




































©2025 Ada Academica https://adac.eeEur. J. Math. Anal. 5 (2025) 15doi: 10.28924/ada/ma.5.15
Three Step Inverse Free Kurchatov-Like Methods of Convergence Order Close to Four for

Equations

Ioannis K. Argyros1,∗ , Stepan Shakhno2 , Halyna Yarmola3,∗ , Samundra Regmi4 , NirjalShrestha5
1Department of Computing and Mathematics Sciences, Cameron University, Lawton, OK 73505, USA

iargyros@cameron.edu
2Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Lviv, Ukraine

stepan.shakhno@lnu.edu.ua
3Department of Computational Mathematics, Ivan Franko National University of Lviv, Lviv, Ukraine

halyna.yarmola@lnu.edu.ua
4Department of Mathematics, University of Houston, Houston, TX 77004, USA

sregmi5@uh.edu
5Department of Mathematics, University of Florida, Gainesville, FL 32603, USA

n.shrestha@ufl.edu
∗Correspondence: iargyros@cameron.edu, halyna.yarmola@lnu.edu.ua

Abstract. An inverse free Kurchatov-like methods with three steps is introduced of convergence orderclose to four to generate sequences approximating solutions of equations defined on complete normedspaces. The local analysis shows R-convergence close to four under conditions controlling the divideddifference. Numerous experiments demonstrate the performance of the method.

1. Introduction
Let E1, E2 represent complete normed spaces [1] and Ω ⊂ E1 be open and convex. A plethoraof applications from different fields can be formulated as

F (x) = 0. (1.1)
Here F : Ω→ E2 is a continuos operator. A solution of equation (1.1), which is denoted by x∗ ∈ Ωis given in analytical form only in rare cases. That leads to the development of iterative methodsgenerating sequence converging to x∗ provided some conditions are satisfied involving the initialinformation.One of the most time-consuming part of iterative methods for solving nonlinear problems isfinding the inverse operator or solving the corresponding linear problem. To avoid this, methods

Received: 9 Mar 2025.
Key words and phrases. complete normed space; divided difference; Kurchatov method; local convergence; order four.1

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Eur. J. Math. Anal. 10.28924/ada/ma.5.15 2with approximation of the inverse operator were developed. One of the first such methods wasproposed by Ulm in [17]
xn+1 = xn − TnF (xn),

Tn+1 = 2Tn − TnF ′(xn+1)Tn, n = 0, 1, 2,. . . . (1.2)
Here x0 ∈ Ω and T0 ∈ L(E2, E1) are given initial approximations for the solution x∗ and the inverseoperator F ′(x∗)−1, respectively. The Ulm method (1.2) was studied under conditions of differenttypes and it was shown that it converges with second order [2, 3, 10, 12, 13, 15, 17]. Similar methodwas prosed by Moser [13,14]

xn+1 = xn − TnF (xn),

Tn+1 = 2Tn − TnF ′(xn)Tn, n = 0, 1, 2,. . . . (1.3)
In (1.3) F ′(xn) appears instead of F ′(xn+1) in (1.2). The convergence order for (1.3) is equal to
1+
√
5

2 . The methods with approximation of the inverse operator with higher convergence order werestudied in [6, 9].In this paper we propose the three step Kurchatov-like method (TSKLM). This method is definedfor T0 ∈ L(E2, E1) and each n = 0, 1, 2, . . . by
yn = xn − TnF (xn),

zn = yn − TnF (yn),

xn+1 = zn − TnF (zn),

Kn+1 = [2yn − xn, xn;F ],

Mn = 2Tn − TnKn+1Tn,
Tn+1 = Mn +Mn(2I −Kn+1Mn)(I −Kn+1Mn),

(1.4)

where [·, ·;F ] : Ω × Ω → E2 is a divided difference of order one [1, 7] and L(E2, E1) is the spaceof linear operators mapping E2 into E1, x0 ∈ Ω.
Definition 1.1. [1, 7] Let F be a nonlinear operator defined on a subset Ω of a Banach space E1
with values in a Banach space E2, and let x, y , be two different points of Ω. A linear operator from
E1 to E2 which is denoted by [x, y ;F ] and satisfies the following conditions

[x, y ;F ](x − y) = F (x)− F (y)

is called a first-order divided difference of F at the points x and y .

