Study of frozen Newton-like method in a Banach space with dynamics
UDC 519.6 The main objective of this work is investigation of positives and negatives of the three steps iterative frozen-type Newtonlike method for solving nonlinear equations in a Banach space. We perform a local convergence analysis by Taylor’s expansion and semilocal convergence by recurrence re...
Gespeichert in:
| Datum: | 2022 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2022
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6764 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512516957798400 |
|---|---|
| author | Singh, M. K. Singh, A. K. Singh, M. K. Singh, A. K. Singh, Manoj Kumar |
| author_facet | Singh, M. K. Singh, A. K. Singh, M. K. Singh, A. K. Singh, Manoj Kumar |
| author_sort | Singh, M. K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:45:58Z |
| description | UDC 519.6
The main objective of this work is investigation of positives and negatives of the three steps iterative frozen-type Newtonlike method for solving nonlinear equations in a Banach space. We perform a local convergence analysis by Taylor’s expansion and semilocal convergence by recurrence relations technique under the conditions of Kantorovich theorem for the Newton’s method. The convergence results are examined by comparing the proposed method with the Newton’s method and the fourth order Jarratt’s method using some test functions. We discuss the corresponding conjugacy maps for quadratic polynomials along with the extraneous fixed points. Additionally, the theoretical and numerical results are examined byusing the dynamical analysis of a selected test function. It not only confirms the theoretical and numerical results, but also reveals some drawbacks of the frozen Newton-like method. |
| doi_str_mv | 10.37863/umzh.v74i2.6764 |
| first_indexed | 2026-03-24T03:30:02Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i2.6764
UDC 519.6
M. K. Singh, A. K. Singh (Inst. Sci., Banaras Hindu Univ., Varanasi, India)
STUDY OF FROZEN-TYPE NEWTON-LIKE METHOD
IN A BANACH SPACE WITH DYNAMICS
ВИВЧЕННЯ МЕТОДУ НЬЮТОНА ЗАМОРОЖЕНОГО ТИПУ
У БАНАХОВОМУ ПРОСТОРI З ДИНАМIКОЮ
The main objective of this work is investigation of positives and negatives of the three steps iterative frozen-type Newton-
like method for solving nonlinear equations in a Banach space. We perform a local convergence analysis by Taylor
expansion and semilocal convergence by recurrence relations technique under the conditions of Kantorovich theorem for
the Newton’s method. The convergence results are examined by comparing the proposed method with the Newton method
and the fourth order Jarratt method using some test functions. We discuss the corresponding conjugacy maps for quadratic
polynomials along with the extraneous fixed points. Additionally, the theoretical and numerical results are examined by
using the dynamical analysis of a selected test function. It not only confirms the theoretical and numerical results, but also
reveals some drawbacks of the frozen-type Newton-like method.
Мета цiєї роботи — вивчення плюсiв та мiнусiв трикрокового iтерацiйного методу Ньютона замороженого типу для
розв’язання нелiнiйних рiвнянь у банаховому просторi. Проведено аналiз локальної збiжностi за допомогою рядiв
Тейлора та напiвлокальної збiжностi за допомогою рекурентних спiввiдношень за умов теореми Канторовича для
методу Ньютона. Отриманi результати збiжностi перевiрено шляхом порiвняння запропонованого методу з методом
Ньютона та методом Джарратта четвертого порядку з використанням деяких тестових функцiй. Обговорено вiдпо-
вiднi спряженi вiдображення для квадратичних полiномiв, а також додатковi нерухомi точки. Крiм того, отриманi
теоретичнi та числовi результати перевiрено за допомогою методiв динамiчного аналiзу певної тестової функцiї.
Тим самим не тiльки пiдтверджено теоретичнi та числовi результати, але й виявлено деякi недолiки запропонованого
методу Ньютона замороженого типу.
1. Introduction. Convergence analysis results are the central part of a paper related to the solution
of the nonlinear equations of the form
F (x) = 0. (1)
Generally, three type of convergence analysis are used for the numerical solution of nonlinear equa-
tions. First one is local convergence analysis in which we start with the assumption of the existence
of the particular solution, around this solution, there exists a neighborhood starting with any vec-
tor in this neighborhood leads to a sequence which converges to the solutions under some suitable
conditions [13, 15]. Second one is global convergence analysis, it also start with the assumption
of existence of the solution but it does not requires any local neighborhood for any initial vector to
converge to the solution. Third and last is semilocal convergence analysis, it does not requires the
knowledge of the existence of a solution, rather than demands that some conditions around the initial
vector.
Newton method is commonly used and basic method for solving the nonlinear equation (1). It is
only of order two under some conditions [1 – 24]. It is defined as follows:
xn+1 = xn - F \prime (xn)
- 1F (xn), n = 0, 1, 2, . . . . (2)
c\bigcirc M. K. SINGH, A. K. SINGH, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2 233
234 M. K. SINGH, A. K. SINGH
The semilocal convergence of Newton method in Banach spaces was established by Kantorovich [16],
where it was supposed that the second Fréchet derivative should be bounded in some domain. The
convergence of the sequence obtained by the iterative expression was derived from the convergence
of majorizing sequences. This technique has been used by many authors in order to establish the
order of convergence of the variants of Newton methods (see, for example, [7, 23]). In [11], Rall
introduced a different technique for the semilocal convergence of these methods, which is based on
recurrence relations. Parida and Gupta in [10], Madhu in [17], Ezquerro et al. in [8] and Chun et al.
in [6] have used this idea to prove the semilocal convergence for several Newton-like method of
different orders.
The earlier study of the multistep Newton-like methods for the solution of (1) in real space and
Banach space uses the higher order derivatives, while the Newton-like methods, e.g., the proposed
method (3) involves only the first-order derivative, hence, it limits the applications of the method (3):
yn = xn - F \prime (xn)
- 1F (xn),
zn = yn - F \prime (xn)
- 1F (yn),
xn+1 = zn - F \prime (xn)
- 1F (zn), n = 0, 1, 2, . . . .
(3)
In this paper first we have performed the local convergence analysis of the method (3) around the local
root x\ast , which requires fourth-order Fréchet derivative of operator F. Then we analyze the semilocal
convergence analysis using the recurrence relations technique which requires only the first-order
Fréchet derivative. In this technique, we generate a sequence of positive real numbers that guarantees
the convergence of the iterative scheme in Banach spaces, providing a suitable convergence domain.
The main advantage of this technique is that we get a result of semilocal convergence under the
same conditions of Kantorovich theorem for Newton method, which has quadratic convergence. This
allows us to apply the fourth-order convergence method for solving nonlinear equations F (x) = 0
under the same conditions that assure us the convergence of Newton method. Moreover, the semilocal
convergence analysis by recurrence relations technique is important in other aspects, especially if F
has no third order Fréchet derivative. As a motivational example, let us define function M on
X = Y = R, D = [ - 1, 1] by
M(t) =
\left\{ t5 \mathrm{s}\mathrm{i}\mathrm{n}
1
t
+ t5 - t4, if t \not = 0,
0, if t = 0.
