Open AccessArticle

Convergence of a Family of Methods with Symmetric, Antisymmetric Parameters and Weight Functions

Ramandeep Behl

¹ and

Ioannis K. Argyros

^2,*

Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

Author to whom correspondence should be addressed.

Symmetry 2024, 16(9), 1179; https://doi.org/10.3390/sym16091179

Submission received: 31 July 2024 / Revised: 28 August 2024 / Accepted: 2 September 2024 / Published: 9 September 2024

(This article belongs to the Section Mathematics)

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Abstract

Many problems in scientific research are reduced to a nonlinear equation by mathematical means of modeling. The solutions of such equations are found mostly iteratively. Then, the convergence order is routinely shown using Taylor formulas, which in turn make sufficient assumptions about derivatives which are not present in the iterative method at hand. This technique restricts the usage of the method which may converge even if these assumptions, which are not also necessary, hold. The utilization of these methods can be extended under less restrictive conditions. This new paper contributes in this direction, since the convergence is established by assumptions restricted exclusively on the functions present on the method. Although the technique is demonstrated on a two-step Traub-type method with usually symmetric parameters and weight functions, due to its generality it can be extended to other methods defined on the real line or more abstract spaces. Numerical experimentation complement and further validate the theory.

Keywords:

semi-local convergence; order of convergence; iterative solver; local convergence

MSC:

65G99; 65H10

1. Introduction

A nonlinear equation or system of nonlinear equations is commonly used to solve a wide range of applied science problems from several domains, including economics, engineering, applied mathematics, health science, physics, and chemistry. An example of such a system is given below:

H (x) = 0,

(1)

where

H : D \subset S \to R, S = R or S = C

[1,2,3,4,5]. The exact solution of Equation (1) is almost nonexistent. Therefore, we have to obtain the solutions to such equations by relying on iterative methods.

A popular technique for finding approximations to the roots of a nonlinear equation is Newton’s method, which has a quadratic order of convergence for simple roots. Newton’s method considers an initial point; the method uses the function’s value and its derivative to iteratively refine the approximation. This process is repeated until we reach the desired level of accuracy. Newton’s method is renowned for its rapid convergence, especially when the initial point is close to the required root. With the advancement of computer algebra, many researchers have proposed modified and extended versions of Newton’s method.

Specifically, we define the following two-step method for each

θ = 0, 1, 2, 3, \dots

\begin{matrix} y_{θ} & = x_{θ} - α \frac{1}{H_{θ}^{'}} H_{θ}, \\ A_{θ} & = β H_{θ} + γ H_{y_{θ}}, \\ C_{θ} & = λ H_{θ}^{2} + μ H_{θ} H_{y_{θ}} + ξ H_{y_{θ}}^{2}, \\ x_{θ + 1} & = x_{θ} - \frac{1}{A_{θ}} \frac{1}{B_{θ}} C_{θ} \frac{1}{H_{θ}^{'}} H_{θ}, \end{matrix}

(2)

where

α \neq 0, β \neq 0, δ \neq 0, λ, μ

and

ξ

are parameters,

A_{θ}, B_{θ}

are non-vanishing functions for each point in their domain, and

C_{θ}

is any function. In addition, we assume that

H_{θ} = H (x_{θ}), H_{θ}^{'} = H_{θ}^{'}

and

H_{y_{θ}} = H_{y_{θ}}

in the whole study.

There exist specializations of the method (2). We have listed some of them.

Newton’s method (order two) [1,6,7,8]

Set

α = 1

C_{θ} = A_{θ} B_{θ}

in (2) to obtain

x_{θ + 1} = x_{θ} - \frac{1}{H_{θ}^{'}} H_{θ} .

(3)

Sharma’s Method (order three) [9]

Set

α = 1

A_{θ} = H_{θ} - H_{y_{θ}}

, i.e.,

β = 1, γ = - 1

, any

B_{θ}

and

C_{θ} = B_{θ} H_{θ}

to obtain

\begin{matrix} y_{θ} & = x_{θ} - \frac{1}{H_{θ}^{'}} H_{θ}, \\ x_{θ + 1} & = x_{θ} - \frac{H_{θ}^{2}}{(H_{θ} - H_{y_{θ}}) H_{θ}^{'}} . \end{matrix}

(4)

Traub–Kou Method (order three) [9,10,11,12,13]

Set

α = 1

A_{θ} = (τ^{2} - τ + 1) H_{θ} - H_{y_{θ}}

, i.e.,

β = (τ^{2} - τ + 1), τ = 1, - 1, - \frac{1}{2}

, any

B_{θ}

and

C_{θ} = τ^{2} B_{θ} H_{θ}

to obtain

\begin{matrix} y_{θ} & = x_{θ} - \frac{1}{H_{θ}^{'}} H_{θ} \\ x_{θ + 1} & = x_{θ} - \frac{τ^{2} H_{θ}^{2}}{((τ^{2} - τ + 1) H_{θ} - H_{y_{θ}}) H_{θ}^{'}} . \end{matrix}

(5)

Traub–Potra–Pták Method (order three) [14,15,16]

Take

α = 1

A_{θ} = H_{θ}

, i.e.,

β = 1, γ = 0

, any

B_{θ}

and

C_{θ} = B_{θ} (H_{θ} - H_{y_{θ}})

to obtain

\begin{matrix} y_{θ} & = x_{θ} - \frac{1}{H_{θ}^{'}} H_{θ}, \\ x_{θ + 1} & = x_{θ} - \frac{H_{θ} + H_{y_{θ}}}{H_{θ}^{'}} . \end{matrix}

(6)

A New Fourth Order Method

Set

α = β = δ = γ = μ = ξ = 1

γ = - 1, and ϵ = λ = - 3

, to obtain

\begin{matrix} y_{θ} & = x_{θ} - \frac{1}{H_{θ}^{'}} H_{θ}, \\ x_{θ + 1} & = x_{θ} - \frac{H_{θ}^{2} - 3 H_{θ} H_{y_{θ}} + H_{y_{θ}}^{2}}{(H_{θ} - H_{y_{θ}}) (H_{θ} - 3 H_{y_{θ}})} \frac{H_{θ}}{H_{θ}^{'}} . \end{matrix}

(7)

The fourth-order of this method is shown in Section 4.

Taylor expansion approaches are mostly used to find the convergence order. Generally, the proofs require high-order derivatives.

However, there are other problems under the Taylor series approach.