If there exists a Fréchet derivative F ′(x), then
[x, x ;F ] = F ′(x).

Notice that there are other selection for the Kurchatov operator Kn+1 such as Kn+1 = [2xn+1−
zn, zn;F ] or Kn+1 = [2xn+1 − xn, xn;F ] or Kn+1 = F ′(xn+1) or Kn+1 = L(xn+1), where L(xn+1) is

https://doi.org/10.28924/ada/ma.5.15


Eur. J. Math. Anal. 10.28924/ada/ma.5.15 3an approximation to F ′(xn+1), or other options [1,4,5,8,16]. Denote the corresponding methods by
yn = xn − TnF (xn),

zn = yn − TnF (yn),

xn+1 = zn − TnF (zn),

Kn+1 = [2xn+1 − zn, zn;F ],

Mn = 2Tn − TnKn+1Tn,
Tn+1 = Mn +Mn(2I −Kn+1Mn)(I −Kn+1Mn),

(1.5)

yn = xn − TnF (xn),

zn = yn − TnF (yn),

xn+1 = zn − TnF (zn),

Kn+1 = [2xn+1 − xn, xn;F ],

Mn = 2Tn − TnKn+1Tn,
Tn+1 = Mn +Mn(2I −Kn+1Mn)(I −Kn+1Mn),

(1.6)

yn = xn − TnF (xn),

zn = yn − TnF (yn),

xn+1 = zn − TnF (zn),

Kn+1 = F ′(xn+1),

Mn = 2Tn − TnKn+1Tn,
Tn+1 = Mn +Mn(2I −Kn+1Mn)(I −Kn+1Mn),

(1.7)

yn = xn − TnF (xn),

zn = yn − TnF (yn),

xn+1 = zn − TnF (zn),

Kn+1 = L(xn+1),

Mn = 2Tn − TnKn+1Tn,
Tn+1 = Mn +Mn(2I −Kn+1Mn)(I −Kn+1Mn),

(1.8)

respectively. Notice that method (1.8) specializes to (1.7) if L = F ′. A possible choice for L ispresented in the numerical section.To test numerically the order of convergence of the iterative methods very often use computationalorder of convergence (COC) and approximated computational order of convergence (ACOC) [11].COC is denoted by δ2, ACOC can be computed by formulas denoted by δ1 and δ3:
δ1 ≈

ln
(
‖xn+1−xn‖
‖xn−xn−1‖

)
ln
(
‖xn−xn−1‖
‖xn−1−xn−2‖

) , δ2 ≈
ln
(
‖xn+1−x∗‖
‖xn−x∗‖

)
ln
(
‖xn−x∗‖
‖xn−1−x∗‖

) , δ3 ≈
ln
(
‖F (xn+1)‖
‖F (xn)‖

)
ln
(
‖F (xn)‖
‖F (xn−1)‖

) .
In this article, we provide the local convergence analysis of the method (1.4) under assumptionsthat Fréchet derivative and first-order divided differences satisfy classical Lipschitz conditions (seeSection 2). Sections 3 and 4 present results of numerical experiments and conclusions, respectively.

https://doi.org/10.28924/ada/ma.5.15


Eur. J. Math. Anal. 10.28924/ada/ma.5.15 42. Convergence
The local analysis of convergence is very important since it provides the degree of difficulty inselecting the initial points x0 from a ball centered at the solution x∗ and of a certain specifiedradius.The symbol S(x, r) is denoting an open ball centered at x ∈ E1 and of a radius r > 0.Define the parameter ρ by

ρ = sup{t > 0 : S(x∗, t) ⊂ Ω}.