Then we have t\ast \approx 0.5169 and
M \prime (t) = 5t4 \mathrm{s}\mathrm{i}\mathrm{n}
1
t
- t3 \mathrm{c}\mathrm{o}\mathrm{s}
1
t
+ 5t4 - 4t3,
M \prime \prime (t) = 20t3 \mathrm{s}\mathrm{i}\mathrm{n}
1
t
- 8t2 \mathrm{c}\mathrm{o}\mathrm{s}
1
t
- t \mathrm{s}\mathrm{i}\mathrm{n}
1
t
+ 20t3 - 12t2,
M \prime \prime \prime (t) = 60t2 \mathrm{s}\mathrm{i}\mathrm{n}
1
t
- 36t \mathrm{c}\mathrm{o}\mathrm{s}
1
t
- 9 \mathrm{s}\mathrm{i}\mathrm{n}
1
t
+
1
t
\mathrm{c}\mathrm{o}\mathrm{s}
1
t
+ 60t2 - 24t.
It is clear that M \prime \prime \prime (t) is not bounded on D.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
THE STUDY OF FROZEN-TYPE NEWTON-LIKE METHOD IN BANACH SPACE WITH DYNAMICS 235
Numerical results consist of the comparative study of the efficiency of the proposed scheme with
the Newton method and Jarratt method (see [9]) by using the classical efficiency index defined by
Ostrowski in [19]. In addition, we include the comparative study of the Newton method and Jarratt
method for the solutions of a nonlinear equation and a system of nonlinear equations. One important
aspect of this study is the discussion of the extraneous fixed points and the corresponding conjugacy
maps for quadratic polynomials as well as the comparative study of the dynamical analysis of the
proposed frozen-type Newton-like method along with fourth-order Jarratt method. The basins of
attraction reveal some other defects of these types of modified frozen-type Newton-like method.
2. Local convergence analysis.
Theorem 1. Let D be a convex subset of \Re n and F : D \rightarrow \Re n be a function such that
(1) it has simple zero x\ast \in I,
(2) Jacobian matrix F \prime (x\ast ) is non singular at the zero x\ast ,
(3) F is a fourth-order Fréchet differential in the convex set D at some neighborhood S of the
zero x\ast .
Then the iterative method (3) has convergence of fourth-order to the zero x\ast .
Proof. Let x\ast \in I be a simple zero of a function F, en = xn - x\ast and
Ak =
\biggl(
1
k!
\biggr)
F \prime (x\ast ) - 1F (k)(x\ast ).
Using Taylor expansion of F around x\ast and taking into account F (x\ast ) = 0, we get
F (xn) = F \prime (x\ast )
\bigl[
en +A2e
2
n +A3e
3
n +O(e4n)
\bigr]
, (4)
F \prime (xn) = F \prime (x\ast )
\bigl[
1 + 2A2en + 3A3e
2
n + 4A4e
3
n +O(e4n)
\bigr]
. (5)
Now from (4) and (5), we get
F \prime (xn)
- 1F (xn) = en - A2e
2
n +
\bigl(
2A2
2 - 2A3
\bigr)
e3n +O(e4n).
Since yn = xn - F \prime (xn)
- 1F (xn), we obtain
yn = x\ast +A2e
2
n +
\bigl(
2A3 - 2A2
2
\bigr)
e3n +
\bigl(
4A3
2 - 7A2A3 + 3A4
\bigr)
e4n +O(e5n),
F (yn) = F \prime (x\ast )
\Bigl[
A2e
2
n - 2(A2
2 - A3)e
3
n +
\bigl(
5A3
2 - 7A2A3 + 3A4
\bigr)
e4n -
- 2
\bigl(
6A4
2 - 12A2
2A3 + 3A2
3 + 5A2A4 - 2A5
\bigr)
e5n +O(e6n)
\Bigr]
.
Next
zn = 2A2
2e
3
n + ( - 9A3
2 + 7A2A3)e
4
n + (30A4
2 - 44A2
2A3 + 6A2
3 + 10A2A4)e
5
n +O(en)
6,
F (zn) = F \prime (x\ast )
\Bigl[
2A2
2e
3
n + ( - 9A3
2 + 7A2A3)e
4
n+
+2
\bigl(
15A4
2 - 22A2
2A3 + 3A2
3 + 5A2A4
\bigr)
e5n +O(e6n)
\Bigr]
.
Hence, using the proposed method (3), we have
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
236 M. K. SINGH, A. K. SINGH
xn+1 = 4A3
2e
4
n +
\bigl(
- 26A4
2 + 20A2
2A3
\bigr)
e5n +O(e6n).
Therefore,
en+1 = 4A3
2e
4
n +
\bigl(
- 26A4
2 + 20A2
2A3
\bigr)
e5n +O(e6n). (6)
Equation (6) confirms that the proposed method (3) converges with fourth-order to the root of (1)
locally, if there exist a fourth-order Fréchet differentiable operator in an open convex domain D.
3. Recurrence relations. Let X, Y be Banach spaces and F : D \subseteq X \rightarrow Y be a nonlinear
twice Fréchet differentiable operator in an open convex domain D. Now we study the semilocal
convergence analysis for the fourth-order method (3):
yn = xn - \tau nF (xn),
zn = yn - \tau nF (yn),
xn+1 = zn - \tau nF (zn), n = 0, 1, 2, . . . ,
(7)
where \tau n = [F \prime (xn)]
- 1 for n \in N. We assume that the inverse of F \prime at x0, [F
\prime (x0)]
- 1 = \tau 0 \in
\in L(Y,X) exists at some x0 \in D, where L(Y,X) is the set of bounded linear operators from Y
into X. In the following we assume that y0, z0 \in D and
1) \| \tau 0\| \leq \beta ,
2) \| \tau 0F (x0)\| \leq \eta ,
3) \| F \prime (x) - F \prime (y)\| \leq k\| x - y\|
in order to obtain the recurrence relations which satisfy the steps that appear in the iterative pro-
cess (7). Notice that these are the classical Kantorovich conditions [16] for the semilocal convergence
of Newton method. Let us also denote by a0 = \beta \eta k and define the sequence an+1 = anf(an)
2g(an),
where
f(x) =
1
1 - x(h(x) + 1)
, (8)
g(x) =
x
2
+ (x+ 1)h(x) +
x
2
h(x)2, (9)
and
h(x) =
x
2
+
x2
2
+
x3
8
. (10)
To study the convergence of xn defined by (7) to a solution of F (x) = 0 in a Banach space,
we have to prove that xn is a Cauchy sequence. To do this, we need to analyze some properties of
sequence an and, previously, of the real functions described in (8) – (10), respectively.
Lemma 1. Let f(x), g(x) and h(x) be the real functions described in (8) – (10). Then:
1) f is increasing and f(x) > 1 for x \in (0, 0.6),
2) h and g are increasing for x \in (0, 0.6).
Lemma 2. Let f(x) and g(x) as before and a0 \in (0, b) for b = 0.2990377177778545. Then:
1) f(a0)
2g(a0) < 1,
2) f(a0)g(a0) < 1,
3) the sequence an is decreasing and an < b for n \geq 0.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
THE STUDY OF FROZEN-TYPE NEWTON-LIKE METHOD IN BANACH SPACE WITH DYNAMICS 237
Proof. From the definition of functions f and g (i) follows trivially. From (i) and f(a0) > 1,
we obtain (ii). Now we prove (iii) by induction method for n \geq 0. Firstly, from (i) and the definition
of a1, we have a1 < a0. Now, we suppose that ak < ak - 1 for k \leq n. Then
an+1 = anf(an)
2g(an) < an - 1f(an)
2g(an) < an - 1f(an - 1)
2g(an - 1) = an.
As f and g are increasing, f(x) > 1. Finally, for all n \geq 0, an < b, since an is a decreasing
sequence and a0 < b.
