comparison with existing methods, numerical experiments have been conducted and the results are presented in this section. To achieve this goal, we consider the following nonlinear problems, most of which are the same as in [13, 19, 21].

Example 1. Considering the following system of 20 equations:

                                               𝑦_𝑖     = 0,           𝑖 = 1,2,…,20.

 

Computational Efficiency Index

n

Figure 1. Computational Efficiency Index for 𝑛 = 10 to 100 (logarithmic scale)

Computational Efficiency Index

n

Figure 2. Computational Efficiency Index for 𝑛 = 10 to 100 (logarithmic scale)

 

The solution is 𝑦^∗ = {−0.89,−0.89,…,−0.89}^𝑇. For this solution, we choose the starting vector 𝑥₀ = {−0.9,−0.9,…,−0.9}^𝑇.

Example 2. Consider the system of twenty equations

20

                                    −𝑦_𝑖 − 3 + ∑ 𝑦_𝑗 − 𝑒^𝑦𝑖 + 4 cos(2 ln(|1 + 𝑦_𝑖|)) = 0,                𝑖 = 1,2,…,20.

𝑗=1

The exact solution 𝑦^∗ = {0,0,…,0}^𝑇 of above system. For this solution, we choose the starting vector 𝑥₀ = {0.01,0.01,…,0.01}^𝑇.

In Tables 3 and 4, we present the residual error of the example function ‖𝐹(𝑥_𝑘+1)‖, the error between two consecutive iterations ‖𝑥_𝑘+1 − 𝑥_𝑘‖ and the computational order of convergence 𝜌_𝑐𝑜. The computational order of convergence (𝑝_𝑐𝑜) is calculated using the formula [4]

ln(‖𝑥𝑘+1 − 𝑥𝑘‖/‖𝑥𝑘 − 𝑥𝑘−1‖)

𝜌𝑐𝑜 = ln(‖𝑥𝑘 − 𝑥𝑘−1‖/‖𝑥𝑘−1 − 𝑥𝑘−2‖). The following stopping criterion is used in these experiments:

‖𝑥_𝑘+1 − 𝑥_𝑘‖ + ‖𝐹(𝑥_𝑘)‖ ⩽ 10⁻⁶⁰.

Tables3and4reportthenumericalperformanceoftheconsideredderivative-freeiterativemethods. The first column lists the names of the methods under comparison. The second column shows the total CPU time (in seconds) required by each method to reach the prescribed stopping criterion. The third column indicates the number of iterations (Iter) needed for convergence.

The fourth column presents the absolute error measured by the norm ‖𝑥_𝑘+1 − 𝑥_𝑘‖, while the fifth column reports the residual norm ‖𝐹(𝑥_𝑘+1)‖ at the final iteration, which reflects the accuracy of the computed solution. The last column displays the approximate computational order of convergence (ACOC), confirming the theoretical convergence order of each method.

Table 3 Comparison numerical results on Example 1

Methods	CPUTime	Iter	‖𝑥𝑘+1 − 𝑥𝑘‖	‖𝐹(𝑥𝑘+1)‖	ACOC
𝑀14	0.356	4	1.6016 × 10−76	4.4188 × 10−300	4.00
𝑀34	0.344	4	1.6016 × 10−76	4.4188 × 10−300	4.00
𝑀25	0.344	4	5.8123 × 10−166	1.9572 × 10−822	5.00
𝑀45	0.625	4	5.8123 × 10−166	1.9572 × 10−822	5.00
𝑀56	0.343	4	3.4780 × 10−228	3.2447 × 10−1359	6.00
𝑀66	0.640	4	9.7496 × 10−287	1.8481 × 10−1711	6.00
𝑀76	0.703	4	1.2693 × 10−274	1.7032 × 10−1638	6.00
𝑀116	0.672	4	7.8812 × 10−283	4.2801 × 10−1688	6.00
𝑀126	0.735	4	9.7496 × 10−287	1.8481 × 10−1711	6.00
𝑀136	0.782	4	1.2693 × 10−274	1.7032 × 10−1638	6.00
𝑀87	0.656	4	9.7872 × 10−388	3.0845 × 10−2523	7.00
𝑀97	0.640	4	2.2881 × 10−402	2.9211 × 10−2524	7.00
𝑀147	0.734	4	9.7872 × 10−388	9.3521 × 10−2524	7.00
𝑀157	0.732	4	1.5722 × 10−400	8.5164 × 10−2524	7.00
𝑀108	0.585	3	1.2259 × 10−79	2.7310 × 10−624	8.00
𝑀168	0.532	3	1.2259 × 10−79	2.7310 × 10−624	8.00
𝑀6,2 [18]	1.614	4	1.4692 × 10−107	6.3590 × 10−633	6.00
NM7 [19]	2.750	3	3.8545 × 10−71	1.2384 × 10−487	7.00
PM1 [20]	1.984	4	2.9442 × 10−271	1.0079 × 10−2152	8.00

