1. Introduction
In this article, the function is differentiable, where or and is an open nonempty set.
The nonlinear equation
is studied in this article. An analytic form of a solution
is preferred. However, this form is not always available. So, mostly iterative solution methods have been applied to approximate the solution
In particular, King’s [
1] fourth-order method (KM) has been used;
where
is a parameter and
As motivation consider the real function
However, then, the third derivative is unbounded. So, the convergence of KM is not assured by previous analyses in [
1,
2,
3,
4,
5,
6,
7,
8].
This is the case, since Taylor series requiring derivatives of high order (not in KM) are utilized in the analysis for convergence. This is a common observation for other methods, such as Traub’s, Jarratt’s, and the Kung–Traub method to mention some [
2,
3,
5,
6,
7,
8,
9,
10]. On the top of these concerns, some other problems exist with earlier studies. No computable data are provided for distances
or
or the uniqueness and location of solution
All these concerns are addressed utilizing conditions involving only the first derivative in the method (
2) [
9,
10,
11,
12,
13,
14,
15,
16].
The next four sections include semi-local analysis, local analysis, the experiment, and conclusions, respectively.
2. Semi-Local Analysis
Set
and
to be positive parameters. Set
and
Let the sequence
be given as
where
Sequence
shall be shown to be majorizing for KM.
Lemma 1. Then, the following assertions holdandwhere is the unique least upper bound of sequence Proof. Assertions (
5) and (
6) follow immediately by (
3) and (
4). □
Another result is given for the sequence
using stronger conditions but which are easier to verify than (
4). However, first, we need to introduce some concepts. Let
and
Develop polynomials defined on the interval
as
and
Notice that polynomials
and
are independent of
In particular, say
Then, condition
needed in the next Lemma holds if
The left side of this estimate is a positive multiple of However, the right side of it is positive but independent of So, this estimate certainly holds for sufficiently small The same observation is made for polynomial and condition
An auxiliary result connects these polynomials.
Lemma 2. The following items hold:
- (i)
;
- (ii)
;
- (iii)
if;
and
- (iv)
if
Proof. By the definition of these polynomials, we get in turn:
- (i)
- (ii)
- (iii)
This estimate follows immediately from the first two;
- (iv)
It follows similarly from the definition of polynomials and since
□
Define the parameters
and
Notice that
Lemma 3. Suppose:andhold for someThen, sequenceis convergent toNotice, criteria (7)–(11) determine the “smallness” of η to force convergence of the method. Proof. Mathematical induction is used to show
and
These estimates are true for
by (
7) or (
8) and the definition of sequence
Then, it follows
and
Suppose:
and
Evidently, (
12) holds if
or
It can be shown instead from Lemma 2 that
However, by (
15) and (
20),
Then, (
21) holds by (
10) and (
22). Moreover, instead of (
13), we can show
since
and
hold. Indeed, (
24) holds if
or
However, this holds because of the choice of
and (
9). Moreover, estimate (
25) holds if
which is true by the choice of
and (
9). Then, (
23) holds if
or
or
or
However, this holds by (
11). By sequence
(
12) and (
13), the estimate (
14) also holds. Therefore, the induction for estimates (
12)–(
14) is terminated. Hence,
is bounded by
which is non-decreasing. Hence, it converges to
. □
The semi-local convergence analysis of KM uses conditions (H). Suppose that there exist:
- (H1)
and ;
- (H2)
for all Set ;
- (H3)
and
for all
;
- (H4)
The conditions in Lemma 1 or in Lemma 3 are true;
- (H5)
Theorem 1. Assume conditions H hold. Then, KM is well defined in lies in for all and converges to a solution of Equation (1), soand Proof. We have by
and (H1)
So, (
29) is true if
and
Pick
By (H1), (H2) and
then
That is
with
By the Banach lemma on functions [
11,
12,
13], iteration
is well-defined. Suppose
Then, we can write
By (H1), (H3), we get
so
and
Then, by (H3), (
3), (
31) (for
), (
32) and (
33), we obtain
so (
30) holds, where we also used that (
29) and (
30) hold for all
k smaller than
We also get
and
We also have
so
Then, we write
Then, by the first substep of KM
and
Therefore, (
29) holds and
The induction is finished. So,
is Cauchy in
Hence, there exists
such that
By letting
k approach
∞ in (
35),
□
Notice that under conditions of Lemma 1 or under conditions of Lemma 3 provided in closed form may be used for in Theorem 1.