Note that a0 = b, is the value of the solution of equation f(a0)
2g(a0) - 1 = 0. By using Taylor
expansion of F (y0) around x0, we get
z0 - x0 = y0 - x0 - \tau 0F (y0) = y0 - x0 - \tau 0
1\int
0
\bigl(
F \prime (x0 + t(y0 - x0)) - F \prime (x0)
\bigr)
(y0 - x0) dt,
\| z0 - x0\| \leq \| y0 - x0\| +
1
2
K\beta \| y0 - x0\| 2.
In a similar way, \| z0 - y0\| \leq 1
2
\| y0 - x0\| . Now, by using Taylor expansion of F (z0) and (1), we
have
\| x1 - x0\| = \| - \tau 0
\bigl(
F (x0) + F (y0) + F (z0)
\bigr)
\| =
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| y0 - x0 - \tau 0
y0\int
x0
(F \prime (x) - F \prime (x0))dx - \tau 0
z0\int
x0
(F \prime (x) - F \prime (x0))dx
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq \| y0 - x0\| +
1
2
K\beta \| y0 - x0\| 2 +
1
2
K\beta \| z0 - x0\| 2 \leq
\leq
\biggl(
1 +
a0
2
+
1
2
a20 +
1
8
a30
\biggr)
\eta =
\bigl(
1 + h(a0)
\bigr)
\eta .
Assuming that a0 < 0.6 and applying assumptions (i) – (iii), we have
\| I - \tau 0F
\prime (x1)\| \leq \| \tau 0\| \| F \prime (x1) - F \prime (x0)\| \leq
\leq \beta K\| x1 - x0\| \leq \beta K\eta (1 + h(a0)) \leq
\leq a0(1 + h(a0)) < 1.
Next, by the Banach lemma, \tau 1 exists and
\| \tau 1\| \leq \tau 0
1 - \tau 0K\| x1 - x0\|
\leq 1
1 - a0(1 + h(a0))
\tau 0 = f(a0)\| \tau 0\| .
Note that we need a0 < 0.6 in order to guaranty a0(1+h(a0)) < 1. We also note that K\tau 0\| y0 -
- x0\| \leq a0, so it can be deduced that x1 is well defined and
\| x1 - x0\| \leq \| \tau 0\| \| F (x0) + F (y0) + F (z0)\| \leq (h(a0) + 1)\| \tau 0F (x0)\| . (11)
Again we assume that xn, yn, zn \in D and an < 0.6 for n \geq 1. Then the following estimations can
be proved by induction for n \geq 1:
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
238 M. K. SINGH, A. K. SINGH
(\mathrm{I}n) \| \tau n\| \leq f(an - 1)\| \tau n - 1\| ,
(\mathrm{I}\mathrm{I}n) \| yn - xn\| = \| \tau nF (xn)\| \leq f(an - 1)g(an - 1)\| yn - 1 - xn - 1\| ,
(\mathrm{I}\mathrm{I}\mathrm{I}n) \| zn - yn\| \leq 1
2
K\beta (f(a0))
n\| yn - xn\| 2,
(\mathrm{I}\mathrm{V}n) K\| \tau n\| \| yn - xn\| \leq an,
(\mathrm{V}n) \| xn - xn - 1\| \leq (1 + h(an - 1))\| yn - 1 - xn - 1\| .
Let us consider n = 1. So, (\mathrm{I}1) has been proved before.
(\mathrm{I}\mathrm{I}1) By using Taylor formula
F (x1) = F (y0) + F \prime (y0)(x1 - y0) +
x1\int
y0
(F \prime (x) - F \prime (y0))dx =
=
1\int
0
\bigl(
F \prime (x0 + t(y0 - x0)) - F \prime (x0)
\bigr)
(y0 - x0) dt - (F \prime (y0) -
- F \prime (x0) + F \prime (x0))\tau 0
\bigl(
F (y0) + F (z0)
\bigr)
-
- \tau 0(F (y0) + F (z0))
1\int
0
\bigl(
F \prime (y0 + t(x1 - y0)) - F \prime (y0)
\bigr)
dt.
On the other hand,
F (y0) + F (z0) \leq \eta
ha0
\beta
.
Then we have
\| F (x1)\| \leq 1
2
K\eta 2 +K\eta 2h(a0) + \eta
h(a0)
\beta
+
k
2
\eta 2h(a0)
2,
\| y1 - x1\| = \| \tau 1\| \| F (x1)\| \leq \| f(a0)\| \| \tau 0\| \| F (x1)\| \leq f(a0)g(a0)\| y0 - x0\| .
(\mathrm{I}\mathrm{I}\mathrm{I}1) It is clear that
\| z1 - y1\| = \tau 1\| F (y1)\| \leq
\leq \beta f(a0)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
\bigl(
F \prime (x1 + t(y1 - x1)) - F \prime (x1)
\bigr)
(y1 - x1) dt
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq 1
2
\beta Kf(a0)\| (y1 - x1)\| 2.
(\mathrm{I}\mathrm{V}1) By using (\mathrm{I}1) and (\mathrm{I}\mathrm{I}1), we get
K\| \tau 1\| \| y1 - x1\| \leq Kf(a0)\| \tau 0\| f(a0)g(a0)\| y0 - x0\| = a1.
(\mathrm{V}1) It has been shown in (11) that
\| x1 - x0\| \leq (1 + h(a0))\| y0 - x0\| .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
THE STUDY OF FROZEN-TYPE NEWTON-LIKE METHOD IN BANACH SPACE WITH DYNAMICS 239
By considering that the induction hypothesis of items (\mathrm{I}n) to (\mathrm{V}n) are true for a fixed n \geq 1,
it can be proved (\mathrm{I}n+1) to (\mathrm{V}n+1) in a similar way. Note that condition an < 0.6, for n \geq 1, is
necessary for the existence of operators \tau n, n \geq 1. The above recurrence relations for the proposed
method given in (3) allow us to establish a new semilocal convergence result for this method under
mild conditions.
4. Semilocal convergence analysis. From the Lemmas 1 and 2 and the recurrence relations
proved in the previous section, we are in position to prove the semilocal convergence result for
method (3) under mild conditions. In the previous results we have used different conditions for
parameter a0. In the following, we consider the most restrictive one in order to prove the semilocal
convergence.
Theorem 2. Let X and Y be Banach spaces and F : D \subseteq X \rightarrow Y be a twice Fréchet diffe-
rentiable nonlinear operator in an open convex domain D. Let \tau 0 = F \prime (x0)
- 1 \in B(Y,X) exists at
some x0 \in D and
(i) \| \tau 0\| \leq \beta ,
(ii) \| \tau 0F (x0)\| \leq \eta ,
(iii) \| F \prime (x) - F \prime (y)\| \leq k\| x - y\| , x, y \in D,
are satisfied. Let a0 = \beta \eta k, a0 < 0.29903 . . . and B(x0, R\eta ) = \{ x \in X : \| x - x0\| < R\eta \} \subset D,
where R =
1
2
a0 +
1 + h(a0)
1 - f(a0)g(a0)
. Then the following conditions hold:
(a) the solution x\ast and the iterates xn, yn and zn belong to B(x0, R\eta ),
(b) the sequence \{ xn\} generated by (7) is well defined furthermore with initial point x0 con-
verges to a solution x\ast of operator F (x) = 0,
(c) the solution x\ast of F (x) = 0 is unique and belong to B
\biggl(
x0,
2
K\beta
- R\eta
\biggr)
\cap D.