Motivation

(P₁): We consider an academic but motivational example that contradicts the above factors. We choose $D = [- 1.2, 1.2]$ and the function H on the interval $D$ by $H (t) = σ_{1} t^{2} l o g t + σ_{2} t^{5} + σ_{3} t^{4}$ if $t \neq 0$ , and $H (t) = 0$ , if $t = 0$ , where $σ_{1} \neq 0, σ_{2}, σ_{3}$ are parameters satisfying $σ_{1} + σ_{2} + σ_{3} = 0$ . It is straightforward to see that $t_{*} = 1$ solves the equation $H (t) = 0$ . However, the third derivative of function $H^{(3)}$ is not continuous at $t = 0 \in D$ . However, the results utilizing Taylor series which require the continuity of $H^{(3)}$ or even higher cannot guarantee convergence to it.
(P₂): There are no prior computable error estimates for $| x_{*} - x_{θ} |$ . Therefore, the number of iterations required to achieve a specified error tolerance cannot be determined in advance.
(P₃): No computable neighborhood of $x_{*}$ does not contain another solution to (1).
(P₄): The choice of $x_{0}$ to assure convergence of these method to $x_{*}$ is very difficult or impossible, since the radius of convergence is not usually determined.
(P₅): The convergence of iterative methods is established only for the real line.
(P₆): For the majority of these techniques, the challenging semi-local examination of convergence is not mentioned in the earlier studies.

Hence, the problem

(P_{1})

–

(P_{6})

constitute the motivation for writing the paper. These issues are addressed positively in this study. In particular, we have, respectively:

Novelty

$(P_{1}^{'})$: The local convergence is assured using on the derivatives on the method, i.e., $H^{'}$
$(P_{2}^{'})$: A prior error estimates those which are computable, and are provided.
Thus, the number of iterations required to achieve the tolerance is known in advance.
$(P_{3}^{'})$: A computable neighbourhood of $x_{*}$ is given that contains no other solution.
$(P_{4}^{'})$: The radius of convergence can be computed.
$(P_{5}^{'})$: The convergence is assured on S.
$(P_{6}^{'})$: Real majorizing sequences are utilised to establish the convergence [1,2,3]. The derivative $H^{'}$ is controlled using generalised continuity [1].

The approach is applied to the method (2). However, it can be used on other methods not necessarily defined in S, but in more general spaces such as Hilbert, Euclidean or Banach, along the same lines [2,3,4,5,6,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. Therefore, we can expand the usages of these methods for solving Equation (1) in cases not covered before. That is,

(P_{1}^{'})

–

(P_{6}^{'})

constitute the novelty of the paper.

The rest of the paper is organized as follows: Local and semi-local analyses are presented in Section 2 and Section 3, respectively. The convergence order of the method is discussed in Section 4. Examples are provided in Section 5, and conclusions are drawn in Section 6. It is also worth noting that (2) unifies the study of methods. Moreover, their convergence study is carried out under the same conditions. Consequently, a direct comparison between these methods becomes possible.

2. Convergence 1: Local Analysis

Some real functions defined on interval

T = [0, \infty]

play a role in analysis. Suppose:

(E₁): There is a function $Φ_{0} : T \to T$ that is continuous and non-decreasing (CD), such that the function $Φ_{0} (t) - 1$ has zeros in the interval $(0, + \infty)$ . Let the smallest such zero be denoted as $ρ_{0}$ . Define $T_{0} = [0, ρ_{0}]$ .
(E₂): There exists a CD function $Φ : T_{0} \to T$ such that for $h_{1} : T_{0} \to T$ , defined by

$h_{1} (t) = \frac{\int_{0}^{1} Φ ((1 - κ) t) d κ + | 1 - α | (1 + \int_{0}^{1} Φ_{0} (κ t) d κ)}{1 - Φ_{0} (t)},$

The function $h_{1} (t) - 1$ has a $S Z$ , which is denoted by $r_{1}$ .
(E₃): Functions $p : T_{0} \to T$ , $q : T_{0} \to T$ are given as:

$p (t) = \frac{1}{| β |} [| β | \int_{0}^{1} Φ_{0} (κ t) d κ + | γ | (1 + \int_{0}^{1} Φ_{0} (κ h_{1} (t) t)) d κ) h_{1} (t)], β \neq 0,$

and

$q (t) = \frac{1}{| δ |} [| δ | \int_{0}^{1} Φ_{0} (κ t) d κ + | ϵ | (1 + \int_{0}^{1} Φ_{0} (κ h_{1} (t) t) d κ) h_{1} (t)], δ \neq 0,$

are such that functions $p (t) - 1$ and $q (t) - 1$ have $(S Z)$ in interval $(0, ρ_{0})$ , which are denoted by $ρ_{p}$ and $ρ_{q}$ , respectively.
Let $ρ = min [ρ_{p}, ρ_{q}]$ and $T_{1} = [0, ρ]$ .
Let $δ_{1} = β δ - λ$ , $δ_{2} = β ϵ + γ - μ$ , and $δ_{3} = γ ϵ - ξ$ . Define the functions $g : T_{1} \to T$ and $h_{2} = T_{1} \to T$ by

$\begin{matrix} g (t) = & | δ_{1} | {(1 + \int_{0}^{1} Φ_{0} (κ t) d κ)}^{2} + | δ_{2} | (1 + \int_{0}^{1} Φ_{0} (κ t) d κ) (1 + \int_{0}^{1} Φ_{0} (κ h_{1} (t) t) d κ) h_{1} (t) \\ + | δ_{3} | {(1 + \int_{0}^{1} Φ_{0} (κ h_{1} (t) t) d κ)}^{2} h_{1}^{2} (t), \\ h_{2} (t) = & \frac{\int_{0}^{1} Φ ((1 - κ) t) d κ}{1 - Φ_{0} (t)} + \frac{g (t) (1 + \int_{0}^{1} Φ_{0} (κ t) d κ)}{| β | | δ | (1 - p (t)) (1 - q (t)) (1 - Φ_{0} (t))} . \end{matrix}$
(E₄): The function $h_{2} (t) - 1$ has a $S Z$ , which is denoted by $r_{2}$ . Set

$r = min {r_{j}}, j = 1, 2, and T_{2} = [0, r] .$

(8)

Then, we have for each $t \in T_{2}$

$0 \leq Φ_{0} (t) < 1,$

(9)

$0 \leq p (t) < 1,$

(10)

$0 \leq q (t) < 1,$

(11)

$0 \leq g (t)$

(12)

and

$0 \leq h_{j} (t) < 1 .$

(13)

The functions $Φ_{0}$ and $Φ$ relate to the derivative $H^{'}$ on the method (2).
(E₅): There exists a non-zero number L, $x_{*} \in D$ solving (1), and for each $u \in D$ , we obtain

$∥L^{- 1} (H^{'} (u) - L)∥ \leq Φ_{0} (∥ u - x_{*} ∥) .$

Let $D_{0} = B (x_{*}, ρ_{0}) \cap T$ .
The notation $B (x, \bar{ρ})$ is used to denote an open real interval with center x and of radius $\bar{ρ}$ . Moreover, the symbol $B (x, \bar{ρ})$ is the closure of interval $B (x, \bar{ρ})$ .
(E₆): For each $u_{1}, u_{2} \in D_{0}$ , we have

$∥L^{- 1} (H^{'} (u_{2}) - H^{'} (u_{1}))∥ \leq Φ (∥ u_{2} - u_{1} ∥)$

and
(E₇): $B [x_{*}, r] \subset D$ .

Remark 1.

