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Updated and Weaker Convergence Criteria of Newton Iterates for Equations

Samundra Regmi1, Ioannis K. Argyros2,∗, Santhosh George3, Michael I. Argyros4
1Department of Mathematics, University of Houston, Houston, TX 77204, USA

sregmi5@uh.edu
2Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

iargyros@cameron.edu
3Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka,

India-575 025
sgeorge@nitk.edu.in

4University of Oklahoma, Department of Computer Science, Norman, OK 73019, USA
michael.i.argyros-1@ou.edu

∗Correspondence: iargyros@cameron.edu

Abstract. Newton iteration is often used as a solver for nonlinear equations in abstract spaces.Some of the main concerns are general: Criteria for convergence, error estimations on consecutiveiterates, and the location of a solution. A plethora of authors has addressed these concerns by pre-senting results based on the celebrated Kantorovich theory. This article contributes in this directionby extending earlier results but without additional conditions. These extensions become possibleusing a more precise majorization than the one given in earlier articles. Numerical experimentationcomplements the theoretical results involving a partial differential and an integral equation.

1. Introduction
Nonlinear equation

F (x) = 0, (1.1)
plays a important role due to the fact that many applications can be brought to look like it. Thecelebrated Newton Iteration (NI) in the following form

xn+1 = xn −F ′(xn)−1F (xn), ∀ n = 0, 1, 2, . . . (1.2)
is often applied to solve equation (1.1) iteratively. Here, F : Ω ⊂ M1 −→ M2 is differentiable perFréchet and operates between Banach spaces M1 and M2, whereas set Ω 6= ∅.Kantorovich inaugurated the semi-local convergence of NI (SLCNI) analysis of NI in abstractspaces by applying the contraction mapping principle due to Banach. He presented two different

Received: 29 Apr 2022.
Key words and phrases. iterative processes; Newton iteration; Banach space; semi-local convergence.1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.5 2
proofs based on majorization and recurrent relations [12]. The Newton-Kantorovich Theorem givesthe SLCNI. Numerous authors applied this result, in applications and also as a theoretical tool.Even a simple equation given in [1–4, 7, 10, 11] shows that convergence criteria may not besatisfied. However, NI may be convergent (see the numerical Section, Example 4.1). That iswhy these criteria are weakened in [2–4]. But no new conditions are added. In this study twoadditional features are presented. One involves an explicit upper bound on the smallness of initialapproximation. Moreover by choosing a bit larger bound the convergence order of NI is recovered.Consequently, new results can always replace corresponding ones by Kantorovich [7] and others[5, 8–11], since preceding results imply the one in this study but not necessarily vice versa. Methodin this study uses smaller Lipschitz or Hölder parameters to achieve these extensions which arespecializations of earlier ones. That is no additional effort is needed. The generality of this ideaallows its application on other processes [3, 4, 11].Contributions by others can be found in Section 4, where comparisons take place. The majoriza-tion of NI is discussed in Section 2. SLCNI appears in Section 3. The numerical experimentationis given in Section4. Conclusions complete this study in Section 5.

2. Majorization of NI
Let K0,K,L0,L denote positive numbers, q ∈ (0, 1] and t stand for a positive variable. Theseparametrs are connected in Section 3 to initial data D = (Ω,y,F,F ′,x0). Define sequence {sn}by s0 = 0,s1(t) = s1 = t

s2(t) = s2 = s1 +
K(s1 − s0)1+q

(1 + q)(1 −K0s
q
1 )
,

sn+2(t) = sn+2 = sn+1 +
L(sn+1 − sn)1+q

(1 + q)(1 −L0s
q
n+1)

, ∀n = 1, 2, . . . . (2.1)
Sequence {xn} is majorized by {sn} (see Section 3). That is why convergence is studied first forsequence {sn}.
LEMMA 2.1. Suppose

K0t
q < 1 and L0sqn+1 < 1 ∀n = 0, 1, 2, . . . . (2.2)

Then, sequence {sn} is strictly increasing and converges to some limit point s∗ ∈ (0, ( 1L0 )
1
q ]. The

point s∗ is the unique least upper bound of sequence {sn}.