Proof. Firstly, let us recall that \tau n exists for n \geq 1, since a0 < 0.29903 . . . . Moreover, we are
going to prove that yn and zn belong to B(x0, R\eta ) \subset D. By recurrence relation (\mathrm{V}n), it is easy to
observe that
\| xn - x0\| \leq \| xn - xn - 1\| + \| xn - 1 - xn - 2\| + . . .+ \| x1 - x0\| \leq
\leq (1 + h(a0))\| y0 - x0\|
n - 1\sum
k=0
\bigl(
f(a0)g(a0)
\bigr) k
.
Hence
\| yn - x0\| \leq \| yn - xn\| + \| xn - x0\| \leq
\leq (1 + h(a0))
\bigl(
f(a0)g(a0)
\bigr) n\| y0 - x0\| + (1 + h(a0))\| y0 - x0\|
n - 1\sum
k=0
\bigl(
f(a0)g(a0)
\bigr) k
<
< (1 + h(a0))
1 -
\bigl(
f(a0)g(a0)
\bigr) n+1
1 - f(a0)g(a0)
\eta < R\eta .
Now, by applying recurrence relations (\mathrm{I}n) and (\mathrm{I}\mathrm{I}n), we have
\| zn - yn\| \leq \tau nF (yn) \leq
1
2
K\beta (f(a0))
n\| yn - xn\| 2 \leq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
240 M. K. SINGH, A. K. SINGH
\leq 1
2
a0
\bigl(
(f(a0))
3(g(a0))
2
\bigr) n\| y0 - x0\| .
Therefore,
\| zn - x0\| \leq \| zn - yn\| + \| yn - x0\| \leq
\leq 1
2
a0
\bigl(
(f(a0))
3(g(a0))
2
\bigr) n\| y0 - x0\| + (1 + h(a0))
1 -
\bigl(
f(a0)g(a0)
\bigr) n+1
1 - f(a0)g(a0)
\| y0 - x0\| <
<
\Biggl(
1
2
a0 + (1 + h(a0))
1 -
\bigl(
f(a0)g(a0)
\bigr) n+1
1 - f(a0)g(a0)
\Biggr)
\eta < R\eta .
In order to prove the convergence of the sequence xn, let us state that
\| xn+1 - xn\| \leq (1 + h(an))\| yn - xn\| \leq
\leq (1 + h(an))f(an - 1)g(an - 1)\| yn - 1 - xn - 1\| \leq . . .
. . . \leq (1 + h(an))
\Biggl[
n - 1\prod
j=0
f(aj)g(aj)
\Biggr]
\| y0 - x0\| (12)
by (\mathrm{V}n) and (\mathrm{I}\mathrm{I}n).
Then, from (12), we obtain
\| xn+m - xn\| \leq \| xn+m - xn+m - 1\| + \| xn+m - 1 - xn+m - 2\| + . . .+ \| xn+1 - xn\| \leq
\leq
\bigl(
1 + h(an+m - 1)
\bigr)
\eta
n+m - 2\prod
j=0
f(aj)g(aj) +
\bigl(
1 + h(an+m - 2)
\bigr)
\eta
n+m - 3\prod
j=0
f(aj)g(aj) + . . .
. . .+
\bigl(
1 + h(an)
\bigr)
\eta
n - 1\prod
j=0
f(aj)g(aj),
and, as h is increasing and an is decreasing by Lemmas 1 and 2,
\| xn+m - xn\| \leq
\bigl(
1 + h(a0)
\bigr)
\eta
m - 1\sum
l=0
\Biggl[
n+l - 1\prod
j=0
f(aj)g(aj)
\Biggr]
\leq
\leq
\bigl(
1 + h(a0)
\bigr)
\eta
m - 1\sum
l=0
\bigl(
f(a0)g(a0)
\bigr) l+n
.
Since f and g are also increasing. Therefore, by applying the partial sum of a geometric sequence,
we have
\| xn+m - xn\| \leq
\bigl(
1 + h(a0)
\bigr) 1 - \bigl( f(a0)g(a0)\bigr) m
1 - f(a0)g(a0)
\bigl(
f(a0)g(a0)
\bigr) n
\eta .
Then we conclude that \{ xn\} is a Cauchy sequence if f(a0)g(a0) < 1.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
THE STUDY OF FROZEN-TYPE NEWTON-LIKE METHOD IN BANACH SPACE WITH DYNAMICS 241
In order to prove that x\ast is a solution of F (x) = 0, we start with the bound of \| F \prime (xn)\| ,
\| F \prime (xn)\| \leq \| F \prime (x0)\| + \| F \prime (xn) - F \prime (x0)\| \leq
\leq \| F \prime (x0)\| + K\| xn - x0\| \leq \| F \prime (x0)\| +KR\eta , (13)
by applying hypothesis (ii) and Lemmas 1 and 2. Then, by (12), we get
\| F (xn)\| \leq \| F \prime (xn)\| \| \tau nF (xn)\| \leq
\leq \| F \prime (xn)\| \leq f(an - 1)g(an - 1)\| yn - 1 - xn - 1\| \leq
\leq \| F \prime (xn)\|
\Biggl[
n - 1\prod
j=0
f(aj)g(aj)\eta
\Biggr]
,
and, as f and g are increasing and an is decreasing,
\| F (xn)\| \leq \| F \prime (xn)\|
\bigl(
f(a0)g(a0)
\bigr) n
\eta .
Since \| F \prime (xn)\| is bounded (see (13)) and
\bigl(
f(a0)g(a0)
\bigr) n
tends to zero when n \rightarrow \infty , we conclude
that \| F (xn)\| \rightarrow 0. By continuity of F in D, F (x\ast ) = 0.
Let us observe that, if a0 \in (0, 0.29903 . . .),
2
K\beta
- R\eta > 0. So, we are going to prove the
uniqueness of x\ast in B
\biggl(
x0,
2
K\beta
- R\eta
\biggr)
\cap D. Let us assume that y\ast is the another solution of
F (x) = 0 in B
\biggl(
x0,
2
K\beta
- R\eta
\biggr)
\cap D. Then, in order to prove that y\ast = x\ast , and taking into account
the Taylor expansion
0 = F (y\ast ) - F (x\ast ) =
1\int
0
F \prime (x\ast + t(y\ast - x\ast )) dt(y\ast - x\ast ),
it is necessary to show that the operator P =
\int 1
0
F \prime (x\ast +t(y\ast - x\ast )) dt is invertible. So, by applying
hypothesis (iii), we have
\| I - \tau 0P\| = \| \tau 0(F \prime (x0) - P )\| =
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \tau 0
1\int
0
\bigl(
F \prime (x\ast + t(y\ast - x\ast )) - F \prime (x0)
\bigr)
dt
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq K\beta
1\int
0
\bigm\| \bigm\| (x\ast + t(y\ast - x\ast )) - x0
\bigm\| \bigm\| dt \leq
\leq K\beta
1\int
0
\bigl(
(1 - t)\| x\ast - x0\| + t\| y\ast - x0\|
\bigr)
dt <
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
242 M. K. SINGH, A. K. SINGH
Table 1. Efficiency indices for Newton, Jarratt and proposed methods
m Newton method (2) Jarratt method (14) Proposed method (3)
2 1.1220 1.1487 1.1487
3 1.0595 1.0682 1.0801
4 1.0353 1.0393 1.0508
5 1.0234 1.0255 1.0353
6 1.0166 1.0179 1.0260
7 1.0125 1.0133 1.0200
8 1.0097 1.0102 1.0159
<
1
2
K\beta
\biggl(
R\eta +
2
K\beta
- R\eta
\biggr)
= 1.
Therefore, by the Banach lemma, the integral operator P is invertible and, hence, y\ast = x\ast .