(i): The number r given by the formula (8) is shown in Theorem 1 to be a convergence radius for the method (2).
(ii): $L = H^{'} (x_{*})$ is the usual but not the most appropriate (necessarily) selection. However, in this case, $x_{*}$ is a simple solution of Equation (1). In Theorem 1, we do not assume $x_{*}$ is a simple solution. So, the method (2) is also useful to find solutions of multiplicity greater than the one provided that $L \neq H^{'} (x_{*})$ . The preceding notation, together with conditions $(E_{1})$ – $(E_{7})$ , is used in the local analysis of convergence for method (2). Related work on other methods for Banach space valued operators can be found in [1,7].

Theorem 1.

Suppose that the conditions

(E_{1})

–

(E_{7})

hold. Then, if

x_{0} \in B (x_{*}, r) - (x_{*})

, the sequence

x_{θ}

is well defined in the interval

B (x_{*}, ϵ)

, such that the following items hold

x_{θ} \subseteq B (x_{*}, r),

(14)

∥ ρ_{y_{θ}} ∥ \leq h_{1} (∥ ρ_{x_{θ}} ∥) ∥ ρ_{x_{θ}} ∥ \leq ∥ ρ_{x_{θ}} ∥ < r,

(15)

∥ ρ_{x_{θ + 1}} ∥ \leq h_{2} (∥ ρ_{x_{θ}} ∥) ∥ ρ_{x_{θ}} ∥ \leq ∥ ρ_{x_{θ}} ∥,

(16)

where

x_{θ} - x_{*} = ρ_{x_{θ}}

y_{θ} - x_{*} = ρ_{y_{θ}}

(for

θ = 0, 1, 2, \dots)

and

x_{*} = lim_{x \to \infty} x_{θ}

Proof.

The items (14)–(16) are shown using induction on n. By hypothesis

x_{0} \in B (x_{*}, r) - (x_{*})

, the item (14) clearly holds if

θ = 0

, since

B (x_{*}, r) - {x_{*}} \subset B (x_{*}, r)

. Pick

u \in B (x_{*}, r)

. It follows (8), (9) and

(E_{5})

in turn that

∥L^{- 1} (H^{'} (u) - L)∥ \leq Φ_{0} (∥ u - x_{*} ∥) \leq Φ_{0} (r) < 1 .

So,

H^{'} (u) \neq 0

by perturbation lemma due to Banach standard and

∥\frac{1}{H^{'} (u)} L∥ \leq \frac{1}{1 - Φ_{0} (∥ u - x_{*} ∥)} .

(17)

Then, notice that iterate

y_{0}

is defined by first substep of method (2) if

θ = 0

, since

H_{x_{0}}^{'} \neq 0

and

\begin{matrix} y_{0} - x_{*} = & x_{0} - x_{*} - \frac{1}{H_{x_{0}}^{'}} H_{x_{0}} + (1 - α) H_{x_{0}}^{- 1} H_{x_{0}} \\ = & \int_{0}^{1} \frac{1}{H_{x_{0}}^{'}} (H^{'} (x_{*} + κ (ρ_{x_{0}})) - H_{x_{0}}^{'}) d κ (ρ_{x_{0}}) + (1 - α) H_{x_{0}}^{'} H_{x_{0}} . \end{matrix}

(18)

The application of condition

(E_{6})

, (8), (17) (for

u = x_{0}

), (13) (for j = 1), and (18), we provide

\begin{matrix} ∥ ρ_{y_{0}} ∥ = & \frac{[\int_{0}^{1} Φ ((1 - κ) ∥ ρ_{x_{0}} ∥) d κ + | 1 - α | (1 + \int_{0}^{1} Φ_{0} (κ ∥ ρ_{x_{0}} ∥) d κ)] ∥ ρ_{x_{0}} ∥}{1 - Φ_{0} (∥ ρ_{x_{0}} ∥)} \\ = & h_{1} (∥ ρ_{x_{0}} ∥) ∥ ρ_{x_{0}} ∥ \leq ∥ ρ_{x_{0}} ∥ < r . \end{matrix}

(19)

Thus, item (15) holds if

θ = 0

, and the iterate

δ_{0} \in B (x_{0}, r)

Then, we show

A_{0} \neq 0

and

B_{0} \neq 0

. We obtain in turn if

x_{0} \neq x_{*}

∥{(β L (ρ_{x_{0}}))}^{- 1} (β (H_{x_{0}} - H_{x_{*}} - L (ρ_{x_{0}}))) + γ H_{y_{0}}∥,

where

H (x_{0}) = H_{x_{0}}

and we use

H_{x_{0}} = H_{x_{0}} - H_{x_{*}} = \int_{0}^{1} H^{'} (x_{*} + κ (ρ_{x_{0}})) d κ (ρ_{x_{0}}) .

Thus, we obtain

\begin{matrix} ∥ L^{- 1} H_{x_{0}} ∥ & = ∥L^{- 1} [\int_{0} (H^{'} (x_{*} + κ (ρ_{x_{0}})) - L + L) d κ] (ρ_{x_{0}})∥ \\ \leq (1 + \int_{0}^{1} Φ_{0} (κ ∥ x_{0} - x_{*}) d κ) ∥ ρ_{x_{0}} ∥ \\ \leq \frac{1}{| β | ∥ x_{0} - x_{*} ∥} [| β | \int_{0}^{1} Φ_{0} (κ ∥ x_{0} - x_{*}) d κ ∥ ρ_{x_{0}} ∥ \\ + | γ | (1 + \int_{0}^{1} Φ_{0} (κ ∥ y_{0} - x_{*}) d κ) ∥ ρ_{y_{0}} ∥)] \\ \leq \frac{1}{| β |} [| β | \int_{0}^{1} Φ_{0} (κ ∥ x_{0} - x_{*}) d κ + | γ | (1 + \int_{0}^{1} Φ_{0} (κ ∥ y_{0} - x_{*}) d κ) h_{1} (∥ ρ_{x_{0}} ∥)] \\ \leq p_{0} < 1, \end{matrix}

so,

∥\frac{1}{A_{0}} L∥ = \frac{1}{| β | (1 - p_{0} (∥ ρ_{x_{0}} ∥))} .

(20)

Similarly, for

q_{0} = \frac{1}{| δ |} [| δ | \int_{0}^{1} Φ_{0} (κ ∥ ρ_{x_{0}} ∥) d κ + | ϵ | (1 + \int_{0}^{1} Φ_{0} (κ ∥ ρ_{y_{0}} ∥) d κ) (h_{1} ∥ ρ_{x_{0}} ∥)],

we have

∥\frac{1}{B_{0}} L∥ \leq \frac{1}{| δ | (1 - q_{0} (∥ ρ_{x_{0}} ∥))} .