Proof. The result follows from definition of sequence {sn} and hypothesis (2.1).
�Let � be a positive constant. Moreover, introduce parameters by α = 1+�, β = L

(1+q)
(1+�), γ =

�
(1+�)L0

, δ = β(s2 − s1), λ = γ
1
q , h = δ1+q and u = ( 1

K0
)
1
q . Furthermore, consider functions withcommon domain in T = [0,u) given as

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Eur. J. Math. Anal. 10.28924/ada/ma.3.5 3
f1(t) =

(
Ktq

(1 + q)(1 −K0tq)
+ t

)q
−γ,

f2(t)
KLαt1+q

(1 + q)2(1 −K0tq)
− 1

and
f3(t) = (s2 +

β
−1
q h

1 −h
)q −γ.

It follows by these definitions f1(0) = −γ < 0, f2(0) = −1 < 0, f3(0) = −γ < 0 and f1(t) −→
∞, f2(t) −→ ∞ and f3(t) −→ ∞ as t −→ u−. So, function fi, i = 1, 2, 3 have zeros in interval
T by IVT (Intermediate Value Theorem). Let ηi denote the smallest such zero of functions fi ininterval T0 = (0,u), respectively.It also follows by these choices of zeros ηi

K0s
q
1 < 1, s

q
2 < γ,f1(t) < 0 at t = η1 (2.3)

δ < 1, f2(t) < 0 at t = η2 (2.4)
and

f3(t) < 0 at t = η3. (2.5)
Define parameter

η0 = min{ηi}. (2.6)
Suppose

η ≤ η0. (2.7)
If η0 = η1 or η0 = η2, suppose hypothesis (2.7) holds as a strict inequality.A second stronger convergence result follows. But hypotheses are easier to verify.
LEMMA 2.2. Suppose hypothesis (2.7) holds. Then, sequence {sn} is strictly increasing and
convergent to some s∗ ∈ (0,γ0), where γ0 = s2 + β

−1q h
1−h . Moreover, for σn+2 = sn+2 − sn+1 ∀n =

0, 1, 2, . . .

σn+2 ≤ βσ
1+q
n+1 ≤ β

−1
q δ(1+)

n (2.8)
and

s∗ − sn+1 ≤ β
−1
q
δ(1+q)

n

1 −δ1+q
. (2.9)

Proof. The assertions
(Ij) : 0 <

1

1 −L0s
q
j+1

≤ α (2.10)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.5 4
is shown using induction. Assertion (I1) is true by the choice of η1 and estimates (2.3). It followsby (I1) and sequence {sn} that

0 < s3 − s2 ≤ β(s2 − s1)1+q,

or
s3 < s2 + β

−1
q δ(1+q)

1

≤ γ.

So, assertion (2.8) holds for n = 1. Suppose assertion (2.10) holds ∀j = 1, 2, . . .n. Then,
0 < σn+1 ≤ βσ1+qn

and
sn+1 ≤ sn + βσ1+qn ≤ . . .

≤ s2 + β
(1+q)−1
1+q−1 σ

1+q
2 + β

(1+q)2−1
1+q−1 σ

(1+q)2

2 + . . . + β
(1+q)n−1−1
1+q−1 σ

(1+q)n−1

2

= s2 + β
−1
q (δ1+q + δ2(1+q) + . . . + δ(n−1)(1+q)) (δ < 1)

= s2 + β
−1
q δ1+q

1 − (δ1+q)n−1

1 −δ1+q

< s2 + β
−1
q

δ1+q

1 −δ1+q

= s2 + β
−1
q

h

1 −h
= γ0.

Hence, estimate
s
q
n+1 ≤ γholds if f3(t) ≤ 0 at t = η3, which is estimate (2.5). The induction for assertion (2.10) is completed.It follows that estimate (2.8) holds. Notice

δ = βσ2 =
Lα

1 + q

Kσ
1+q
1

(1 + q)(1 −K0s
q
1 )
< 1 (2.11)

also holds since it is equivalent to the second estimate in (2.4). Let n = 2, 3, . . . . Then, it followsin turn by assertion (2.8)
sj+n − sj+1 ≤ σj+n + σj+n−1 + . . . + σj+2

≤ β−
1
q (δ(1+q)

j+n−2
+ δ(1+q)

j+n−1
+ . . . + δ(1+q)

j

)

≤ β−
1
q δ(1+q)

n

δ1+q
1 −δ2n−1

1 −δ1+q
. (2.12)

Then, assertion (2.9) follows from estimate (2.12) if n −→∞.
�

REMARK 2.3. An at least as large parameter as η3 can replace it in condition (2.7) as follows.
Define sequences of functions ϕn on the interval T by