From Tables 3 and 4, we observe that the 𝑀₃⁴ iterative method is faster than the considered fourthand fifth-order methods. Furthermore, Tables 3 and 4 indicate that the proposed 𝑀₅⁶ method is the fastest among the considered methods with orders 𝜌 = 6,7 and 8. This finding is consistent with the results presented in Section 4. From these tables, it follows that 𝑀₁₆⁸ is not only faster but also more accurate than the considered seventh- and eighth-order methods. Thus, the eighth-order method 𝑀₁₆⁸ can be highly useful in practical applications that require high accuracy. In conclusion, the numerical results clearly demonstrate that the proposed derivative-free methods with vector and scalar coefficients are superior to those employing matrix coefficients, both in terms of computational time and overall computational cost.

Conclusions

We obtain family of two-step derivative-free iterations of order 4 and 5 and three-step derivativefree iterations of order 6, 7 and 8 with vector and scalar parameter. The specific of these iterations is that they include vector or even scalar parameter of iteration instead of matrix parameter that inherent to other existing iterative methods. The theoretical conclusions are confirmed by numerical

Table 4 Comparison numerical results on Example 2

Methods	CPUTime	Iter	‖𝑥𝑘+1 − 𝑥𝑘‖	‖𝐹(𝑥𝑘+1)‖	ACOC
𝑀14	20.672	4	7.0707 × 10−76	1.6644 × 10−300	4.00
𝑀34	20.594	4	7.0707 × 10−76	1.6644 × 10−300	4.00
𝑀25	35.572	4	5.9230 × 10−172	4.1262 × 10−858	5.00
𝑀45	37.563	4	5.9230 × 10−172	4.1262 × 10−858	5.00
𝑀56	20.282	4	2.4149 × 10−219	1.0347 × 10−1310	6.00
𝑀66	37.782	4	1.1978 × 10−315	2.8416 × 10−1893	6.00
𝑀76	37.532	4	4.4919 × 10−302	1.5809 × 10−1811	6.00
𝑀116	29.656	3	7.6581 × 10−66	1.9409 × 10−394	6.00
𝑀126	37.469	4	1.1978 × 10−315	2.8416 × 10−1893	6.00
𝑀136	37.375	4	4.4919 × 10−302	1.5809 × 10−1811	6.00
𝑀87	42.219	4	9.1502 × 10−380	4.7631 × 10−2654	7.00
𝑀97	38.344	4	1.7236 × 10−399	2.0037 × 10−2792	7.00
𝑀147	37.922	4	9.1502 × 10−380	4.7631 × 10−2654	7.00
𝑀157	37.641	4	2.5776 × 10−400	3.3521 × 10−2798	7.00
𝑀108	29.906	3	6.0681 × 10−78	1.3860 × 10−620	8.00
𝑀168	29.391	3	6.0681 × 10−78	1.3860 × 10−620	8.00
𝑀6,2 [18]	56.801	4	7.8477 × 10−204	1.4880 × 10−1216	6.00
NM7 [19]	186.703	3	3.4339 × 10−83	5.5892 × 10−288	7.00
PM1 [20]	57.985	3	5.8566 × 10−78	9.7060 × 10−617	8.00

experiments. Based on numerical examples, one can conclude that our proposed iterations are the most efficient and faster than the existing ones of similar nature.

About the authors

Tugal Zhanlav

Academician, Professor, Doctor of Sciences in Physics and Mathematics

Author for correspondence.
Email: tzhanlav@yahoo.com
ORCID iD: 0000-0003-0743-5587
Scopus Author ID: 24484328800

Full member of Mongolian Academy of Sciences, professor, doctor of sciences in physics and mathematics, Honorary Doctor of JINR

Russian Federation, Ulaanbator, 13330, Mongolia; Ulaanbator, 14191, Mongolia

Khuder Otgondorj

Mongolian University of Science and Technology

Email: otgondorj@gmail.com
ORCID iD: 0000-0003-1635-7971
Scopus Author ID: 57209734799

Associate Professor of Department of Mathematics at School of Applied Sciences

Russian Federation, Ulaanbator, 14191, Mongolia

Vandandoo Ulziibayar

Mongolian University of Science and Technology

Email: v_ulzii@must.edu.mn
ORCID iD: 0000-0003-2279-0755

Professor of Department of Mathematics at School of Applied Sciences

Ulaanbator, 14191, Mongolia

Khangai Enkhbayar

Mongolian University of Science and Technology

Email: eegii33@must.edu.mn
ORCID iD: 0000-0002-1259-5502

Senior Lecturer of Department of Mathematics at School of Applied Sciences

Ulaanbator, 14191, Mongolia

References

Supplementary files