(21)

It follows by (20) and (21) that the iterate

x_{*}

is well-defined by the second substep of method (2) if

θ = 0

, and

\begin{matrix} x_{1} - x_{*} & = x_{0} - x_{*} - \frac{1}{H_{x_{0}}^{'}} H_{x_{0}} + (I - \frac{1}{A_{0}} \frac{1}{B_{0}} C_{0}) \frac{1}{H_{x_{0}}^{'}} H_{x_{0}} \\ = x_{0} - x_{*} - \frac{1}{H_{x_{0}}^{'}} H_{x_{0}} + \frac{1}{A_{0}} \frac{1}{B_{0}} (A_{0} B_{0} - C_{0}) \frac{1}{H_{x_{0}}^{'}} H_{x_{0}} \end{matrix}

(22)

We need an estimate on the norm of

∥L^{- 2} (A_{0} B_{0} - C_{0})∥

. Notice that

\begin{matrix} A_{0} B_{0} - C_{0} = & (β H_{x_{0}} + γ H_{y_{0}}) (δ H_{x_{0}} + ϵ H_{y_{0}}) - (λ H_{x_{0}}^{2} + μ H_{x_{0}} H_{y_{0}} + ξ H_{y_{0}}^{2}) \\ = & (β δ - λ) H_{x_{0}}^{2} + (β ϵ + γ - μ) H_{x_{0}} H_{y_{0}} + (γ ϵ - ξ) H_{y_{0}}^{2} \\ = & δ_{1} H (x_{0}^{2}) + δ_{2} H_{x_{0}} H_{y_{0}} + δ_{3} H (y_{0}^{2}) \end{matrix}

leading to

\begin{matrix} ∥L^{- 2} (A_{0} B_{0} - C_{0})∥ & \leq [| δ_{1} | {(1 + \int_{0}^{1} Φ_{0} (κ ∥ ρ_{x_{0}} ∥) d κ)}^{2} \\ + | δ_{2} | (1 + \int_{0}^{1} Φ_{0} (κ ∥ ρ_{x_{0}} ∥) d κ) (1 + \int_{0}^{1} Φ_{0} (κ ∥ ρ_{y_{0}} ∥) d κ) \\ + | δ_{3} | {(1 + \int_{0}^{1} Φ_{0} (κ ∥ ρ_{y_{0}} ∥) d κ)}^{2} h_{1} (∥ ρ_{x_{0}} {∥)}^{2}] {∥ ρ_{x_{0}} ∥}^{2} \\ = g_{0} | | x_{0} - x_{*} {| |}^{2} \end{matrix}

(23)

By adopting (8), (13) (for j = 1), (20), (22), it follows

\begin{matrix} | | x_{1} - x_{*} | | & \leq [\frac{\int_{0}^{1} Φ (κ | | x_{0} - x_{*} | |) d κ}{1 - Φ_{0} (| | x_{0} - x_{*} | |)} \\ + \frac{g_{0} (1 + \int_{0}^{1} Φ_{0} (κ ∥ x_{0} - x_{*}) d κ) | | x_{0} - x_{*} {| |}^{2}}{| b | | δ | (1 - p_{0}) (1 - q_{0}) (1 - Φ_{0} (| | x_{0} - x_{*} | |)) | | x_{0} - x_{*} {| |}^{2}}] | | x_{0} - x_{*} | | \\ \leq h_{2} (∥ ρ_{x_{0}} ∥) ∥ ρ_{x_{0}} ∥ \leq ∥ ρ_{x_{0}} ∥ . \end{matrix}

(24)

Therefore, the item (16) holds if

θ = 0

, and the iterate

x_{1} \in B (x_{*}, r)

, i.e., (14) is true for

θ = 1

. The induction for items (14)–(16) if simply terminated if iterates

x_{k}, y_{k}, x_{k + 1}

are used for

x_{0}, y_{0}, x_{1}

, respectively in the above computations. Then, we obtain

∥ x_{k + 1} - x_{*} ∥ \leq δ_{*} ∥ x_{k + 1} - x_{*} ∥ \leq δ_{*}^{k + 1} ∥ x_{k + 1} - x_{*} ∥ \leq r

(25)

where

δ_{*} = h_{2} (∥ x_{k + 1} - x_{*} ∥) \in [0, 1]

, we conclude that iterate

x_{k + 1} \in B (x_{*}, r)

as well as

lim_{k \to \infty} x_{k} = x_{*}

. □

An interval is found that contains only

x_{*}

as solution.

Proposition 1.

Suppose there exist

ρ_{1} > 0

, such that condition

(E_{5})

holds in the interval

B (x_{*}, ρ_{1})

and

ρ_{2} \geq ρ_{1}

, such that

\int_{0}^{1} Φ_{0} (κ ρ_{2}) d κ < 1 .

(26)

Set

D_{1} = B [x_{*}, ρ_{2}] \cap D

. Then, the only solution of Equation (1) in the set

D_{1}

x_{*}

Proof.

Let

\bar{x} \in D_{1}

with

H (\bar{x}) = 0

. Set

L_{1} = \int_{0}^{1} H^{'} (x_{*} + κ (\bar{x} - x_{*})) d κ

. The application of condition

(E_{5})

on the interval

B (x_{*}, ρ_{1})

and (25) imply

\begin{matrix} ∥ L^{- 1} (L_{1} - L) ∥ & \leq \int_{0}^{1} Φ_{0} (κ ∥ \bar{x} - x_{*} ∥) d κ \\ \int_{0}^{1} Φ_{0} (κ ρ_{2}) d κ < 1 . \end{matrix}

(27)

So,

L_{1} \neq 0

. Then, from approximations

\bar{x} - x_{*} = L^{- 1} (H (\bar{x}) - H_{x_{*}}) = L^{- 1} (0) = 0,

it follows

\bar{x} = x_{*}

. □

Remark 2.

Note that if all the conditions of Theorem 1 are satisfied in Proposition 1, we can set

δ_{1} = r

3. Convergence 2: Semi-Local Analysis

The calculations are similar to those in the previous section, but the roles of the solution

x_{*}

and the functions

Φ_{0}

and

Φ

are now taken by the starting point

x_{0}

and the functions

Ψ_{0}

and

Ψ

, respectively.

Suppose:

(M₁): There exists a CD function $Ψ_{0} : T \to T$ so that $Ψ_{0} (t) - 1$ has a $S Z$ which is denoted by $s_{0}$ . Set $T_{3} = [0, s_{0})$ .
(M₂): There exists a CD function $Ψ : T_{3} \to T$ .
It is convenient to set $γ_{1} = α β δ - λ, γ_{2} = α β ϵ + α γ$ and $γ_{3} = α γ ϵ - ξ$ .
Define the sequences ${a_{θ}}, {b_{θ}}, {\hat{g_{θ}}}, {\hat{p_{θ}}}, {\hat{q_{θ}}}$ and ${c_{θ + 1}}$ for $a_{0} = 0$ , $b_{0} \geq 0$ and each $θ = 0, 1, 2, 3 . . .$ by