ϕn(t) = (s2(t) + β
−1
q (δ(t)1+q + δ(t)(1+q)

2

+ . . . + δ(t)(1+q)
n−1

)q −γ. (2.13)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.5 5
It follows by these definition that

ϕn+1(t) −ϕn(t) ≥ 0,

so
ϕn(t) ≤ ϕn+1(t) ∀t ∈ T. (2.14)

Moreover these functions have zeros in T0. These zeros are assured to exist by (IVT), since by
the definitions of fucntions ϕn give ϕn(0) = −γ < 0 and ϕn(t) −→ ∞ as t −→ u−. Denote the
smallest such zeros of functions ϕn in T by rn, respectively. According to the proof of Lemma 2.2,

lim
n−→∞

ϕn(t) ≤ f3(t) ∀t ∈ T. (2.15)
So, this limit exists as a well defined function denoted by ψ. Then, this function has zeros in T0,
since ψ(0) = −γ and ψ(t) −→ ∞ as t −→ u−. Denote by η4 the smallest such zero in (0,u).
Clearly, the proof of Lemma 2.2 goes through if instead of f3(t) ≤ 0, it is shown that

ψ3(t) ≤ 0 ∀t ∈ T. (2.16)
Define parameter

η̄0 = min{η1,η2,η4}.

Notice that by (2.15) ψ(η3) ≤ f3(η3) = 0, so η3 ≤ η4 and consequenlty η0 ≤ η̄0. If η̄0 replaces η0
in condition (2.6), then assertion (2.16) follows. Condition (2.7) becomes

η ≤ η̄0. (2.17)
Hence, the range of initial approximations η is further extended.

3. Convergence of NM
The notation U(w,ρ),U[w,ρ] means the open and closed balls with radius ρ > 0 and center

w ∈ X, respectively. The parameters K0,L0,K,L and t are connected with operator F as follows.Consider conditions (A):Suppose
(A1) There exist x0 ∈ Ω, t ≥ 0 such that F ′(x0)−1 ∈ L(M2,M1),

‖F ′(x0)−1F (x0)‖≤ t.

‖F ′(x0)−1(F ′(x1) −F ′(x0))‖≤ K0‖x1 −x0‖qand
‖F ′(x0)−1(F ′(x0 + τ(x1 −x0)) −F ′(x0))‖≤ K‖τ(x1 −x0)‖q.(A2) ‖F ′(x0)−1(F ′(x) −F ′(x0))‖≤ L0‖x −x0‖q, ∀x ∈ Ω. Set B1 = U(x0, ( 1L0 ) 1q ) ∩ Ω.(A3) ‖F ′(x0)−1(F ′(x + τ(y −x)) −F ′(x))‖≤ L‖τ(y −x)‖q ∀x,y ∈ B1 and ∀τ ∈ [0, 1).(A4) Conditions of Lemma 2.1 or Lemma 2.2 hold(A5) U[x0,t∗] ⊂ Ω.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.5 6
Notice that K0 ≤ K ≤ L0.Next, conditions A are applied to show the main convergence result for NI.
THEOREM 3.1. Under conditions A sequence NI is convergent to a solution x∗ ∈ U[x0,s∗] of
equation F (x∗) = 0. Moreover, upper bounds

‖x∗ −xn‖≤ s∗ − sn (3.1)
hold ∀n = 0, 1, 2, . . . .

Proof. The items
‖xi+1 −xi‖≤ si+1 − si, (3.2)and

U[xi+1,s
∗ − si+1] ⊆ U[xi,s∗ − si ], (3.3)are shown by induction ∀i = 0, 1, 2, . . . . Let u ∈ U[x1,s∗ − s1]. It follows by condition (A1)

‖x1 −x0‖ = ‖F ′(x0)−1F (x0)‖≤ t = s1 − s0,

‖u −x0‖≤‖u −x1‖ + ‖x1 −x0‖≤ s∗ − s1 + s1 − s0 = s∗.

Hence, point u ∈ U[x0,s∗−s0]. That is items (3.2) and (3.3) hold for i = 0. Assume these assertionshold if i = 0, 1, . . . ,n. It follows for each ξ ∈ [0, 1]
‖xi + ξ(xi+1 −xi ) −x0‖ ≤ si + ξ(si+1 − si ) ≤ s∗,

and
‖xi+1 −xi‖≤

i+1∑
j=1

‖xj −xj−1‖≤
i+1∑
j=1

(sj − sj−1) = si+1.