5. Numerical results. In this section, we consider the situation X = Y = Rm to study the
efficiency of iterative method (3). Notice that we have proved in Theorem 2 of previous section,
that the method (3) has a fourth-order of convergence. Nevertheless, it is not the only advantage
of the scheme: the number of evaluations of the nonlinear function F and its associated Jacobian
matrix are also lower than the respective one of known methods. The most used tool to compare the
efficiency of different iterative methods is the efficiency index, defined by Ostrowski as \mathrm{E}\mathrm{I} = p1/d,
where p is the order of convergence and d is the total number of functional evaluations per iteration.
The efficiency index of proposed frozen-type Newton-like method (3) is \mathrm{E}\mathrm{I}\mathrm{P}\mathrm{M} = 41/(m
2+3m). We
compare it in Table 1 with not only the index of classical Newton method, \mathrm{E}\mathrm{I}\mathrm{N} = 21/(m
2+m), but
also with fourth-order Jarratt method, \mathrm{E}\mathrm{I}\mathrm{J} = 41/(2m
2+m), whose iterative expression is given by
yn = xn - 2/3F \prime (xn)
- 1F (xn),
xn+1 = xn -
\bigl[
6F \prime (yn) - 2F \prime (xn)
\bigr] - 1\bigl[
3F \prime (yn) + F \prime (xn)
\bigr]
F \prime (xn)
- 1F (xn),
(14)
where n = 0, 1, 2, . . . .
In Table 1, the efficiency indices for systems of size m \leq 8 can be observed. We remark that the
best efficiency index is the one of method (3). In a similar way, the same conclusion can be reached
for higher sizes of the system.
Example 1. Let X = R, D = ( - 1, 1) and F : D \rightarrow R be an operator defined by
F (x) = ex - 1 \forall x \in D.
Then its Fréchet derivative F
\prime
(x) at any point x \in D is given by
F \prime (x) = ex.
We have computed the numerical results with the help of MATLAB 2007 and the stopping criterion
used for the computation is | xn+1 - x\ast | + | f(xn+1)| < 10 - 14. The initial approximation is 0.1 and
approximate solution is 0. The numerical solution of Example 1 by 2nd order Newton method (2),
4th order Jarratt method (14) and proposed frozen-type Newton-like method (3) are given in Table 2.
Numerical results in Table 2 reveals that the proposed method (3) is converging to the root 0 in much
better way in comparison to the other methods.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
THE STUDY OF FROZEN-TYPE NEWTON-LIKE METHOD IN BANACH SPACE WITH DYNAMICS 243
Table 2. Comparison of the different methods for Example 1
Method N x f(x)
Newton method (2)
1 0.1000000 0.10517091807565
2 0.00483741803596 0.00484913723185
3 1.168146299657721e-005 1.168153122521609e-005
4 6.822793885920731e-011 6.822786779991930e-011
5 7.106394304750309e-017 0.00000000
Jarratt method (14)
1 0.1000000 0.10517091807565
2 4.469864996595185e-006 4.469874986368083e-006
3 8.853103661407222e-017 0.00000000
Proposed method (3)
1 0.1000000 0.10517091807565
2 4.270663384022157e-005 4.270754578139524e-005
3 5.636437603011429e-017 0.00000000
Example 2. Let X = R, D = ( - 2, 2) and F : D \rightarrow R be an operator defined by
F (x) = x3 - 1 \forall x \in D.
Then F is Fréchet differentiable and its Fréchet derivative F
\prime
(x) at any point x \in D is given by
F \prime (x) = 3x2.
We have computed the numerical results with the help of MATLAB 2007 and the stopping criterion
used for the computation is | xn+1 - x\ast | + | f(xn+1)| < 10 - 14. The initial approximation is - 2.0
and the approximate solution is 1.0. The numerical solution of Example 2 by 2nd order Newton
method (2), 4th order Jarratt method (14) and proposed method (3) are given in Table 3. Numerical
results in Table 3 reveals that the proposed method (3) is converging to the root 1.0 in much better
way in comparison to the others starting with the point - 2.0.
Example 3. Let D = X = Y = R2. Consider an operator F : R2 \rightarrow R2 defined by
F (x, y) =
\bigl(
- x2 + 1/3, - y2 + 1/3
\bigr)
\forall (x, y) \in R2.
The starting vector is [0.1, 0.1] and approximate solution is [0.57735, 0.57735]. The numerical
solution of Example 3 by 2nd order Newton method (2), 4th order Jarratt method (14) and proposed
method (3) are shown in Table 4. Numerical results show that the proposed method (3) is converging
to the root much faster in comparison to the other method.
Example 4. Consider the boundary problem
x\prime \prime + 3xx\prime = 0, x(0) = 0, x(2) = 1.
We take t0 = 0 < t1 < t2 < t3 < . . . < tn - 1 < tn = 2, ti+1 = ti+h, h =
2
n
. Here x0 = x(t0) = 0,
x1 = x(t1), x2 = x(t2), x3 = x(t3), . . . , xn - 1 = x(tn - 1) and xn = x(tn) = 1.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
244 M. K. SINGH, A. K. SINGH
Table 3. Comparison of the different methods for Example 2
Method N x f(x)
Newton method (2)
1 –2.0000000 –9.0000000
2 –1.25000000000000 –2.95312500000000
3 –0.62000000000000 –1.23832800000000
4 0.45381893860562 –0.90653525030627
5 1.92104897791877 6.08949519579644
6 1.37102304993910 1.57711779001700
7 1.09134823246687 0.29984045141206
8 1.00743271644716 0.02246429578528
9 1.00005470281893 1.641174341520113e–004
10 1.00000000299218 8.976540177840775e–009
11 1.0000000000 0.00000000
Jarratt method (14)
1 –2.0000000 –9.0000000
2 –0.53409090909091 –1.15235108705860
3 3.58636394131284 45.12783567736838
4 1.60490771004722 3.13380694233384
5 1.02464061498428 0.07575828546674
6 1.00000023207008 6.962104164287553e–007
7 1.0000000000 0.00000000
Proposed method (3)
1 –2.0000000 –9.0000000
2 –0.83625920116901 –1.58482068849867
3 0.84307872254916 –0.40075504502418
4 0.84307872254916 0.01634336199084
5 1.00000000335886 1.007659089502511e–008
6 1.0000000000 0.00000000
We discretize the above problem by using the central difference schemes for the first and second
order derivatives, i.e.,
x\prime \prime i =
xi - 1 - 2xi + xi+1
h2
, i = 1, 2, 3, . . . , n - 1,
x\prime i =
xi+1 - xi - 1
2h
, i = 1, 2, 3, . . . , n - 1,
xi =
xi+1 - xi - 1
2
, i = 1, 2, 3, . . . , n - 1.