Proposition 1. Suppose
- (1)
The point is a solution of Equation (1) with and condition (H2) holds;
- (2)
Set Then, b uniquely solves Equation (1) in
Proof. Let
satisfy
Set
Then, by (H2) and (
40), we obtain in turn that
Therefore, follows from and □
3. Local Convergence
Set
, and
to be positive parameters. Define function
by
Notice that
is a radius of convergence for Newton’s method provided by us in [
11,
12,
13]. This point
also solves the equation
Develop
by
and
Then, we have
and
The intermediate value theorem assures
Q has zeros in
Let
stand for the smallest zero in
Define functions
and
by
and
It follows
and
as
Let
be the smallest such zero of
on
Set
Then, the definition of
implies that for all
and
The local convergence of KM uses conditions (C). Suppose that there is
- (C1)
a solution of Equation (1) with ;
- (C2)
so that
for all
Define
;
- (C3)
There exist
such that
and
for all
;
- (C4)
Theorem 2. Choose Then, under conditions (C), sequence generated by KM converges to so thatandwhere and the functions , were previously defined. Proof. Pick
Then, by (C1) and (C2)
So, we have
and
If
we see that iterate
is well-defined by KM for
Moreover, we can write
By (
42), (
48) (for
), (C3) and (
46), we have in turn that
Hence, iterate
and (
42) holds if
Next, we show that
If
we obtain by (C1), (C2), and (
46)
It follows that
and
Then, using (
44), (C3), (
48), (
50), and (
51)
That is iterate
and (
43) holds for
Simply switch
by
in the above calculations to terminate the induction for (
42) and (
43). Then, it follows from the estimate
where
. We conclude
and
□
A uniqueness of the solution result follows next.
Proposition 2. Suppose
- (1)
Element solves Equation (1), , and (C2) holds;
- (2)
There exists such that Set Then, element λ uniquely solves Equation (1) in .
Proof. Let
with
Set
Then, using (C2) and (
54), we get in turn that
Hence, follows from and □
Next, the fourth-order convergence is shown using only the first derivative. Suppose:
and
hold for all
for some constants
and
Further, suppose
Let where Then, and Hence, by the intermediate value theorem, has positive solutions. Let be the smallest such solution.
Theorem 3. Suppose conditions (55)–(57) hold. Then, sequence given in (2) is convergent to with order four, i.e.,where Proof. The first substep of (2) and (
56) gives
Note
so, since
and
Therefore, (
55) and (
56) give
□
4. Numerical Example
We verify convergence criteria using KM.
Example 1. Let us consider a scalar function F defined on the set for byChoose and Then, we obtain the estimates for each so for each and so for each so and Then, for , we have .
According to the information taken from Table 1, the conditions of Lemma 1 hold. Consequently, the sequence converges and the interval of initial points has been further extended. Example 2. Set function asNotice that solves equation Choose Then, conditions of Theorem 3 hold for Then, the radius is Example 3. The example used in the introduction gives Then, for the radius is Recall that it was shown in the Introduction that earlier articles cannot be used to solve this problem. The method used is a specialization of KM for
5. Conclusions
In this article, the extension of KM is presented. The convergence of KM has been shown by assuming the existence of a fifth derivative which was not considered before. This observation holds true for other high-convergence order methods such as Traub’s and Jarratt’s method. Other such methods can be found in [
1,
2,
3,
4,
5,
6,
7,
8] and the references therein. Therefore, these results cannot assure convergence. However, these methods may converge. Other concerns involve the absence of error estimates or uniqueness results that can be computed. This is our motivation for presenting a convergence analysis based on the first derivative used in KM. The generality of the technique allows its usage in other methods mentioned previously. This can be a fruitful direction of future research.
Author Contributions
Conceptualization, S.R., C.I.A., I.K.A. and S.G.; methodology, S.R., C.I.A., I.K.A. and S.G.; software, S.R., C.I.A., I.K.A. and S.G.; validation, S.R., C.I.A., I.K.A. and S.G.; formal analysis, S.R., C.I.A., I.K.A. and S.G.; investigation, S.R., C.I.A., I.K.A. and S.G.; resources, S.R., C.I.A., I.K.A. and S.G.; data curation, S.R., C.I.A., I.K.A. and S.G.; writing—original draft preparation, S.R., C.I.A., I.K.A. and S.G.; writing—review and editing, S.R., C.I.A., I.K.A. and S.G.; visualization, S.R., C.I.A., I.K.A. and S.G.; supervision, S.R., C.I.A., I.K.A. and S.G. project administration, S.R., C.I.A., I.K.A. and S.G.; funding acquisition, S.R., C.I.A., I.K.A. and S.G. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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Table 1.
Sequence (3) and condition (4).
Table 1.
Sequence (3) and condition (4).
n | 1 | 2 | 3 | 4 | 5 | 6 |
---|
| 0 | 0.1004 | 0.1033 | 0.1033 | 0.1033 | 0.1033 |
| 0.0167 | 0.0172 | 0.0172 | 0.0172 | 0.0172 | 0.0172 |
| 0.0167 | 0.0172 | 0.0172 | 0.0172 | 0.0172 | 0.0172 |
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