A high-order Chebyshev-type method for solving nonlinear equations: local convergence and applications

1.   Introduction

In this paper, our main purpose is to estimate an approximate solution $\gamma_{*}$ of the equation

where $P: \Omega \subseteq T_{1}\rightarrow T_{2}$ is a scalar function in an open convex interval $\Omega$.

Solving problems of nonlinear equations is widely used in many fields, such as physics, chemistry, and biology ^[1]. Usually, the analytical solution of nonlinear equations is difficult to obtain in general cases. Therefore, in most situations, iterative methods are applied to find approximate solutions ^[2]. The most famous and fundamental iterative method is Newton's method ^[3]. Currently, many methods are constructed on the basis of Newton's method, and they are called Newton-type methods ^[4,5]. Convergence analysis is an important part of the research of the iterative method ^[6]. The issue of local convergence is, based on the information surrounding a solution, to find estimates of the radii of the convergence balls ^[7]. At present, many scholars study the local convergence analysis of iterative methods, such as Argyros et al. studied the local convergence of a third-order iterative method ^[8] and Chebyshev-type method ^[9]. In addition, some iterative methods and their local convergence are used in the study of diffusion equations ^{[10,11,12,13]}. The domain of convergence is an important problem in the study of iterative process; see ^[14]. Generally, the domain of convergence is small, which limits the choice of initial points. Thus, it is crucial that the domain of convergence is expanded without additional conditions. This paper will study the local convergence of a Chebyshev-type method without the second derivative in order to broaden its applied range.

The classical Chebyshev-Halley type methods of third-order convergence ^[15], which improves Newton's methods are defined by

where

This method includes Halley's method ^[16] for $\lambda = \frac{1}{2}$, Chebyshev's method ^[17] for $\lambda = 0$, and the super-Halley method for $\lambda = 1$. Since these methods need to calculate the second derivative, they have an expensive computational cost. To avoid the second derivative, some scholars have proposed some variants of Chebyshev-Halley type methods free from the second derivative ^[18,19]. Cordero et al. ^[20] proposed a high-order three-step form of the modified Chebyshev–Halley type method:

where $\beta\in\mathbb{R}$ denotes a parameter, and $s_{0}\in \Omega$ denotes an initial point. $[., .;P]$ and $[., ., .;P]$ denote divided difference of order one and two, in particular, the second-order divided difference cannot be generalized to Banach spaces. So, we study the local convergence of method (1.3) in real spaces. The order of convergence of the above method is at least six, and if $\beta = 1$, it is optimal order eight.

However, earlier proofs of the analysis of convergence required third or higher derivatives. This limits the applicability of the above method. For example, define $P(s)$ on $\Omega = [0, 1]$ by

Then, $P'''(s) = 6\ln s^{2}-60s^{2}+24s+22$ is unbounded on $\Omega$. So when using the iterative method to solve the equation (1.4), the convergence order of the iterative method cannot be guaranteed. In this paper, the analysis of local convergence for method (1.3) only uses the first-order derivative.In particular, using Lipschitz continuity conditions based on the first derivative, the applicability of method (1.3) is extended.

The rest part of this paper is laid out as follows: Section 2 is devoted to the study of local convergence for method (1.3) by using assumptions based on the first derivative. Also, the uniqueness of the solution and the radii of convergence balls are analyzed. In Section 3, according to the different parameter values, the fractal graphs of the family are drawn. The convergence and stability of the iterative method are analyzed by drawing the attractive basins. In Section 4, the convergence criteria are verified by some numerical examples. Finally, conclusions appear in Section 5.