Thus we get an (n - 1)\times (n - 1) nonlinear system
4(xi - 1 - 2xi + xi+1) + 3h(x2i+1 - x2i - 1) = 0, i = 1, 2, 3, . . . , n - 1. (15)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
THE STUDY OF FROZEN-TYPE NEWTON-LIKE METHOD IN BANACH SPACE WITH DYNAMICS 245
Table 4. Comparison of the different methods for Example 3
Method N x y f(x, y) g(x, y)
Newton method (2)
1 0.1000000 0.1000000 0.323333 0.323333
2 1.71667 1.71667 –2.61361 –2.61361
3 0.955421 0.955421 –0.579495 –0.579495
4 0.652154 0.652154 –0.091971 –0.091971
5 0.58164 0.58164 –0.00497212 –0.00497212
6 0.577366 0.577366 - 0.000018269 - 0.000018269
7 0.57735 0.57735 - 2.50303\times 10 - 10 - 2.50303\times 10 - 10
8 0.57735 0.57735 0.00000000 0.00000000
Jarratt method (14)
1 0.1000000 0.1000000 0.323333 0.323333
2 0.955421 0.955421 –0.579495 –0.579495
3 0.58164 0.58164 –0.00497212 –0.00497212
4 0.57735 0.57735 –2.50303\times 10 - 10 - 2.50303\times 10 - 10
5 0.57735 0.57735 0.00000000 0.00000000
Proposed method (3)
1 0.1000000 0.1000000 0.323333 0.323333
2 0.348886 0.348886 0.211612 0.211612
3 0.577366 0.577366 –0.000018269 –0.000018269
4 0.57735 0.57735 0.00000000 0.00000000
Table 5. Solution of Example 4 by proposed method
N x1 x2
1 0.7642513878376436 0.9813462685344896
2 0.7321437776456221 0.9820633448941537
3 0.7321436796857499 0.9820632479169275
N f(x1, x2) g(x1, x2)
1 –0.2625450310300572 0.038075035451129
2 –1.481897515809294\times 10 - 8 –6.708611455241709\times 10 - 7
3 –2.220446049250313\times 10 - 16 0.000000
Next, we solve the above problem for n = 3 by the proposed method using the initial approximations
x0 = [0.1, 0.1]. The solution of the problem is shown in Table 5 with x = [x1, x2] and F =
= [f, g]. We use the numerical iterations up to 3 and solution comes out to be [0.7321436796857499,
0.9820632479169275].
6. Corresponding conjugacy maps for quadratic polynomials. In this section, we have
discussed the rational map R(z) arising from various methods applied to a generic polynomial with
simple roots.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
246 M. K. SINGH, A. K. SINGH
Theorem 3 (Newton method). For a rational map R(z) arising from Newton method applied
to P (z) = (z - a)(z - b), a \not = b, R(z) is conjugate via the Mobius transformation given by
M(z) = (z - a)/(z - b) to
S(z) = MoRoM - 1(z) = M
\biggl(
R
\biggl(
zb - a
z - 1
\biggr) \biggr)
,
S(z) = z2.
Theorem 4 (Jarratt method [9]). For a rational map R(z) arising from Jarratt method (14)
applied to P (z) = (z - a)(z - b), a \not = b, R(z) is conjugate via the Mobius transformation given by
M(z) = (z - a)/(z - b) to
S(z) = z4M(z),
where M(z) = 1.
Theorem 5 (proposed Newton-like method). For a rational map R(z) arising from proposed
Newton-like method (3) applied to P (z) = (z - a)(z - b), a \not = b, R(z) is conjugate via the Mobius
transformation given by M(z) = (z - a)/(z - b) to
S(z) = z4M(z),
where M(z) = (4 + 14z + 14z2 + 6z3 + z4)/(1 + 6z + 14z2 + 14z3 + 4z4).
Theorem 6 (Newton-like method). For a rational map R(z) arising from any Newton-like
method of order p applied to P (z) = (z - a)(z - b), a \not = b, R(z) is conjugate via the Mobius
transformation given by M(z) = (z - a)/(z - b) to
S(z) = zpM(z),
where M(z) is either unity or a rational function and p is the order of the Newton-like method.
7. Extraneous fixed points. The Newton-like iterative methods discussed in earlier sections can
be written in the fixed-point iteration form as
xn+1 = xn - Ef (xn)
f(xn)
f \prime (xn)
, n = 0, 1, 2, . . . .
Clearly, the root x\ast of f(x) = 0 is a fixed point of the method. However, the points \xi \not = x\ast at
which Ef (\xi ) = 0 are also fixed points of the method as, with Ef (\xi ) = 0, second term on right-hand
side of (15) vanishes. These points are called extraneous fixed points (see [24]). In this section,
we have discussed the extraneous fixed points of some Newton-like method for the polynomial
z3 - 1.
Theorem 7. Newton method given by xn+1 = xn -
f(xn)
f \prime (xn)
, n = 0, 1, 2, . . . , has no extraneous
fixed points.
Proof. For Newton method, we have Ef (xn) = 1. Hence, it has no extraneous fixed point.
Theorem 8. Jarratt method [9] given by equation (14) has 6 extraneous fixed points.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
THE STUDY OF FROZEN-TYPE NEWTON-LIKE METHOD IN BANACH SPACE WITH DYNAMICS 247
Proof. For Jarratt method (14), Ef (xn) given by the equation (1 + 7z3 + 19z6)/(2 + 14z3 +
+ 11z6). In this equation numerator is of degree 6 and hence, Jarratt method has 6 extraneous fixed
points
z = - 0.5983578868038158646404450602561 - 0.1293213674687494778306860664455i,
z = - 0.5983578868038158646404450602561 + 0.1293213674687494778306860664455i,
z = 0.1871833539218283973810430511561 + 0.5828538142612528060331642836771i,
z = 0.1871833539218283973810430511561 - 0.5828538142612528060331642836771i,
z = 0.4111745328819874672594020091000 - 0.4535324467925033282024782172316i,
z = 0.4111745328819874672594020091000 + 0.4535324467925033282024782172316i.
These fixed points are repelling (the magnitude of the derivative at these points are greater than 1).
Theorem 9. There are 27 extraneous fixed points for the proposed frozen-type Newton-like
method (3).
Proof. For the proposed Newton-like method (3) we have Ef (xn) given by the equation
- (1/(1594323z26))( - 1 - 18z3 + 18z6 + 1434z9 + 900z12 -
- 38376z15 - 21495z18 + 87246z21 + 561861z24 + 1002754z27).
In this equation numerator is of degree 27 and hence, proposed Newton-like method (3) has 27
extraneous fixed points:
z = - 0.817521886913956291382533799779,
z = - 0.709816021821800370519780121431 - 0.303973876829089487675153098673i,
z = - 0.709816021821800370519780121431 + 0.303973876829089487675153098673i,
z = - 0.48909766002136655288123896330,
z = - 0.45537054547975304662985690711 - 0.01694655852238461177201778938i,
z = - 0.45537054547975304662985690711 + 0.01694655852238461177201778938i,
z = - 0.327728143334710450688745469971 - 0.567641795325934006419800381417i,
z = - 0.327728143334710450688745469971 + 0.567641795325934006419800381417i,
z = - 0.316119935047690763754218947392 - 0.456037024774720125076116948043i,
z = - 0.316119935047690763754218947392 + 0.456037024774720125076116948043i,
z = - 0.236879680997335665242581064393 - 0.501786406781346974501530338165i,
z = - 0.236879680997335665242581064393 + 0.501786406781346974501530338165i,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
248 M. K. SINGH, A. K. SINGH
z = 0.091658911490066742981618441188 - 0.766705645325433323511620927240i,
z = 0.091658911490066742981618441188 + 0.766705645325433323511620927240i,
z = 0.21300912255277176994946657842 - 0.40283573978183552090630445168i,
z = 0.21300912255277176994946657842 + 0.40283573978183552090630445168i,
z = 0.24236142292698127668039032869 - 0.38588918125945090913428666229i,
z = 0.24236142292698127668039032869 + 0.38588918125945090913428666229i,
z = 0.24454883001068327644061948165 - 0.42357099851002806413216010987i,
z = 0.24454883001068327644061948165 + 0.42357099851002806413216010987i,
z = 0.408760943456978145691266899889 - 0.707994722217275186239453660464i,
z = 0.408760943456978145691266899889 + 0.707994722217275186239453660464i,
z = 0.552999616045026428996800011785 - 0.045749382006626849425413390123i,
z = 0.552999616045026428996800011785 + 0.045749382006626849425413390123i,
z = 0.618157110331733627538161680243 - 0.462731768496343835836467828567i,
z = 0.618157110331733627538161680243 + 0.462731768496343835836467828567i,
z = 0.655456286669420901377490939942.