Let a > 0, b0 ≥ 0, b > 0, d1 > 0, d2 > 0, λ ≥ 0 and l ≥ 0 be given parameters. It is convenient todefine the parameters
q =

b

a
,

ρ0 = min

{
1, ρ,

1

4lb(l + d1)d2

}
,

ρ̄ ∈ (0, ρ0),

c1 = 4lb(l + d1),

c2 = 1 + 2bc1,

c3 = c2 + 2lb, (2.1)
c4 = c2 + 4lb(1 + c3),

c5 = c4 + 4lb(1 + c3),

c6 = 1 + 4bc1 + 8lb,

b0 =
d2

1− c1d2ρ̄
f or ρ̄ ∈ (0, ρ0),

a = min

{
1,

1

c3c4 + 2lbc23
,

1

c5 + 2lb
,

1

c66

}
,

and

b = min{ρ̄, a}.

Let x∗ ∈ Ω be a solution of the equation F (x) = 0. We assume from now on that for each
u1, u2, v1, v2 ∈ S(x∗, ρ) there exists l > 0 such that

‖[u2, u1;F ]− [v2, v1;F ]‖ ≤ l(‖u2 − v2‖+ ‖u1 − v1‖). (2.2)
It follows by (2.2) that F ′ exists, [x, x ;F ] = F ′(x) and for each u, v ∈ S(x∗, ρ̄)

‖F ′(u)− F ′(v)‖ ≤ 2l‖u − v‖. (2.3)
Moreover, assume F ′(x∗) is invertible and set

‖F ′(x∗)‖ ≤ d1, ‖F ′(x∗)−1‖ ≤ d2. (2.4)

https://doi.org/10.28924/ada/ma.5.15


Eur. J. Math. Anal. 10.28924/ada/ma.5.15 5Furthermore, assume T0, K0 satisfy
‖T0‖ ≤ b0 ≤ b, ‖I − T0K0‖ ≤ λ, b ≥ b0 and λ ∈ [0, b]. (2.5)

Finally, assume
S(x∗, ρ∗) ⊂ Ω, ρ∗ = max{ρ̄, 3b}. (2.6)The conditions (2.2), (2.4)-(2.6) are called (A) from now on.Next, the main local analysis of convergence is presented under the conditions (A) and thepreceding notation.

THEOREM 2.1. Assume that the conditions (A) hold and b < a. Then, the sequence {xn}
generated by (TSKLM) converges to the solution of the equation F (x) = 0 provided that
x0 ∈ S(x∗, b). Moreover, the following assertions hold for en = ‖xn − x∗‖, hn = ‖I − TnKn‖,

en ≤ aq4
n (2.7)

and
hn ≤ aq4

n−1
, h0 ≤ b, (2.8)

n = 0, 1, 2, . . ., where ρ, a, b, q are given by (2.1).

Proof. The assertions (2.7) and (2.8) are shown by induction. If n = 0, assertion (2.7) for n = 0holds, since e0 ≤ b. Then, by (2.5), we get a ∈ [0, 1] and h0 ≤ λ ≤ b, so (2.8) holds if n = 0.Assume (2.7) and (2.8) hold if n = i . That is
ei ≤ aq4

i

< b < ρ̄ (2.9)
and

hi ≤ aq4
i−1
. (2.10)We need the estimate

‖Ki+1 − F ′(xi)‖ = ‖[2yi − xi , xi ;F ]− [xi , xi ;F ]‖

≤ l(‖2yi − xi − xi‖+ ‖xi − xi‖) = 2l‖yi − xi‖ (2.11)
≤ 2l‖TiF (xi)‖ ≤ 2l‖Ti‖(l + d1)‖xi − x∗‖

≤ c1ei ≤ c1aq4
i

,

where we used (2.1), (2.2) (i.e (2.3)), (2.5), the induction hypotheses (2.9), (2.10) and
F (xi) = F (xi)− F (x∗) =

1∫
0

F ′(x∗ + θ(xi − x∗))dθ(xi − x∗)

=

1∫
0

[
F ′(x∗ + θ(xi − x∗))− F ′(x∗) + F ′(x∗)

]
dθ(xi − x∗)

https://doi.org/10.28924/ada/ma.5.15


Eur. J. Math. Anal. 10.28924/ada/ma.5.15 6leading to
‖F (xi)‖ ≤ (l + d1)‖xi − x∗‖. (2.12)