It follows by induction hypotheses, Lemmas and conditions (A1) and (A2)
‖F ′(x0)−1(F ′(xi+1) −F ′(x0))‖ ≤ K̄‖xi+1 −x0‖q,

≤ K̄(si+1 − s0)q ≤ K̄s
q
i+1

< 1.

Hence, the inverse of linear operator F ′(xi+1) exists. Therefore, Hence, F ′(v)−1 ∈ L(M2,M1) and
‖F ′(xi+1)−1F ′(x0)‖≤

1

1 − K̄sq
i+1

)
, (3.4)

follows as a consequence of a Lemma on invertible linear operators due to Banach [2, 7], where
K̄ =

{
K0, i = 0

L0, i = 1, 2, . . . .NI gives

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Eur. J. Math. Anal. 10.28924/ada/ma.3.5 7

F (xi+1) = F (xi+1) −F (xi ) −F ′(xi )(xi+1 −xi ),

=

∫ 1
0

(F ′(xi + ξ(xi+1 −xi ))dξ−F ′(xi ))(xi+1 −xi ). (3.5)
Then, using induction hypotheses, identity (A3) and condition (??)

‖F ′(x0)−1F (xi+1)‖ ≤ L̄
∫ 1
0

(‖xi+1 −xi‖)q (3.6)
≤

L̄

1 + q
(si+1 − si )1+q,

where L̄ = { K, i = 0
L, i = 1, 2, . . . .It follows by NI, estimates (3.4), (3.6) and the definition (2.1) of sequence {sn}

‖xi+2 −xi+1‖ ≤ ‖F ′(xi+1)−1F ′(x0)‖‖F ′(x0)−1F (xi+1)‖,

≤
K̃(si+1 − si )2

2(1 − L̃si+1)
= si+2 − si+1,

where K̃ = { K, i = 0
L, i = 1, 2, . . . .

and L̃ = { K0, i = 0
L0, i = 1, 2, . . . .

Moreover, if v ∈ U[xi+2,s∗ −
si+2] it follows

‖v −xi+1‖ ≤ ‖v −xi+2‖ + ‖xi+2 −xi+1‖

≤ s∗ − si+2 + si+2 − si+1 = s∗ − si+1.

Hence, point w ∈ U[xi+1,s∗ − si+1] completing the induction for items (3.2) and (3.3). Noticethat scalar majorizing sequence {si} is fundamental as convergent. Hence, the sequence {xi} isalso convergent to some x∗ ∈ U[x0,s∗]. Furthermore, let i −→ ∞ in estimate (3.6), to conclude
F (x∗) = 0.

�Next, the uniqueness ball for a solution is presented. Notice that not all condition A are used.
PROPOSITION 3.2. Under center-Lipschitz condition (A2) further suppose the existence of a
solution p ∈ U(x0, r) ⊂ Ω of equation (1.1) such that operator F ′(p) is invertible for some r > 0;
a parameter r1 ≥ r given by

r1 =

(
1 + q

L0
− rq

)1
q

. (3.7)
Then, the poiny p solves uniquely equation F (x) = 0 in the domain B2 = U(x0, r1) ∩ Ω.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.5 8
Proof. Define linear operator Q = ∫1

0
F ′(p̄ + ξ(p − p̄))dξ for some point p̄ ∈ B2 satisfying

F (p̄) = 0. By using the definition of r1, set B2 and condition (A2)
‖F ′(x0)−1(F ′(x0) −Q)‖ ≤

∫ 1
0

L0((1 −ξ)‖x0 −p‖q + ξ‖x0 − p̄‖q)dξ,

<
L0

1 + q
(r
q
1 + r

q) = 1,

concluding that p = p̄, where the invertability of linear operator is also used together with theapproximation 0 = F (p) −F (p̄) = Q(p− p̄).
�

REMARK 3.3. (1) If conditions A hold, set p = x∗ and r = s∗ in Proposition 3.2.
(2) Lipschitz condition (A3) can be replaced by

‖F ′(x0)−1(F ′(z1 + τ(z2 −z1)) −F ′(z1))‖≤ d‖τ(z1 −z2)‖q (3.8)
for all z1 ∈ B1 and z2 = z1 −F ′(z1)−1F (z1) ∈ B1. This even smaller parameter d can replace L
in the previous results. The existence of iterate z2 is assured by (A2).