$\begin{matrix} d_{θ} & = \int_{0}^{1} γ ((1 - κ) (b_{θ} - a_{θ})) d κ + |1 - \frac{1}{α}| (1 + Ψ_{0} (a_{θ})), \\ \hat{g_{θ}} & = | γ_{1} | {(1 + Ψ_{0} (a_{θ}))}^{2} + | γ_{2} | (1 + Ψ_{0} (a_{θ})) d_{θ} + | γ_{3} | (d_{θ}^{2}), \\ \hat{p_{θ}} & = \frac{1}{| β |} [| β | (1 + \int_{0}^{1} Ψ_{0} (a_{θ} + κ (b_{θ} - a_{θ})) d κ) + | β + γ | d_{θ}], \\ \hat{q_{θ}} & = \frac{1}{| δ |} [| β | (1 + \int_{0}^{1} Ψ_{0} (a_{θ} + κ (b_{θ} - a_{θ})) d κ) + | δ + ϵ | d_{θ}], \\ a_{θ + 1} & = b_{θ} + \frac{\hat{h_{θ}}}{α β δ | (1 - \hat{p_{θ}} (1 - \hat{q_{θ}}))} \\ a n d \\ c_{θ + 1} & = (1 + \int_{0}^{1} Ψ_{0} (a_{θ} + κ (a_{θ + 1} - a_{θ})) d κ) (a_{θ + 1} - a_{θ}) + \frac{1}{| α |} (1 + Ψ_{0} (a_{θ})) (b_{θ} - a_{θ}) . \end{matrix}$

(28)

The following conditions assure the convergence of sequences $a_{θ}$ and $b_{θ}$ .
(M₃): There exists $s \in [0, s_{0}]$ , such that for each $θ = 0, 1, 2, . .,$

$Ψ_{0} (a_{θ}) < 1, \hat{p_{θ}} < 1, \hat{q_{θ}} < 1, and a_{θ} \leq s .$

This condition together with (28) imply $0 \leq a_{θ} \leq b_{θ} \leq a_{θ + 1} < s$ and there exists $a_{*} \in [0, s]$ such that $lim_{θ \to \infty} a_{θ} = lim_{θ \to \infty} b_{θ} = a_{*}$ . It is well known that $a_{*}$ is the unique least upper bound of the sequence $a_{θ}$ . As in the local analysis, the functions $Ψ_{0}$ and $Ψ$ are related to $H^{'}$ .
(M₄): There exists $x_{0} \in D$ and a non zero number L, such that for each $u \in D$

$∥ L^{- 1} (H^{'} (u) - L) ∥ \leq Ψ_{0} (∥ u - x_{0} ∥) .$

Set $D_{2} = D \cap B (x_{0}, s_{0})$ if we take $u = x_{0}$ in this condition and use the definition of

$∥L^{- 1} (H_{x_{0}}^{'} - L)∥ \leq Ψ_{0} (0) < 1 .$

Thus, $H_{x_{0}}^{'} \neq 0$ , and the iterate $y_{0}$ exists. So, we can take $b_{0} \geq | α | ∥\frac{1}{H_{x_{0}}^{'}} H_{x_{0}}∥$
(M₅): For each $u_{1}, u_{2} \in D$

$∥ L^{- 1} (H^{'} (u_{2}) - H^{'} (u_{1})) ∥ \leq Ψ (∥ u_{2} - u_{1} ∥)$

and
(M₆): $B [x_{0}, a_{*}] \subset D$ .

Remark 3.

A popular solution for

M = H_{x_{0}}^{'}

. (See also the explanations in Remark 1).

The semi-local analysis of convergence follows, as in the local case, in next set of results.

Theorem 2.

Suppose that the conditions

(M_{1})

–

(M_{6})

hold. Then, the sequence

x_{θ}

is well-defined and the following items hold

x_{k} \subseteq B (x_{0}, a_{*}),

(29)

∥ y_{k} - x_{k} ∥ \leq b_{k} - a_{k},

(30)

∥ x_{k + 1} - y_{k} ∥ \leq a_{k + 1} - b_{k},

(31)

and

x_{*} \in B [x_{0}, a_{*}]

exists as a solution of the equation

H (x) = 0

, such that

∥ x_{*} - x_{k} ∥ \leq a_{*} - a_{k} .

(32)

Proof.

Induction shall establish the items (29)–(31). Clearly, (29) holds if

k = 0

. The choice of

b_{0}

and the first substep of the method (2) imply

∥ y_{0} - x_{0} ∥ = ∥ α ∥ ∥\frac{1}{H_{x_{0}}^{'}} H_{x_{0}}∥ \leq b_{0} = b_{0} - x_{0} < a_{*} .

Thus, the item (30) holds if

k = 0

and the iterate

y_{0} \in B (x_{0}, a_{0})

. Now, we need the estimates that correspond to the local analysis

\begin{matrix} H (y_{k}) & = H (y_{k}) - H (x_{k}) - \frac{1}{α} H^{'} (x_{k}) (y_{k} - x_{k}) \\ = H (y_{k}) - H (x_{k}) - \frac{1}{α} H^{'} (x_{k}) (y_{k} - x_{k}) + (1 - \frac{1}{α} H^{'} (x_{k}) (y_{k} - x_{k})) \\ ∥ L^{- 1} H (x_{k}) ∥ & \leq \int_{0}^{1} v (1 - κ) ∥ y_{k} - x_{k} ∥ d κ ∥ y_{k} - x_{k} ∥ \\ + (1 - \frac{1}{α} (1 + Ψ_{0} ∥ x_{k} - x_{0} ∥)) ∥ y_{k} - x_{k} ∥ \\ \leq \int_{0}^{1} v ((1 - κ) ∥ b_{k} - a_{k} ∥) d κ ∥ y_{k} - x_{k} ∥ + (1 - \frac{1}{α} (1 + Ψ_{0} (a_{k}))) ∥ y_{k} - x_{k} ∥ \\ = d_{k} ∥ y_{k} - x_{k} ∥ \end{matrix}

and for

y_{k} \neq x_{k}

\begin{matrix} \frac{1}{∥ β L (x_{k} - y_{k})} [β (H (x_{k}) - H (y_{k}) - L (x_{k} - y_{k})) + (β + γ) H (y_{k}) ∥] \\ \leq \frac{1}{| β | ∥ y_{k} - x_{k} ∥} [| β | (1 + \int_{0}^{1} Ψ_{0} (a_{k} + κ (b_{k} - a_{k})) d κ) + | β + γ | d_{k}] ∥ y_{k} - x_{k} ∥ \\ = \hat{p_{k}} < 1, \\ ∥\frac{1}{A_{k}} L∥ & \leq \frac{1}{| β | (1 - \hat{p_{k}}) ∥ y_{k} - x_{k} ∥} \end{matrix}

and similarly,

∥\frac{1}{B_{k}} L∥ \leq \frac{1}{| δ | (1 - \hat{q_{k}}) ∥ y_{k} - x_{k} ∥} .