It follows by the Banach Lemma on invertible operators [1], (2.1) and (2.11) that the operator Ki+1is invertible and
‖K−1i+1‖ ≤

‖F ′(x∗)−1‖
1− ‖F ′(x∗)−1‖‖Ki+1 − F ′(xi)‖

≤
d2

1− d2c1ei
≤

d2
1− d2c1ρ̄

= b0 ≤ b, (2.13)
so

‖Ti‖ = ‖TiKiK−1i ‖ = ‖(−I + (I − TiKi))K−1i ‖

≤ (1 + ‖I − TiKi‖)‖K−1i ‖

≤ (1 + aq4
i

)b ≤ (1 + 1)b = 2b, (2.14)
and

ξi = ‖I − TiF ′(xi)‖ = ‖(I − TiKi) + (TiKi − TiF ′(xi))‖

≤ ‖I − TiKi‖+ ‖Ti‖‖Ki − F ′(xi)‖

≤ aq4
i

+ 2bc1aq
4i = c2aq

4i . (2.15)
Then, by the first substep of (TSKLM) we can write in turn

yi − x∗ = xi − x∗ − Ti(F (xi)− F (x∗))

=

1∫
0

[
(I − TiF ′(xi)) + Ti(F

′(xi)− F ′(x∗ + θ(xi − x∗)))
]

(xi − x∗)dθ. (2.16)
It follows by the induction hypotheses (2.9), (2.10), the (2.16) triangle inequality, (2.3), and (2.13)-(2.16), we have in turn that

‖yi − x∗‖ = ξiei + l‖Ti‖e2i

≤ c2aq
4iaq4

i

+ 2lb
(
aq4

i
)2
≤ c3a2q2×4

i

. (2.17)
and since a ∈ [0, 1]

‖xi − yi‖ ≤ ei + ‖yi − x∗‖

≤ aq4
i

+ c3a
2q2×4

i ≤ (1 + c3)aq
4i (2.18)

and
‖I − TiF ′(yi)‖ = ‖(I − TiF ′(xi)) + (TiF

′(xi)− TiF ′(yi))‖

≤ c2aq
4i + 4lb(1 + c3)aq

4i = c4aq
4i . (2.19)

https://doi.org/10.28924/ada/ma.5.15


Eur. J. Math. Anal. 10.28924/ada/ma.5.15 7In an analogous way, we get in turn
‖zi − x∗‖ ≤ ‖(I − TiF ′(yi))(yi − x∗)‖+ l‖Ti‖‖yi − x∗‖2

≤ c4aq
4i c3a

2q2×4
i

+ l(2bc23a
4q4×4

i

)

≤ (c3c4 + 2lbc23 )a2q3×4
i

= a2q3×4
i

, (2.20)
‖yi − zi‖ ≤ ‖yi − x∗‖+ ‖zi − x∗‖

≤ c3a
2q2×4

i

+ a2q3×4
i

= (1 + c3)a
2q2×4

i (2.21)
and

‖I − TiF ′(zi)‖ ≤ ‖I − TiF ′(yi)‖+ ‖Ti‖‖F ′(yi)− F ′(zi))‖

≤ c4aq
4i + 4lb(1 + c3)a

2q2×4
i

= c5aq
4i (2.22)

leading to
ei+1 ≤ ‖I − TiF ′(zi)‖‖zi − x∗‖+ l‖Ti‖‖zi − x∗‖2

≤ c5aq
4ia2q3×4

i

+ l(2ba4q6×4
i

)

≤ (c5 + 2lb)q4
i+1 ≤ aq4i+1 (2.23)

showing (2.7) for n = i + 1. Moreover, we have
‖xi+1 − xi‖ ≤ ‖xi+1 − x∗‖+ ‖xi − x∗‖

≤ aq4
i+1

+ aq4
i ≤ 2aq4

i (2.24)
Notice that 2yi − xi ∈ S(x∗, 3b) by (2.6) and ei+1 ≤ q4i+1 < b < ρ̄.Hence, we can have