These fixed points are repelling (the magnitude of the derivative at these points are greater than 1).
Remark . Similarly we may calculate the extraneous fixed points for other Newton-like method.
These fixed points are repelling (the derivative at these points has its magnitude > 1). These fixed
points can be seen in the basin of attractions plot for Example 2 (z3 - 1), Fig. 2 (see dynamics of
methods in Subsection 8.2).
8. Dynamics of methods. In numerical and theoretical sections, we have seen the advantages
of the fourth-order frozen-type Newton-like method. Now we have disclosed some defects of the
fourth order frozen-type Newton-like method by the study of the dynamical analysis of the functions
F (z) = (ez - 1) and z3 - 1. For these purpose we have plotted the basins of attraction of above two
examples by using different iterative methods. The dynamics of the function by iterative methods
usually help us to study the important information about the convergence, divergence and stability of
the methods. The basic definitions and dynamical concepts of function can be found in [1, 4].
8.1. For Example 1. We have taken a square R \times R = [ - 5.0, 5.0] \times [ - 5.0, 5.0] of 500 \times
\times 500 points to study the dynamics of function F (z) = (ez - 1). If with every starting point z0 in
the above squares numerical iterative methods generate a sequence that converges to a zero z\ast of the
function with a tolerance - | F (zn)| < 5 \times 10 - 2 and a maximum of 21 iterations, then we say that
z0 will lie in the basin of attraction of this zero and we assign a fixed color to this point z0, i.e., z0
would be the part of the basin if both the above criterion are satisfied. We have described the basins
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
THE STUDY OF FROZEN-TYPE NEWTON-LIKE METHOD IN BANACH SPACE WITH DYNAMICS 249
500
450
400
350
300
250
200
150
100
100 200 300 400 500
50
(a)
500
450
400
350
300
250
200
150
100
100 200 300 400 500
50
(b)
500
450
400
350
300
250
200
150
100
100 200 300 400 500
50
(c)
Fig. 1. Basin of attraction for ex - 1 by Newton method (a), Jarratt method (b) and proposed method (c).
500
400
300
200
100
600
500400300200100 600700700
(a)
450
400
350
300
250
200
150
100
50
400300200100
(b)
450
400
350
300
250
200
150
100
50
400300200100
(c)
Fig. 2. Basin of attraction for f2 = z3 - 1 by Newton 2nd order method (a), Jarratt 4th order method (b), proposed
4th order method (c).
of attraction for 2nd order Newton method (2), 4th order Jarratt method (14) and 4th order proposed
frozen-type Newton-like method (3) for finding complex roots of above mentioned function in Fig. 1.
1. The basins of attraction for all the iterative methods contains fractal Julia set and basin of
Newton method looks almost similar to that of proposed method.
2. The Fatou set with basins of attraction of the Jarratt method is larger in comparison to the
other methods shown in blue color.
3. Again the Fatou set with bigger orbits of proposed method in comparison to the other is
showing the faster convergence of the proposed method to the roots.
4. Julia set with blue color shows the chaotic behavior and instability in the case of Newton
method and proposed frozen-type Newton-like method.
8.2. For Example 2. We have also considered Example 2 for the illustrations of the dynamics of
the iterative methods under the same previous conditions. We have plotted the fractal patterns graph
of the Example 2 (F (z) = z3 - 1) for the different iterative methods with a fixed different color to
each root of the basins of attraction.
Following points may be concluded by the study of the basins of attraction for 2nd order Newton
method (2), 4th order Jarratt method (14) and 4th order proposed frozen-type Newton-like method (3)
in Fig. 2.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
250 M. K. SINGH, A. K. SINGH
500
450
400
350
300
250
200
150
100
100 200 300 400 500
50
(a)
500
450
400
350
300
250
200
150
100
100 200 300 400 500
50
(b)
500
450
400
350
300
250
200
150
100
100 200 300 400 500
50
(c)
500
450
400
350
300
250
200
150
100
100 200 300 400 500
50
(d)
500
450
400
350
300
250
200
150
100
100 200 300 400 500
50
(e)
Fig. 3. Dynamics of the proposed method with 03 (a), 06 (b), 09 (c), 15 (d), and 21 (e) iterations for f2 = z3 - 1.
1. We can see the extraneous fixed points for the Newton-like methods in the basins of attraction
plot of Example 2 (z3 - 1) (see Fig. 2). These fixed points are repelling (the derivative at these
points has its magnitude > 1). Clearly there is no extraneous fixed point for Newton method. Again
there are 6 extraneous fixed points for Jarratt 4th order method and 27 extraneous fixed points for
proposed 4th order frozen-type Newton-like method.
2. The basins in the 4th order proposed frozen method contains a lot of black area with black
points which shows some defects of the method.
8.3. Effect of stop criterion on the dynamics of the proposed method. To study the effect of the
stopping criterion such that the tolerance | F (zn)| < 5 \times 10 - 2 and maximum number of iterations,
we have potted the fractal patterns graph for the proposed method with a fixed value of tolerance,
e.g., | F (zn)| < 5 \times 10 - 2 and variable value of iterations in the Fig. 3. We can observe from this
figure the following.
1. In the proposed method with only three iterations, very few number of starting points are
fulfilling the tolerance and hence the Fatou set with the basin having the colored region is very small
and the Juia set with non converging area having the black region is very large (Fig. 3(a)).
2. As the number of iterations increases Fatou set with the basin get increases and the Juia set
with non converging area decreases.
3. It happen because of the fact that with increasing number of iterations large number of starting
points pass the stopping criterion and become the part of the basin.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
THE STUDY OF FROZEN-TYPE NEWTON-LIKE METHOD IN BANACH SPACE WITH DYNAMICS 251
4. Thus we conclude that the tolerance - | F (zn)| < 5\times 10 - 2 and number of iterations both are
affecting the dynamics of the proposed method.
9. Compliance with ethical standards. Author M. K. Singh declares that he has no conflict
of interest. This article does not contain any studies with animals performed by any of the authors.
This article does not contain any studies with human participants or animals performed by any of the
authors.
10. Conclusion. We have discussed a three-steps 4th order frozen-type Newton-like method
for solving nonlinear equation in Banach space. We have performed the local and semilocal conver-
gence analysis for the method. Local convergence analysis demand the fourth-order differentiability
while the semilocal convergence analysis need only the second-order derivative using the recurrence
relations technique under the same conditions of Kantorovich theorem for Newton method. We have
studied about the extraneous fixed points and they are repulsive. Theoretical results are checked by
the numerical examples and numerical results are examined with the basins for some selected exam-
ples. Theoretical and numerical results show about the faster convergence and ease of not calculating
the inverse of the Jacobian of the proposed method at each step, but dynamical analysis divulge some
internal hidden defects of frozen-type Newton-like method.
References
1. S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical point of view, Sci.
Ser. A, 10, 3 – 35 (2004).