2.   Local convergence

In this Section, we study the local convergence analysis of method (1.3) under Lipschitz continuity conditions. There are some parameters and scalar functions to be used to prove local convergence of method (1.3). $\beta\in \mathbb{R}$ and $\theta \geq 0$ are parameters. Suppose the continuous function $\upsilon_{0}: [0, +\infty)\rightarrow \mathbb{R}$ is nondecreasing, $\upsilon_{0}(0) = 0$, and

has a smallest solution $\gamma_{0}\in [0, +\infty)-\{0\}.$

Let the continuous function $\upsilon: [0, \gamma_{0})\rightarrow \mathbb{R}$ be nondecreasing and $\upsilon(0) = 0$. Functions $h_{1}$ and $g_{1}$ on the interval $[0, \gamma_{0})$ are defined by

and

Then we obtain

and $g_{1}(\xi)\rightarrow \infty$ as $\xi\rightarrow \gamma_{0}^{-}$. According to the intermediate value theorem, the equation $g_{1}(\xi) = 0$ has roots in $(0, \gamma_{0})$. Let $r_{1}$ be the smallest root. Suppose continuous function $\omega_{1}: [0, \gamma_{0}) \rightarrow \mathbb{R}$ is nondecreasing and $\omega_{1}(0) = 0$. Functions $h_{2}$ and $g_{2}$ on the interval $[0, \gamma_{0})$ are defined by

and

Then we have

and $g_{2}(\xi)\rightarrow \infty$ as $\xi\rightarrow \gamma_{0}^{-}$. Similarly, the equation $g_{2}(\xi) = 0$ has roots in $(0, \gamma_{0})$. Let $r_{2}$ be smallest root. Functions $h_{3}$ and $g_{3}$ on the interval $[0, r_{2})$ are defined by

and

Then we have

and $g_{3}(\xi)\rightarrow \infty$ as $\xi\rightarrow r_{2}^{-}$. Similarly, the equation $g_{3}(\xi) = 0$ has roots in $(0, r_{2})$. Let $r_{3}$ be the smallest root. Suppose continuous functions $\omega_{0}, \omega_{2}: [0, \gamma_{0})^{2}\rightarrow \mathbb{R}$ and $\omega_{3}: [0, \gamma_{0})^{3}\rightarrow \mathbb{R}$ are nondecreasing with $\omega_{0}(0, 0) = 0$, $\omega_{2}(0, 0) = 0$, and $\omega_{3}(0, 0, 0) = 0$. Functions $h_{4}$ and $g_{4}$ on the interval $[0, \gamma_{0})$ are defined by

and

Then we have

and $g_{4}(\xi)\rightarrow \infty$ as $\xi\rightarrow r_{3}^{-}$. Similarly, the equation $g_{4}(\xi) = 0$ has roots in $(0, r_{3})$. Let $r_{4}$ be the smallest root. Functions $h_{5}$ and $g_{5}$ on the interval $[0, r_{4})$ are defined by

and

We have

and $g_{5}(\xi)\rightarrow \infty$ as $\xi\rightarrow r_{4}^{-}$. Similarly, the equation $g_{5}(\xi) = 0$ has roots in $(0, r_{4})$. Let $r_{5}$ be the smallest root.

Set

Then, for each $\xi\in[0, r)$, we have that

Applying the above conclusions, the analysis of local convergence for method (1.3) can be proved.

Theorem 2.1. Suppose $P:\Omega\subset T_{1}\rightarrow T_{2}$ is a scalar function. $[., .;P]: \Omega^{2}\rightarrow L(T_{1}, T_{2})$ and $[., ., .;P]: \Omega^{3}\rightarrow L(T_{1}, T_{2})$ are divided differences of one and two. Let $\gamma_{*}\in \Omega$ and continuous function $\upsilon_{0}:[0, +\infty)\rightarrow \mathbb{R}$ be nondecreasing with $\upsilon_{0}(0) = 0$ such that each $x\in \Omega$