‖Ki+1 −Ki‖ ≤ ‖(Ki+1 − F ′(xi)) + (F ′(xi)− F ′(xi−1)) + (F ′(xi−1)−Ki)‖

≤ c1aq
4i + 4laq4

i−1
+ c1aq

4i−1

≤ (2c1 + 4l)aq4
i−1 (2.25)

‖I − TiKi+1‖ = ‖(I − TiKi) + (TiKi+1 − TiKi)‖

≤ aq4
i

+ 2b(2c1 + 4l)aq4
i−1

= c6aq
4i−1 . (2.26)

Next, by the fourth equation of (TSKLM), we can write I −MiKi+1 = (I − TiKi+1)2, so
‖I −MiKi+1‖ ≤ ‖I − TiKi+1‖2 ≤ c26a2q2×4

i−1
. (2.27)

But, we can also write
Ti+1 = Mi +Mi(2I −Ki+1Mi)(I −Ki+1Mi). (2.28)

https://doi.org/10.28924/ada/ma.5.15


Eur. J. Math. Anal. 10.28924/ada/ma.5.15 8Thus, we get
I − Ti+1Ki+1 = I − (Mi +Mi(2I −Ki+1Mi)(I −Ki+1Mi))Ki+1 = (I −MiKi+1)

3. (2.29)
Therefore, since a ∈ [0, 1], we get by the inductions hypotheses and (2.29) that

‖I − Ti+1Ki+1‖ ≤ ‖I −MiKi+1‖3 ≤ c66a6q6×4
i−1 ≤ aq4i

showing (2.8) for n = i + 1.Consequently the induction for (2.7) and (2.8) is completed.Finally, by letting n →∞ in (2.7), we deduce that lim
n→∞

xn = x∗, since q ∈ [0, 1).
REMARK 2.2. It turns out that the proof of Theorem 2.1 can be repeated in the case of the method
(1.8) as long as we add an additional condition of the form for each n = 0, 1, 2, . . .

‖Ln − F ′(xn)‖ ≤ σn‖F (xn)‖, (2.30)
where {σn} is a nonnegative sequence such that sup

n≥0
σn ≤ σ, where σ ≥ 0

3. Numerical Examples
In this section, we present the results of numerical investigation of the three-step Kurchatov-like method for solving the nonlinear equation (1.1). We give errors at each iteration andACOC and COC for considered methods (1.4), (1.5), (1.6), (1.7), (1.8). The computations werecarried out on a PC with 1.00 GHz processor and 8 GB of memory with use of softwareGNU Octave 7.3.0. The Euclidean norm was used. The initial approximation T0 was com-puted by formulas T0 = [2x0 − x−1, x−1;F ]−1 for methods (1.4), (1.5), (1.6), T0 = F ′(x0)

−1– for method (1.7) and T0 = L(x0)
−1 – for method (1.8), where (a) L(x) = [x, x + α;F ] and(b) L(x) = [x + α1F (x), x + α2F (x);F ] with α ∈ R.

EXAMPLE 3.1. Let F : R→ R and consider the nonlinear equation

F (x) = ex−0.1 − 10x |x − 1| − 0.1 = 0

with the exact solution x∗ = 0.1.

EXAMPLE 3.2. Let F : R2 → R2 and consider the system of two equations with x = (ξ; η){
F1(x) = 3ξ2η + η2 + |ξ − 1| − 0.75 = 0,

F2(x) = ξ4 + ξη3 + |η| − 0.5625 = 0,

and the exact solution x∗ = (0.5; −1).

https://doi.org/10.28924/ada/ma.5.15


Eur. J. Math. Anal. 10.28924/ada/ma.5.15 9Table 1. Error’s value at each iteration for Example 3.1.
n Method (1.4) Method (1.5) Method (1.6)
‖xn − x∗‖ ‖F (xn)‖ ‖xn − x∗‖ ‖F (xn)‖ ‖xn − x∗‖ ‖F (xn)‖0 6.0000e-01 7.9488e+00 6.0000e-01 7.9488e+00 6.0000e-01 7.9488e+001 6.0089e-02 4.5850e-01 6.0089e-02 4.5850e-01 6.0089e-02 4.5850e-012 2.3413e-03 1.6447e-02 2.1002e-04 1.4706e-03 1.9390e-04 1.3577e-033 3.4615e-07 2.4231e-06 3.2876e-14 2.3012e-13 1.2934e-14 9.0566e-144 1.3878e-17 8.3267e-17