4. Numerical Experimentation
Three experimenta are considered in this section.

EXAMPLE 4.1. The parameters using example of the introduction are K0 = µ+53 ,K = L0 =
µ+11
6
.

Moreover,Ω0 = U(1, 1 −µ) ∩U(1, 1L0 ) = U(1,
1
L0

). Set L = 2(1 + 1
3−µ) L0 < L1 and L < L1 for

all µ ∈ (0, 0.5). The Kantorovich criterion η ≤ 1
L1

is violated, since η > 1
L1
∀µ ∈ (0, 0.5), where

L1 is the Lipschitz constant on Ω. Interval can be enlarged if condition of Lemma 2.1 is verified.
Then, for µ = 0.4, we have the following; 1

L0
= 0.3846,

Table 1. Sequence (2.1)
n 1 2 3 4 5 6 7
sn+1 0.2000 0.2594 0.2744 0.2755 0.2755 0.2755 0.2755

Hence conditions of Lemma 2.1 hold.
Hence condition (2.2) holds, and the interval is extended form ∅ to [0.4,o.5].

EXAMPLE 4.2. Let us consider the two point PBVP(TPBVP)

u′′ + u
3
2 = 0

u(0) = u(1) = 0.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.5 9
The interval [0, 1] is divided into j subintervals. Set m = 1

j
. Denote by w0 = 0 < w1 < ... < wj =

1 the points of subdivision with corresponding values of the function u0 = u(w0), . . . ,uj = u(wj).
Then, the discretization of u′′ is given by

u′′k ≈
uk−1 − 2uk + uk+1

m2
, ∀k = 2, 3, . . . j − 1.

Notice that u0 = uj = 0. It follows that the following system of equations is obtained

m2u
3
2
1 − 2u1 + u2 = 0,

uk−1 + m
2u

3
2
k
− 2uk + uk+1 = 0, ∀k = 2, 3, . . . , j − 1

uj−2 + m
2u

3
2
j−1 − 2uj−1 = 0.

This system can be converted into an operator equation as follows: Define operator G : Rj−1 −→
Rj−1 whose derivative is given as

G′(u) =



3
2
m2u

1
2
1 − 2 1 0 . . . 0

1 3
2
m2u

1
2
2 − 2 1 0 . . .

0
. . . . . . . . . . . .

...
...

...
...

...

0 · · · 1 0 3
2
m2u

1
2
j−1 − 2


.

Let z ∈ Rj−1 be arbitrary. The norm is ‖z‖ = max1≤k≤j−1‖zk‖, where as the norm for G ∈
Rj−1 ×Rj−1 is given as

‖G‖ = max
1≤k≤j−1

j−1∑
i=1

‖gk,i‖.

Then, if u,z ∈Rj−1 for |uk| > 0, |zk| > 0, ∀k = 1, 2, . . . , j − 1 to obtain in turn

‖G′(u) −G′(z)‖ = ‖diag{
3

2
(u
1
2
k
−z

1
2
k

)}‖

=
3

2
m2
[

max
1≤k≤j−1

|uk −zk|
]1
2

=
3

2
m2‖u −z‖

1
2 .

Choose as an initial guess vector 130 sin πx to obtain after four iterations u0 = [3.35740e +
01, 6.5202e + 01, 9.15664e + 01, 1.09168e + 02, 1.15363e + 02, 1.09168e + 02, 9.15664e +

01, 6.52027e + 01, 3.35740e + 01]tr ]. Then, the parameters are ‖Q′(u0)−1‖≤ 2.5582e + 01,η =
9.15311E − 05, q = 0.5,K0 = L0 = K = L = 3200 = 0.015. Then, K0η

p = 1.4351e − 04 and the
following table shows that the conditions of Lemma 2.1 are satisfied.

EXAMPLE 4.3. Let M1 = M2 = C[0, 1] be the set of continuous real functions on [0, 1]. The
norm-max is used. Set Ω = U[x0, 3]. Consider Hammerstein nonlinear integral operator H [3, 6] on

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Eur. J. Math. Anal. 10.28924/ada/ma.3.5 10Table 2. Sequence (2.1)
n 1 2 3 4 5 6

vn+1 0.1435e-03 0.1435e-03 0.1435e-03 0.1435e-03 0.1435e-03 0.1435e-03

Ω as

H(v)(z1) = v(z1) −y(z1) −
∫ 1
0

V(z1,z2)v3(z2)dz1 = 2, v ∈ C[0, 1],z1 ∈ [0, 1]. (4.1)
where function y ∈ C[0, 1], and V is a kernel related by Green’s function

V(z1,z2) =

{
(1 −z1)z2, z2 ≤ z1
z2(1 −z1), z1 ≤ z2.