Let $\Omega_{0} = \Omega\cap B(\gamma_{*}, \gamma_{0})$. There exist $\beta\in \mathbb{R}$, $M\geq0$, continuous functions $\upsilon, \omega_{1} :[0, \gamma_{0}) \rightarrow \mathbb{R}$, $\omega_{0}, \omega_{2}: [0, \gamma_{0})^{2}\rightarrow \mathbb{R}$, $\omega_{3}: [0, \gamma_{0})^{3}\rightarrow \mathbb{R}$ be nondecreasing such that for each $x, y, z\in \Omega_{0}$

and

Then the sequence $\{s_{n}\}$ produced for $s_{0}\in U(\gamma_{*}, r)-\{\gamma_{*}\}$ by method (1.3) converges to $\gamma_{*}$ and remains in $U(\gamma_{*}, r)$ for each $n = 0, 1, 2\ldots$. Furthermore, the following estimates hold:

and

where functions $h_{i}(i = 1, 3, 5)$ have been defined. Moreover, for $R \geq r$, if there exists that

then, the solution $\gamma_{*}\in \bar{U}(\gamma_{*}, R)\subseteq \Omega$ of equation $P(s) = 0$ is unique.

Proof Using $s_{0}\in U(\gamma_{*}, r)$, (2.8), and the definition of $r$, we obtain

According to the Banach lemma ^[2], we obtain $P'(s_{0})$ is invertible and

Then, $t_{0}$ is well defined. Therefore, we can write that

Using (2.2), (2.3), (2.10), (2.20), and (2.21), we obtain in turn that

which shows the estimate (2.16) for $n = 0$ and $t_{0}\in U(\gamma_{*}, r)$.

Using (2.2), (2.4), (2.10), (2.12), (2.16), and (2.23), we obtain

where

so

and

Thus, $(P(s_{0})-2\beta P(t_{0}))^{-1}\in L(T_{1}, T_{2})$ and

So, $z_{0}$ is well defined.

Using (2.2), (2.5), (2.12), (2.16), (2.21), (2.24), and (2.27), we have that

which shows the estimate (2.17) for $n = 0$ and $z_{0}\in U(\gamma_{*}, r)$.

Next, we shall show that

Using (2.2), (2.6), (2.11), (2.13), and (2.14), we have that

By the Banach lemma, we have that $([z_{0}, t_{0}; P]+2(z_{0}-t_{0})[z_{0}, t_{0}, s_{0};P]-(z_{0}-t_{0})[t_{0}, s_{0}, s_{0};P])$ is invertible and

Denote $\triangle = [z_{0}, t_{0}; P]+2(z_{0}-t_{0})[z_{0}, t_{0}, s_{0};P]-(z_{0}-t_{0})[t_{0}, s_{0}, s_{0};P]$. Thus, $x_{1}$ is well defined.

Using $x_{1}\in U(\gamma_{*}, r)$, (2.2), (2.8), (2.12), (2.28), and (2.31), we have that

which shows the estimate (2.18) for $n = 0$ and $s_{1}\in U(\gamma_{*}, r)$. By substituting $s_{0}, t_{0}, z_{0}, s_{1}$ in the previous estimates with $s_{k}, t_{k}, z_{k}, s_{k+1}$, we get (2.16)–(2.18). Using the estimates

we derive that $s_{k+1}\in U(\gamma_{*}, r)$ and $\mathop{lim}\limits_{k\rightarrow \infty}s_{k} = \gamma_{*}$.

Finally, in order to prove the uniqueness of the solution $\gamma_{*}$, suppose there exists a second solution $y_{*}\in \bar B(\gamma_{*}, R)$, then $P(y_{*}) = 0$. Denote $T = \int_{0}^{1}P'(y_{*}+\theta(\gamma_{*}-y_{*}))d\theta$. Since $T(y_{*}-\gamma_{*}) = P(y_{*})-P(\gamma_{*}) = 0$, if $T$ is invertible then $y_{*} = \gamma_{*}$. In fact, by (2.19), we obtain

Thus, according to the Banach lemma, $T$ is invertible. Since $0 = P(y_{*})-P(\gamma_{*}) = T(y_{*}-\gamma_{*})$, we conclude that $\gamma_{*} = y_{*}$. The proof is over.

3.   Attractive basins

In this section, we study some dynamical properties of the family of the iterative methods (1.3), which are based on their attractive basins on the complex polynomial $f(z)$. The convergence and stability of the iterative methods are compared by studying the structure of attractive basins.