Table 2. Error’s value at each iteration for Example 3.1 (Method (1.8)).
n (a), α = 10−6 (b), α1 = 0, α2 = 0.01

‖xn − x∗‖ ‖F (xn)‖ ‖xn − x∗‖ ‖F (xn)‖0 6.0000e-01 7.9488e+00 6.0000e-01 7.9488e+001 6.0098e-02 4.5857e-01 5.2350e-02 3.9520e-012 2.1022e-04 1.4720e-03 1.2109e-04 8.4780e-043 3.2724e-14 2.2890e-13 3.0254e-15 2.1178e-14
Table 3. Error’s value at each iteration for Example 3.1 (Method (1.8)).

n (b), α1 = −1, α2 = 1 (b), α1 = 0, α2 = 1 (b), α1 = −1, α2 = 0

‖xn − x∗‖ ‖F (xn)‖ ‖xn − x∗‖ ‖F (xn)‖ ‖xn − x∗‖ ‖F (xn)‖0 2.2500e-01 2.1048e+00 2.2500e-01 2.1048e+00 2.2500e-01 2.1048e+001 4.3916e-02 3.2765e-01 2.8627e-04 2.0031e-03 8.9687e-02 7.1215e-012 1.1100e-04 7.7714e-04 3.3741e-12 2.3618e-11 1.4334e-02 1.0249e-013 2.4702e-15 1.7292e-14 6.9350e-05 4.8550e-044 5.5539e-14 3.8877e-13
Table 4. COC, ACOC for Example 3.1.

Method δ1 δ2 δ3Method (1.4) 2.7545 2.7145 2.7309Method (1.5) 3.1543 3.9915 3.9319Method (1.6) 4.0870 4.0870 4.0244Method (1.8) (a), α = 10−6 3.9956 3.9931 3.9935Method (1.8) (b), α1 = 0, α2 = 0.01 3.2266 4.0224 3.9731
Tables 1, 2, 3, 6 and 7 contain the values ‖xn− x∗‖ and ‖F (xn)‖ at each iteration. The iterativeprocess was stopped if ‖F (xn)‖ ≤ 10−10.

https://doi.org/10.28924/ada/ma.5.15


Eur. J. Math. Anal. 10.28924/ada/ma.5.15 10Table 5. COC, ACOC for Example 3.1.
Method δ1 δ2 δ3Method (1.8) (b), α1 = −1, α2 = 1 3.2944 4.1014 4.0583Method (1.8) (b), α1 = 0, α2 = 1 2.0169 2.7384 2.6240Method (1.8) (b), α1 = −1, α2 = 0 3.0250 3.9288 3.9133

Table 6. Error’s value at each iteration for Example 3.2.
n Method (1.4) Method (1.5) Method (1.6)
‖xn − x∗‖ ‖F (xn)‖ ‖xn − x∗‖ ‖F (xn)‖ ‖xn − x∗‖ ‖F (xn)‖0 2.9069e-01 4.9986e-01 2.9069e-01 4.9986e-01 2.9069e-01 4.9986e-011 1.3484e-02 8.8111e-03 1.3484e-02 8.8111e-03 1.3484e-02 8.8111e-032 1.1530e-04 9.6933e-05 1.5071e-05 9.5531e-06 7.9188e-05 5.5281e-053 1.5872e-10 9.9579e-11 1.1102e-16 1.1102e-16 4.7092e-11 4.4613e-11
Table 7. Error’s value at each iteration for Example 3.2 (Method (1.8)).