Moreover, by subtracting first, in the second substep we obtain

\begin{matrix} x_{k + 1} - y_{k} & = (α - \frac{1}{A_{k}} \frac{1}{B_{k}} C_{k}) \frac{1}{H^{'} (x_{k})} H (x_{k}) \\ = \frac{1}{A_{k}} \frac{1}{B_{k}} (α A_{k} B_{k} - C_{k}) - \frac{1}{α} (y_{k} - x_{k}) \\ = \frac{1}{α} \frac{1}{A_{k} B_{k}} (C_{k} - A_{k} B_{k}) (y_{k} - x_{k}) . \end{matrix}

(33)

The following estimates are also needed:

\begin{matrix} α A_{k} B_{k} - C_{k} & = (α β δ - λ) H {(x_{k})}^{2} + (α β ϵ + α γ) H (x_{k}) H (y_{k}) + (α γ ϵ - ξ) H {(y_{k})}^{2} \\ = γ_{1} H {(x_{k})}^{2} + γ_{2} H (x_{k}) H (y_{k}) + γ_{3} H {(y_{k})}^{2}, \end{matrix}

leading to

\begin{matrix} ∥ L^{- 2} (C_{k} - α A_{k} B_{k}) ∥ & \leq [| γ_{1} | {(1 + Ψ_{0} (a_{k}))}^{2} + | γ_{2} | (1 + Ψ_{0} (a_{k})) d_{k} + | γ_{3} | d^{2} m] {∥ y_{k} - x_{k} ∥}^{2} \\ = \hat{h_{k}} {∥ y_{k} - x_{k} ∥}^{2} . \end{matrix}

Hence, (32) gives

\begin{matrix} x_{k + 1} - y_{k} & \leq \frac{\hat{h_{k}} {∥ y_{k} - x_{k} ∥}^{3}}{α β δ | (1 - \hat{p_{k}} (1 - \hat{q_{k}})) {∥ y_{k} - x_{k} ∥}^{3}} \\ \leq \frac{\hat{h_{k}} (b_{k} - a_{k})}{| α β δ | (1 - \hat{p_{k}}) (1 - \hat{q_{k}})} = a_{k + 1} - b_{k} \end{matrix}

(34)

and

∥ x_{k + 1} - x_{0} ∥ \leq ∥ x_{k + 1} - y_{k} ∥ + ∥ y_{k} - x_{0} ∥ \leq a_{k + 1} - b_{k} + b_{k} - a_{0} = a_{k + 1} < a_{*} .

Furthermore, the first substep gives

H (x_{k + 1}) = H (x_{k + 1}) - H (x_{k}) - \frac{1}{α} H^{'} (x_{k}) (y_{k} - x_{k}) .

So, we have

\begin{matrix} ∥ L^{- 1} H (x_{k + 1}) ∥ & \leq (1 + \int_{0}^{1} Ψ_{0} (a_{k} + κ (a_{k + 1} - a_{k})) d κ) (a_{k + 1} - a_{k}) + \frac{1}{| α |} (1 + Ψ_{0} (a_{k})) (b_{k} - a_{k}) \\ = c_{k + 1} . \end{matrix}

(35)

Consequently, we obtain

\begin{matrix} | | y_{k + 1} - x_{k + 1} | | & \leq | α | ∥\frac{1}{H^{'} (x_{k + 1}) 1} L∥ ∥ L^{- 1} H (x_{k + 1}) ∥ \\ \leq | α | \frac{c_{k + 1}}{1 - Ψ_{0} (| | x_{k + 1} - x_{0} | |)} \\ \leq | α | \frac{c_{k + 1}}{1 - Ψ_{0} (| | a_{k + 1} | |)} \\ = b_{k + 1} - a_{k + 1} \end{matrix}

(36)

and

\begin{matrix} | | y_{k + 1} - x_{0} | | & \leq | | y_{k + 1} - x_{k + 1} | | + | | x_{k + 1} - x_{0} | | \\ \leq b_{k + 1} - a_{k + 1} + a_{k + 1} - a_{0} \\ = b_{k + 1} \leq a_{*} . \end{matrix}

(37)

The introduction for items (29)–(31) is terminated and all iterates

x_{k}, y_{k}

belong in

B (x_{0}, a_{*})

Therefore, the sequence

x_{k}

is completed, deemed convergent by conditions

(M_{3})

Consequently, there exists

x_{*} \in B (x_{0}, a_{*})

, such that

H_{x_{*}} = 0

. But,

lim_{k \to \infty} c_{k + 1} = 0

, so

H_{x_{*}} = 0

by (35), where we also used a continuation of H. Finally, by

∥ x_{k + i} - x_{k} ∥ \leq a_{k + i} - a_{k}

the item (32) follows, provided that

i \to + \infty

. □

The uniqueness of the solution of Equation (1) is established in the next set of results.

Proposition 2.

Suppose there exists a solution

z \in B (x_{0}, s_{1})

of equation for some

s_{1} > 0

, the condition

(M_{4})

holds the interval

B (x_{0}, s_{1})

and there exist

s_{2} \geq s_{1}

, such that

\int_{0}^{1} Ψ_{0} ((1 - κ) s_{1} + κ s_{2}) d κ < 1

(38)

Set

D_{3} = D \cap B [x_{0}, s_{2}]

. Then, the equation has only one solution in

D_{3}

Proof.

Let

z_{1} \in D_{3}

with

H (z_{1}) = 0

. Define

L_{2} = \int_{0}^{1} H^{'} (z + κ (z_{1} - z)) d κ

. By the hypothesis

(M_{4})

and expression (38); we obtain in turn

∥ L^{- 1} (L_{2} - L) ∥ \leq \int_{0}^{1} Ψ_{0} ((1 - κ) ∥ z - x_{0} ∥ + κ ∥ z_{1} - x_{0} ∥) d κ \leq \int_{0}^{1} Ψ_{0} ((1 - κ) s_{1} + κ s_{2}) d κ < 1,

L_{2} \neq 0

. Then, from approximation

z_{1} - z = L_{2}^{- 1} (0) = 0

, we conclude that

z_{1} = z

. □

Remark 4.

(i): The limit point $a_{*}$ can be exchanged by $s_{0}$ in condition $(M_{6})$ .
(ii): In the Proposition 2, assume $z = x_{*}$ and $s_{1} = a_{*}$ under all conditions $(M_{1})$ – $(M_{6})$ .

4. The Convergence Order of Method (2)

In this section, we show that the convergence order of the method (2) is four, using Taylor series expansions [2,3].

Theorem 3.

Assume that (1) has a simple solution and that the function H is sufficiently differentiable in a neighborhood around

x_{*}

. Then, the convergence order of the method (7) is four.

Proof.

Set

l_{θ} = x_{θ} - x_{*}

and

{\bar{l}}_{θ} = y_{θ} - x_{*}

and

δ_{k} = \frac{H^{k} (x_{*})}{k! H^{'} (x_{*})}

k = 2, 3, 4, \dots

. First, we expand

H_{θ}

about

x_{*}

using Taylor series to obtain in turn

H_{θ} = H^{'} (x_{*}) (l_{θ} + Δ_{2} l_{θ}^{2} + Δ_{3} l_{θ}^{3} + Δ_{4} l_{θ}^{4} + O (l_{θ}^{5})),

H_{θ}^{'} = H^{'} (x_{*}) (1 + 2 Δ_{2} l_{θ}^{2} + 3 Δ_{3} l_{θ}^{3} + 4 Δ_{4} l_{θ}^{4} + O (l_{θ}^{4})) .