2. I. K. Argyros, Convergence and applications of Newton-type iterations, Springer, Berlin (2008).
3. I. K. Argyros, Computational theory of iterative methods, Stud. Comput. Math., 15 (2007).
4. I. K. Argyros, A. A. Magrenan, Iterative methods and their dynamics with applications: a contemporary study, CRC
Press, Taylor and Francis, Boca Raton, Florida (2017).
5. B. B. Mandelbrot, The fractal geometry of nature, Freeman, San Francisco (1983).
6. C. Chun, B. Neta, P. Stanica, Third-order family of methods in Banach spaces, Comput. and Math. Appl., 234, № 61,
1665 – 1675 (2011).
7. J. Chen, I. K. Argyros, R. P. Agarwal, Majorizing functions and two-point Newton-type methods, J. Comput. and
Appl. Math., 234, № 5, 1473 – 1484 (2010).
8. J. A. Ezquerro, M. A. Hernndez, M. A. Salanova, A Newton-like method for solving some boundary value problems,
J. Numer. Funct. Anal. and Optim., 23, № 7-8, 791 – 805 (2002).
9. P. Jarratt, Some fourth order multipoint iterative methods for solving equations, Math. Comp., 20, 434 – 437 (1966).
10. P. K. Parida, D. K. Gupta, Recurrence relations for a Newton-like method in Banach spaces, J. Comput. and Appl.
Math., 206, 873 – 887 (2007).
11. L. B. Rall, Computational solution of nonlinear operator equations, R. E. Krieger, New York (1979).
12. M. A. H. Veron, E. Martinez, On the semilocal convergence of a three steps Newton-type iterative process under
mild convergence conditions, Numer. Algorithms, 70, № 2, 377 – 392 (2015).
13. M. K. Singh, A. K. Singh, Variant of Newton’s method using Simpson’s 3/8th rule, Int. J. Appl. Comput., 6 (2020).
14. M. K. Singh, A. K. Singh, The optimal order Newton-like methods with dynamics, Mathematics, 9, № 5 (2021);
https://doi.org/10.3390/math9050527.
15. M. K. Singh, A six-order variant of Newton’s method for solving non linear equations, Comput. Methods Sci. and
Technol., 15, № 2, 185 – 193 (2009).
16. L. V. Kantorovich, G. P. Akilov, Funtional analysis, Pergamon Press, Oxford (1982).
17. Kalyanasundaram Madhu, Semilocal convergence of sixth order method by using recurrence relations in Banach
spaces, Appl. Math. E-Notes, 18, 197 – 208 (2018).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
252 M. K. SINGH, A. K. SINGH
18. J. Ortega, W. Rheinholdt, Iterative solution of nonlinear equations in several variables, Acad. Press, New York
(1970).
19. A. M. Ostrowski, Solutions of equations and systems of equations, Acad. Press, New York, London (1966).
20. M. Scott, B. Neta, C. Chun, Basin attractors for various methods, Appl. Math. and Comput., 218, № 2, 2584 – 2599
(2011).
21. J. F. Traub, Iterative methods for the solution of equations, Prentice-Hall, Clifford, NJ (1964).
22. K. Wang, J. Kou, C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algorithms,
57, 441 – 456 (2011).
23. Q. Wu, Y. Zhao, Third-order convergence theorem by using majorizing functions for a modified Newton’s method in
Banach spaces, Appl. Math. and Comput., 175, 1515 – 1524 (2006).
24. E. R. Vrscay, W. J. Gilbert, Extraneous fixed points, Basin boundaries and chaotic dynamics for Schröder and König
rational iteration functions, Numer. Math., 52, № 1, 1 – 16 (1987).
Received 01.06.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
|
| id | umjimathkievua-article-6764 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:30:02Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a7/1e556cbb279f8789809c687f9592caa7.pdf |
| spelling | umjimathkievua-article-67642025-03-31T08:45:58Z Study of frozen Newton-like method in a Banach space with dynamics Study of frozen Newton-like method in a Banach space with dynamics Study of frozen Newton-like method in a Banach space with dynamics Singh, M. K. Singh, A. K. Singh, M. K. Singh, A. K. Singh, Manoj Kumar Banach space Newton's method Semilocal convergence Recurrence relations Order of convergence Eciency index UDC 519.6 The main objective of this work is investigation of positives and negatives of the three steps iterative frozen-type Newtonlike method for solving nonlinear equations in a Banach space. We perform a local convergence analysis by Taylor’s expansion and semilocal convergence by recurrence relations technique under the conditions of Kantorovich theorem for the Newton’s method. The convergence results are examined by comparing the proposed method with the Newton’s method and the fourth order Jarratt’s method using some test functions. We discuss the corresponding conjugacy maps for quadratic polynomials along with the extraneous fixed points. Additionally, the theoretical and numerical results are examined byusing the dynamical analysis of a selected test function. It not only confirms the theoretical and numerical results, but also reveals some drawbacks of the frozen Newton-like method. УДК 519.6Вивчення методу Ньютона замороженого типу у банаховому просторi з динамiкоюМета цiєї роботи — вивчення плюсiв та мiнусiв тришагового iтерацiйного методу Ньютона замороженого типу для розв’язання нелiнiйних рiвнянь в банаховому просторi. Зроблено аналiз локальної збiжностi за допомогою рядiв Тейлора та напiвлокальної збiжностi за допомогою рекурентних спiввiдношень за умов теореми Канторовича для методу Ньютона. Отриманi результати збiжностi перевiрено шляхом порiвняння запропонованого методу з методом Ньютона та методом Джарратта четвертого порядку з використанням деяких тестових функцiй. Обговорено вiдповiднi спряженi вiдображення для квадратичних полiномiв, а також додатковi нерухомi точки. Крiм того, отриманi теоретичнi та числовi результати перевiрено за допомогою методiв динамiчного аналiзу певної тестової функцiї.Тим самим не тiльки пiдтверджено теоретичнi та числовi результати, але й виявлено деякi недолiки запропонованого методу Ньютона замороженого типу. Institute of Mathematics, NAS of Ukraine 2022-02-21 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6764 10.37863/umzh.v74i2.6764 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 2 (2022); 233 - 252 Український математичний журнал; Том 74 № 2 (2022); 233 - 252 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6764/9194 Copyright (c) 2022 A. K. Singh, M. K. Singh |
| spellingShingle | Singh, M. K. Singh, A. K. Singh, M. K. Singh, A. K. Singh, Manoj Kumar Study of frozen Newton-like method in a Banach space with dynamics |
| title | Study of frozen Newton-like method in a Banach space with dynamics |
| title_alt | Study of frozen Newton-like method in a Banach space with dynamics Study of frozen Newton-like method in a Banach space with dynamics |
| title_full | Study of frozen Newton-like method in a Banach space with dynamics |
| title_fullStr | Study of frozen Newton-like method in a Banach space with dynamics |
| title_full_unstemmed | Study of frozen Newton-like method in a Banach space with dynamics |
| title_short | Study of frozen Newton-like method in a Banach space with dynamics |
| title_sort | study of frozen newton-like method in a banach space with dynamics |
| topic_facet | Banach space Newton's method Semilocal convergence Recurrence relations Order of convergence Eciency index |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6764 |
| work_keys_str_mv | AT singhmk studyoffrozennewtonlikemethodinabanachspacewithdynamics AT singhak studyoffrozennewtonlikemethodinabanachspacewithdynamics AT singhmk studyoffrozennewtonlikemethodinabanachspacewithdynamics AT singhak studyoffrozennewtonlikemethodinabanachspacewithdynamics AT singhmanojkumar studyoffrozennewtonlikemethodinabanachspacewithdynamics |