n (a), α = 10−6 (b), α1 = −1, α2 = 1 (b), α1 = 0, α2 = 1 (b), α1 = −1, α2 = 0

‖xn − x∗‖ ‖F (xn)‖ ‖xn − x∗‖ ‖F (xn)‖ ‖xn − x∗‖ ‖F (xn)‖ ‖xn − x∗‖ ‖F (xn)‖0 2.9069e-01 4.9986e-01 2.9069e-01 4.9986e-01 2.9069e-01 4.9986e-01 2.9069e-01 4.9986e-011 4.2177e-02 3.3550e-02 1.2130e-01 1.3882e-01 1.2302e-01 1.1058e-01 2.6651e-02 9.4616e-022 4.8296e-04 3.1889e-04 1.9815e-02 1.6501e-02 1.7140e-02 1.1420e-02 7.4376e-04 8.7777e-043 1.2251e-11 7.9157e-12 9.5643e-05 7.1351e-05 5.0435e-05 3.1888e-05 7.0982e-11 9.6911e-114 5.0469e-14 3.6791e-14 3.5108e-15 2.2453e-15
Table 8. COC, ACOC for Example 3.2.

Method δ1 δ2 δ3Method (1.4) 2.8293 2.8343 3.0575Method (1.5) 3.2733 3.7717 3.6881Method (1.6) 2.7872 2.7904 2.7665Method (1.8) (a), α = 10−6 3.9231 3.9130 3.7611Method (1.8) (b), α1 = −1, α2 = 1 3.1016 4.0053 3.9286Method (1.8) (b), α1 = 0, α2 = 1 3.1851 4.0127 3.9750Method (1.8) (b), α1 = −1, α2 = 0 3.1562 4.5167 3.4227
Tables 4, 5 and 8 show COC and ACOC with δ1 was calculated if the condition

‖xn+1− xn‖ ≤ 10−10 was fulfilled δ2 and δ3 were calculated if the condition ‖F (xn)‖ ≤ 10−10 wasfulfilled.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.15 11The initial approximations x0 and x−1 for Example 3.1 are x0 = −0.5, x−1 = −0.6 (Tables 1, 2, 4),
x0 = −0.15, x−1 = −0.25 (Tables 3, 5) and for Example 3.2 – x0 = (0.63; −1.26),
x−1 = (0.73; −1.16).
EXAMPLE 3.3. Let F : Rm → Rm and consider the system of equations with x = (ξ1; . . . ; ξm)

Fi(x) =

m∑
j=1

ξj + eξi − 1 = 0, i = 1, . . . , m.

Here the exact solution x∗ = (0; . . . ; 0).

Table 9. COC, ACOC for Example 3.3.
Method δ1 δ2 δ3Method (1.4) 2.6007 2.5860 2.6621Method (1.5) 3.9096 3.9035 3.7967Method (1.6) 2.0176 2.0059 2.0663Method (1.7) 3.9118 3.9059 3.7993Method (1.8) (a), α = 10−6 3.9117 3.9058 3.7992Method (1.8) (b), α1 = 0, α2 = −0.1 3.9097 3.9064 3.9064Method (1.8) (c) 2.8165 2.8012 2.8803

Table 10. COC, ACOC for Example 3.3.
Method δ1 δ2 δ3Method (1.8) (b), α1 = −1, α2 = 1 4.3139 4.3096 4.2617Method (1.8) (b), α1 = 0, α2 = 1 2.7472 2.7326 2.8050Method (1.8) (b), α1 = −1, α2 = 0 3.9118 3.8946 3.8502

For solving nonlinear equations with differentiable operator can be also used method (1.8) with(c) L(xn+1) = 1
2(F ′(xn+1) + [2xn+1− xn, xn+1;F ]). For this method, the errors decrease faster thanfor (1.4) and (1.6).Tables 9 and 10 show COC and ACOC for Example 3.3 with m = 1, the initial approximation

x0 = 0.5 and x0 = 0.9, respectively.Figures 1 and 2 show changing of ‖xn−xn−1‖, ‖xn−x∗‖ and ‖F (xn)‖ form = 20, x0 = (5; . . . ; 5),
x−1 = (5.1; . . . ; 5.1), Figure 3 – for x0 = (0.05; . . . ; 0.05). The iterative process was stopped underthe condition ‖xn+1 − xn‖ ≤ 10−10.From the obtained results, we see that among methods (1.4)-(1.6), for method (1.5) the errordecreases faster and it has highest computational order of convergence.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.15 12