Consequently, we have

\frac{H_{θ}}{H_{θ}^{'}} = l_{θ} - Δ_{2} l_{θ}^{2} + 2 (Δ_{3} - Δ_{2}^{2}) l_{θ}^{3} + (7 Δ_{2} Δ_{3} - 4 Δ_{2}^{3} - 3 Δ_{4}) l_{θ}^{4} + O (l_{θ}^{5}),

and

{\bar{l}}_{θ} = l_{θ} - \frac{H_{θ}}{H_{θ}^{'}} = Δ_{2} l_{θ}^{2} + 2 (Δ_{3} - Δ_{2}^{2}) l_{θ}^{3} + (4 Δ_{2}^{3} + 3 Δ_{4} - 7 Δ_{2} Δ_{3}) l_{θ}^{4} + O (l_{θ}^{5}) .

Similarly, we obtain

H_{y_{θ}} = H^{'} (x_{*}) ({\bar{l}}_{θ} + Δ_{2} {\bar{l}}_{θ}^{2} + Δ_{3} {\bar{l}}_{θ}^{3} + Δ_{4} {\bar{l}}_{θ}^{4} + O ({\bar{l}}_{θ}^{5})) .

Substituting these formulas in (7), we obtain

l_{θ + 1} = O (l_{θ}^{4}) .

Hence, this ends the proof. □

Remark 5.

Notice that in Theorem 3, the function H must be at least five times differentiable. Therefore, Theorem 3 is not applicable to solve the equation given in the Introduction using method (7).

5. Numerical Experiments

We performed a computational analysis to demonstrate the practical significance of our theoretical results. To this end, we selected four numerical problems to demonstrate our computational findings. These numerical examples are categorized into two groups based on convergence analysis: semi-local area convergence (SLAC) and local area convergence (LAC). The SLAC examples illustrate how the method behaves in a broader region around the initial guess, while the LAC examples focus on the convergence properties in a more restricted, localized area. This distinction helps us to better understand the method’s effectiveness in different ways.

Local area convergence

We investigated local area convergence (LAC) using the first problem as a case study. Detailed information about this example is provided in (1). The radii of convergence are presented in Table 1.

Semi-local area convergence

On the other hand, the remaining examples were selected to explore SLAC. The numerical results for semi-local convergence based on the well-known Planck’s radiation problem (2) are presented in Table 2. Table 3 provides the numerical results for the semi-local convergence of the continuous stirred tank reactor (CSTR), described in example (3), which is another well-known problem in applied science. Finally, the numerical results for semi-local convergence based on the blood rheology model (4) are shown in Table 4.

Furthermore, we also calculated the computational order of convergence (COC) for each example. The COC provides us a measure of how quickly the iterative method approaches the required solution. This can be determined using the following formulas:

η = \frac{ln \frac{∥ x_{κ + 1} - x_{*} ∥}{| x_{κ} - x_{*} ∥}}{ln \frac{∥ x_{κ} - x_{*} ∥}{∥ x_{κ - 1} - x_{*} ∥}}, for κ = 1, 2, \dots

(39)

or the approximate computational order of convergence

(A C O C)

[17,18] by:

η^{*} = \frac{ln \frac{∥ x_{κ + 1} - x_{κ} ∥}{∥ x_{κ} - x_{κ - 1} ∥}}{ln \frac{∥ x_{κ} - x_{κ - 1} ∥}{∥ x_{κ - 1} - x_{κ - 2} ∥}}, for κ = 2, 3, \dots

(40)

The termination criteria for the program are defined as follows: (i)

| x_{κ + 1} - x_{κ} | < ϵ

and (ii)

| J (x_{κ}) | < ϵ

, where

ϵ = 10^{- 300}

. All computations were carried out using Mathematica 11 with multi-precision arithmetic. The specifications of the computer used for programming are as follows:

Device Name: HP
Edition: Windows 10 Enterprise
Version: 22H2
Installed RAM: 8.00 GB (7.89 GB usable)
OS Build: 19045.2006
Processor: Intel(R) Core(TM) i7-4790 CPU @ 3.60 GHz
System type: 64-bit operating system, ×64-based processor

5.1. Examples for LAC

To illustrate the theoretical findings of local convergence, which are provided in Section 2, we select the Example (1).

Example 1.

Let

S = R

and

D = B (0, 1)

. Define the function H on

S

for

v \in S

H (v) = e^{v} - 1 .

(41)

Then, for

L = H^{'} (x_{*})

, where

x_{*} = 0

solves the equation

H (v) = 0

. Moreover, we have

H^{'} (v) = e^{v} .

So,

H^{'} (x_{*}) = I

. Consequently, the conditions

(E_{5})

and

(E_{6})

hold, if we choose

Φ_{0} (t) = (e - 1) t and Φ (t) = e^{\frac{1}{e - 1}} t .

The radii of convergence are given in Table 1.

Table 1. Radii of method (7) for Example (1).

Method	$ρ_{0}$	$r_{1}$	$ρ_{p}$	$ρ_{q}$	$ρ$	$r_{2}$	r
(7)	0.581977	0.38269	0.31685	0.19639	0.19639	0.13829	0.13829

5.2. Examples for SLAC

We consider three examples, (2) through (4), to demonstrate the theoretical results of semi-local convergence proposed in Section 3. The methods under comparison are Newton’s method (3), Sharma’s method (4), the Potra–Pták method (6), and the method (7), which we refer to as (NM), (SM), (PM), and (OM), respectively.

Example 2. Planck’s radiation problem:

The spectral density of electromagnetic radiation emitted by a black body at a specific temperature and in thermal equilibrium is determined using Planck’s radiation equation [19], which is expressed as follows:

G (y) = \frac{8 π c h y^{- 5}}{e^{\frac{c h}{y k T}} - 1},

where

T, y, k, h

and c denote the black-body’s absolute temperature, radiation wavelength, Boltzmann constant, Planck’s constant, and the speed of light in a vacuum, respectively. To determine the wavelength y that maximizes the energy density

G (y)

, we set

G^{'} (y) = 0

. This results in the following formula:

\frac{(\frac{c h}{y k T}) e^{\frac{c h}{y k T}}}{e^{\frac{c h}{y k T}} - 1} = 5 .

By setting

x = \frac{c h}{y k T}

, we have

H_{3} (x) = (e^{- x} - 1 + \frac{x}{5}) .

The required value of the root is

γ = 4.96511423174428

. Using this root, the wavelength y can be determined from the equation

x = \frac{c h}{y k T}

. The Planck problem was tested with an initial guess of

x_{0} = 5.4

, and the computational results are shown in Table 2.

Table 2. Convergence pattern of various approaches to Planck’s radiation problem (2).