(4.2)
It follows by this definition that H′ is

[H′(v)(z)](z1) = z(z1) − 3
∫ 1
0

V(z1,z2)v2(z2)z(z2)dz2 (4.3)
z ∈ C[0, 1],z1 ∈ [0, 1]. Pick x0(z1) = y(z1) = 1. It then follows from (4.1)-(4.3) that H′(x0)−1 ∈
L(M2,M1),

‖I −H′(x0)‖ < 0.375, ‖H′(x0)−1‖≤ 1.6,

η = 0.2, L0 = 2.4, L1 = 3.6,

and Ω0 = U(x0, 3) ∩ U(x0, 0.4167) = U(x0, 0.4167), so L = 1.5. Notice that L0 < L1 and
L < L1. Set K0 = K = L0. The Kantorovich convergence criterion (A3) is not satisfied, since
2L1η = 1.44 > 1. Therefore convergence of NI is not guaranteed. However, the new condition
(2.7) is satisfied, since 2Lη = 0.6 < 1.

5. Conclusions
An updated and weaker unified framework is presented for NI. The new analysis is finer thanbefore. Convergence order 1 + q is also recovered by choosing a larger upper bound on t. NewLipschitz or Hölder parameters are smaller and specilizations of previous parameters. The newtheory can always replace previous ones due to weaker criterion. The strategy can be applied onother iterations [2, 3, 7, 11].

References
[1] J. Appell, E.D. Pascale, J.V. Lysenko, P.P. Zabrejko, New results on newton-kantorovich approximations with appli-cations to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997) 1–17. https://doi.org/10.1080/

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https://doi.org/10.1080/01630569708816744
https://doi.org/10.3390/math9161942
https://doi.org/10.1201/9781003128915
https://doi.org/10.1201/9781003128915


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[4] S. Singh, E. Martínez, P. Maroju, R. Behl, A study of the local convergence of a fifth order iterative method, IndianJ. Pure Appl. Math. 51 (2020) 439–455. https://doi.org/10.1007/s13226-020-0409-5.[5] F. Cianciaruso, E. De Pascale, Newton–Kantorovich approximations when the derivative is Hölderian: Old and newresults, Numer. Funct. Anal. Optim. 24 (2003) 713–723. https://doi.org/10.1081/nfa-120026367.[6] J.A. Ezquerro, M.A. Hernandez, Newton’s scheme: An updated approach of Kantorovich’s theory, Cham Switzerland,2018.[7] L.V. Kantorovich, G.P. Akilov, Functional analysis, Pergamon Press, Oxford, 1982.[8] F.A. Potra, V. Pták, Nondiscrete induction and iterative processes, Research Notes in Mathematics, 103. Pitman(Advanced Publishing Program), Boston, 1984.[9] P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton–Kantorovichtype theorems, J. Complex. 26 (2010) 3–42. https://doi.org/10.1016/j.jco.2009.05.001.[10] R. Verma, New trends in fractional programming, Nova Science Publisher, New York, USA, 2019.[11] T. Yamamoto, Historical developments in convergence analysis for Newton’s and Newton-like methods, J. Comput.Appl. Math. 124 (2000) 1–23. https://doi.org/10.1016/s0377-0427(00)00417-9.[12] P.P. Zabrejko, D.F. Nguen, The majorant method in the theory of newton-kantorovich approximations and the ptákerror estimates, Numer. Funct. Anal. Optim. 9 (1987) 671–684. https://doi.org/10.1080/01630568708816254.

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https://doi.org/10.1007/s13226-020-0409-5
https://doi.org/10.1081/nfa-120026367
https://doi.org/10.1016/j.jco.2009.05.001
https://doi.org/10.1016/s0377-0427(00)00417-9
https://doi.org/10.1080/01630568708816254

	1. Introduction
	2. Majorization of NI
	3. Convergence of NM
	4. Numerical Experimentation
	5.  Conclusions
	References