0 1 2 3 4 5 6
1x10-21

10-20
10-19
10-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-09
10-08
10-07
10-06
10-05
10-04
10-03
10-02
10-01
1000
1001
1002
1003
1004

Method (1.4)

0 1 2 3 4 5
10-20
10-19
10-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-09
10-08
10-07
10-06
10-05
10-04
10-03
10-02
10-01
1000
1001
1002
1003
1004

Method (1.5)

||xn-xn-1|| ||xn-x
*|| ||F(xn)||

0 1 2 3 4 5
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-09
10-08
10-07
10-06
10-05
10-04
10-03
10-02
10-01
1000
1001
1002
1003
1004

Method (1.6)

Figure 1. Example 3.3: error’s value at each iteration.

0 1 2 3 4 5
10-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-09
10-08
10-07
10-06
10-05
10-04
10-03
10-02
10-01
1000
1001
1002
1003
1004

Method (1.7)

0 1 2 3 4 5
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-09
10-08
10-07
10-06
10-05
10-04
10-03
10-02
10-01
1000
1001
1002
1003
1004

Method (1.8) (a)

||xn-xn-1|| ||xn-x
*|| ||F(xn)||

0 1 2 3 4
10-20
10-19
10-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-09
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10-07
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10-01
1000
1001
1002
1003
1004

Method (1.8) (b)

0 1 2 3 4 5
10-19
10-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-09
10-08
10-07
10-06
10-05
10-04
10-03
10-02
10-01
1000
1001
1002
1003
1004

Method (1.8) (c)

Figure 2. Example 3.3: error’s value at each iteration; (b), α1 = 0, α2 = −0.1.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.15 13

0 0.5 1 1.5 2 2.5 3
10-20
10-19
10-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-09
10-08
10-07
10-06
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10-03
10-02
10-01
1000
1001

Method (1.8) (b)-1

0 0.5 1 1.5 2 2.5 3
10-18

10-17

10-16

10-15

10-14

10-13

10-12

10-11

10-10

10-09

10-08

10-07

10-06

10-05

10-04

10-03

10-02

10-01

1000

1001
Method (1.8) (b)-2

||xn-xn-1|| ||xn-x
*|| ||F(xn)||

0 0.5 1 1.5 2 2.5 3
1x10-21

10-20
10-19
10-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-09
10-08
10-07
10-06
10-05
10-04
10-03
10-02
10-01
1000
1001

Method (1.8) (b)-3

Figure 3. Example 3.3: error’s value at each iteration ((b)-1, α1 = −1, α2 = 1;(b)-2, α1 = 0, α2 = 1; (b)-3, α1 = −1, α2 = 0).
4. Conclusion

In this article a three-step Kurchatov-like method with approximation of inverse operator isintroduced and convergence analysis is provided. The R-convergence four is shown theoreti-cally under Lipschitz conditions for Fréchet derivative and first-order divided differences. Nu-merous experiments demonstrate the performance of the method for different cases of Kn+1.Among the methods with divided differences, the best results are demonstrated by the meth-ods (1.5), namely the highest computational order of convergence. The operator L was cho-sen in the form L(xn+1) = [xn+1, xn+1 + α;F ], where α is a small number, and L(xn+1) =

[xn+1 + α1F (xn+1), xn+1 + α2F (xn+1);F ] with α1, α2 ∈ R. These approximating of the deriv-ative give quite good results, in particular if the nonlinear operator is not differentiable. For somevalues α1 and α2 these methods require a good initial approximation. In the case of the differen-tiable operator, the following choice L(xn+1) = 1
2(F ′(xn+1)+[2xn+1−xn, xn+1;F ]) is also possible.It demonstrates advantages over some methods with divided differences.

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	1. Introduction
	2. Convergence
	3. Numerical Examples
	4. Conclusion
	References