Methods	$∥ H (x_{3}) ∥$	$∥ x_{4} - x_{3} ∥$	n	$η$	CPU Timing
NM	$4.7 \times 10^{- 17}$	$2.4 \times 10^{- 16}$	7	2.0000	0.0034916
SM	$2.9 \times 10^{- 58}$	$1.5 \times 10^{- 57}$	5	3.0000	0.0106235
PM	$1.2 \times 10^{- 54}$	$6.1 \times 10^{- 54}$	5	3.0000	0.0081417
OM	$2.3 \times 10^{- 111}$	$1.2 \times 10^{- 110}$	4	4.0000	0.0152859

Example 3. Continuous stirred tank reactor (CSTR):

We assume an isothermal continuous stirred tank reactor (CSTR) (for details, see [20]). The components

M_{1}

and

M_{2}

represent the feed rates to

B_{1}

and

B_{2} - B_{1}

, respectively. The reaction scheme in the reactor is then as follows:

\begin{matrix} M_{1} + M_{2} \to B_{1} \\ B_{1} + M_{2} \to C_{1} \\ C_{1} + M_{2} \to D_{1} \\ C_{1} + M_{2} \to E_{1} \end{matrix}

While creating a basic model for feedback control systems, Douglas [21] also studied the aforementioned model. Then, he transformed it into the subsequent mathematical expression:

R_{C_{1}} \frac{2.98 (x + 2.25)}{(x + 1.45) {(x + 2.85)}^{2} (x + 4.35)} = - 1,

where

R_{C_{1}}

is the gain of the proportional controller. The expression (42) is balanced for negative real values of

R_{C_{1}}

. Specifically, by setting

R_{C_{1}} = 0

, we obtain

H_{4} (x) = x^{4} + 11.50 x^{3} + 47.49 x^{2} + 83.06325 x + 51.23266875 .

(42)

The zeros of the function

H_{4}

correspond to the poles of the open-loop transfer function. The function

H_{4}

has four zeros:

γ = - 1.45, - 2.85, - 2.85, - 4.35

. We select

γ = - 1.45

as the desired zero and choose

x_{0} = - 1.6

as the initial point for

H_{4}

. The computational results are presented in Table 3.

Table 3. Convergence pattern of various approaches to CSTR problem (3).

Methods	$∥ H (x_{3}) ∥$	$∥ x_{4} - x_{3} ∥$	n	$η$	CPU Timing
NM	$3.6 \times 10^{- 4}$	$6.4 \times 10^{- 5}$	9	2.0000	0.0004944
SM	$1.7 \times 10^{- 13}$	$2.9 \times 10^{- 14}$	6	3.0000	0.0006272
PM	$1.2 \times 10^{- 6}$	$2.0 \times 10^{- 7}$	7	3.0000	0.000563
OM	$1.4 \times 10^{- 47}$	$2.5 \times 10^{- 48}$	5	4.0000	0.0008959

Example 4. Blood rheology model:

Finally, we examine the physical and flow properties of blood using the blood rheology model [22]. Blood is called a Caisson fluid because it doesn’t follow Newtonian fluid rules. According to the Caisson fluid model, simple fluids move through tubes in a way that creates a speed difference between the walls and the center, where there is minimal deformation. We study the plug flow of Caisson fluids using the following mathematical expression:

G = - \frac{x^{4}}{21} + \frac{4 x}{3} - \frac{16 \sqrt{x}}{7} + 1 .

We then set

G = 0.40

to calculate the flow rate reduction, which simplifies to the following nonlinear equation:

H_{5} (x) = \frac{x^{8}}{441} - \frac{8 x^{5}}{63} - \frac{2857144357 x^{4}}{50000000000} + \frac{16 x^{2}}{9} - \frac{906122449 x}{250000000} + \frac{3}{10} .

By adopting the proposed methods, we determined the desired zero of the function

H_{5} (x)

to be

x = 0.08643356 \dots

and chose

x_{0} = 0.07

as the starting point for

H_{5}

. The computed results are shown in Table 4.

Table 4. Convergence pattern of various approaches to blood rheology model (4).

Methods	$∥ H (x_{3}) ∥$	$∥ x_{4} - x_{3} ∥$	n	$η$	CPU Timing
NM	$2.1 \times 10^{- 16}$	$6.2 \times 10^{- 17}$	7	2.0000	0.001264
SM	$1.5 \times 10^{- 55}$	$4.6 \times 10^{- 56}$	5	3.0000	0.0032786
PM	$1.1 \times 10^{- 51}$	$3.3 \times 10^{- 52}$	5	3.0000	0.0022388
OM	$2.6 \times 10^{- 164}$	$8.0 \times 10^{- 165}$	4	4.0000	0.0049327

6. Concluding Remarks

The two-step Traub-type iterative method (2) with parameters and weight functions is used as a sample to demonstrate a new convergence technique, which avoids the Taylor series. Method (2) unifies a plethora of other popular iterative methods [2,3,4,5,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. The usage of method (2) is crucial in cases involving the analytic form of the solution

x_{*}

of expression (1). It is hard to find in, for example, the case of fractional order, for a pseudo-hyperbolic telegraph equation [1,19,20,21,22]. Other advantages of this technique include weaker sufficient convergence conditions, information on the number of solutions in a certain interval, and a priori upper error estimates on

∥ ρ_{x_{θ}} ∥

. Further, the semi-local convergence analysis of the iterative approaches was proposed. Moreover, our theoretical aspects are also supported by the numerical results. The generality of this techniques makes it applicable on other methods defined on the real line, the complex plane, or more abstract spaces such as Hilbert or Banach. This is intended to be the direction of our future area of research [4,5,7,8,12,17,18,23,24]. It is also worth noting that the aforementioned references included methods with inverse of linear operators making them suitable for the application of our approach, since they have also used the Taylor series. Consequently, they have the same drawbacks

(P_{1})

–

(P_{6})

Author Contributions

Conceptualization, R.B. and I.K.A.; methodology, R.B. and I.K.A.; software, R.B. and I.K.A.; validation, R.B. and I.K.A., formal analysis, R.B. and I.K.A.; investigation, R.B. and I.K.A.; resources, R.B. and I.K.A.; data curation, R.B. and I.K.A.; writing—original draft preparation, R.B. and I.K.A.; writing—review and editing, R.B. and I.K.A.; visualization, R.B. and I.K.A.; supervision, R.B. and I.K.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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Behl, R.; Argyros, I.K. Convergence of a Family of Methods with Symmetric, Antisymmetric Parameters and Weight Functions. Symmetry 2024, 16, 1179. https://doi.org/10.3390/sym16091179

AMA Style

Behl R, Argyros IK. Convergence of a Family of Methods with Symmetric, Antisymmetric Parameters and Weight Functions. Symmetry. 2024; 16(9):1179. https://doi.org/10.3390/sym16091179

Chicago/Turabian Style

Behl, Ramandeep, and Ioannis K. Argyros. 2024. "Convergence of a Family of Methods with Symmetric, Antisymmetric Parameters and Weight Functions" Symmetry 16, no. 9: 1179. https://doi.org/10.3390/sym16091179

APA Style

Behl, R., & Argyros, I. K. (2024). Convergence of a Family of Methods with Symmetric, Antisymmetric Parameters and Weight Functions. Symmetry, 16(9), 1179. https://doi.org/10.3390/sym16091179

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Convergence of a Family of Methods with Symmetric, Antisymmetric Parameters and Weight Functions

Abstract

1. Introduction

2. Convergence 1: Local Analysis

3. Convergence 2: Semi-Local Analysis

4. The Convergence Order of Method (2)

5. Numerical Experiments

5.1. Examples for LAC

5.2. Examples for SLAC

6. Concluding Remarks

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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