There are some dynamical concepts and basic results to be used later. Let $f:\mathbb{\hat{C}}\rightarrow \mathbb{\hat{C}}$ be a rational function on the Riemann sphere $\mathbb{\hat{C}}$. The orbit of a point $z_{0}\in \mathbb{\hat{C}}$ is defined as

In addition, if $f(z_{0}) = z_{0}$, $z_{0}$ is a fixed point. There are the following four cases:

● If $|f'(z_{0})| < 1$, $z_{0}$ is an attractive point;

● If $|f'(z_{0})| = 1$, $z_{0}$ is a neutral point;

● If $|f'(z_{0})| > 1$, $z_{0}$ is a repulsive point;

● If $|f'(z_{0})| = 0$, $z_{0}$ is an super-attractive point.

The basin of attraction of an attractor $z_{*}$ is defined by

Consider the following four members of the family (1.3): $M_{1}(\beta = 0)$, $M_{2}(\beta = 0.5)$, $M_{3}(\beta = 1)$, $M_{4}(\beta = 2)$. In this study, the complex plane is $\Omega = [-5, 5]\times[-5, 5]$ with $500\times 500$ points. If the sequence converges to roots, it is represented in pink, yellow, and blue. Otherwise, black represents other cases, including non-convergence. When the family (1.3) is applied to the complex polynomials $f(z) = z^{2}-1$ and $f(z) = z^{3}-1$, their attractive basins are shown in Figures 1 and 2.

In Figures 1 and 2, the fractal graphs of the methods $M1$ and $M4$ have some black zones. The black zones indicates non-convergence, and the initial value of the black area causes the iteration to fail; relatively speaking, the method without a black region is better. However, the fractal graphs of the methods $M2$ and $M3$ have a black zone. As a result, the convergence of the methods $M2$ and $M3$ is better than that of the methods $M1$ and $M4$. In addition, the method $M3$ has the largest basins of attraction compared to the other three methods. Thus, the stable parameters are $\beta = 0.5, 1$.

4.   Numerical examples

4.1.   Radius of convergence ball

In this section, we apply the following two numerical examples to compute the above results of convergence for method (1.3).

Example 4.1. Let $\Omega = (0, 2)$; define the function $P: \Omega\rightarrow \mathbb{R}$ by

Thus, a root of $P(x) = 0$ is $\gamma_{*} = 1$. Then,

and

Notice that using conditions (2.9)–(2.15), $\beta = 0$, we obtain

and

Then, according to the above definition of functions $h_{i}(i = 1, 2, 3, 4, 5)$, one have that

Example 4.2. Let $\Omega = (-1, 1)$, define the function $P: \Omega\rightarrow \mathbb{R}$ by

Thus, a root of $P(x) = 0$ is $\gamma_{*} = 0$. Then,

and

Notice that using conditions (2.9)–(2.15), $\beta = 1$, we obtain

and

Then, according to the above definition of functions $h_{i}(i = 1, 2, 3, 4, 5)$, one obtains

4.2.   Application

In this section, the iterative method (1.3) is applied to the following six practical models. For the nonlinear equations obtained from the six models, we can find the solutions of the equations and the data results, such as iterative errors. Therefore, our research is valuable for practical models in various fields.

Example 4.3. Vertical stresses ^[21]: At uniform pressure $t$, the Boussinesq's formula is used to calculate the vertical stress $y$ caused by a specific point within the elastic material under the edge of the rectangular strip footing. The following formula is obtained:

If the value of $y$ is determined, we can find the value of $x$ where the vertical stress $y$ equals 25 percent of the applied footing stress $t$. When $x = 0.4$, the following nonlinear equation is obtained:

Example 4.4. Civil Engineering Problem ^[22]: Some horizontal construction projects, such as the topmost portion of civil engineering beams, are used in the mathematical modeling of the beams. In order to describe the exact position of the beam in this particular case, some mathematical models based on nonlinear equations have been established. The following model is given in ^[22]:

Example 4.5. The trajectory of an electron moving between two parallel plates is defined by

where $m$ and $e$ denote the mass and the charge of the electron at rest, $\nu_{0}$ and $s_{0}$ denote the velocity and position of the electron at time $l_{0}$, and $E_{0}sin(\omega l_{0}+\alpha)$ denotes the RF electric field between the plates. By selecting specific values, one obtains

Example 4.6. Blood rheology model ^[23]: Medical research that concerns the physical and flow characteristics of blood is called blood rheology. Since blood is a non-Newtonian fluid, it is often referred to as a Caisson fluid. Based on the caisson flow characteristics, when the basic fluid, such as water or blood, passes through the tube, it usually maintains its primary structure. When we observe the plug flow of Caisson fluid flow, the following nonlinear equation is considered:

where $s$ is the plug flow of Caisson fluid flow.

Example 4.7. Law of population growth ^[24]: Population dynamics are tested by first-order linear ordinary differential equations in the following way:

where $s$ denotes the population's constant birth rate and $c$ denotes its constant immigration rate. $P(u)$ stands for the population at time $u$. Then, according to solve the above linear differential equation (4.9), the following equation is obtained:

where $P_{0}$ represents the initial population. According to the different values of the parameter and the initial conditions in ^[25], a nonlinear equation for calculating the birth rate is obtained:

Example 4.8. The non-smooth function (1.4) is defined on $\Omega = [0, 1]$ by

The parameter $\beta = 1$ is selected, and the iterative method (1.3) is applied to the above six practical application examples. Table 1 gives the specific results. $k$ denotes the number of iterations. Fun denotes the function $P_{i}(1 = 1, 2, 3, 4, 5)$. $|P(s_{n})-P(s_{n-1})|$ denotes the error values. $|P(s_{n})|$ denotes function values at the last step. Approximated computation order of convergence denotes $ACOC$. $\gamma_{*}$ denotes the root of equation $P_{i}(s) = 0\; (i = 1, 2, 3, 4, 5)$. The stopping criteria is that if the significant digits of the error precision exceed 5, the output will be made. Approximated computation order of convergence (ACOC) is defined by ^[26]

In Table 1, for six models, the error accuracy is from $10^{-10}$ to $10^{-2387}$, and the computational order of convergence is the optimal order 8. When the initial point is 2.5, the error and precision of function $P_{1}$ are higher than those of function $P_{2}$. When the initial point is 4.5, the error and precision of function $P_{3}$ are higher than those of functions $P_{4}$ and $P_{5}$. At the same time, solutions to six decimal places are obtained to improve the accuracy of solutions.

5.   Conclusions

In this paper, local convergence analysis of a high-order Chebyshev-type method free from second derivatives is studied under $\omega$-continuity assumptions. In contrast to the conditions used in previous studies, the new conditions of convergence are weaker. This study extends the applicability of method (1.3). Also, the radii of convergence balls and uniqueness of the solution are also discussed. By drawing the basins of attraction, four methods with different parameter values are compared with each other. Thus, we can find that when the parameter $\beta = 1$ of method (1.3), the method $M3$ is relatively more stable. Then, two numerical examples are used to prove the criteria of convergence. Finally, we apply the method (1.3) to six concrete models. In Table 1, the numerical results such as iterative errors, ACOC, and so on are obtained. Therefore, our research is valuable for practical models in various fields.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 61976027), the Open Project of Key Laboratory of Mathematical College of Chongqing Normal University (No.CSSXKFKTM202005), the Natural Science Foundation of Liaoning Province (Nos. 2022-MS-371, 2023-MS-296), the Educational Commission Foundation of Liaoning Province of China (Nos. LJKMZ20221492, LJKMZ20221498, LJ212410167008), and the Key Project of Bohai University (No. 0522xn078), the Innovation Fund Project for Master's Degree Students of Bohai University (YJC2024-023).

Conflict of interest

The authors declare there are no conflicts of interest.