A New Optimal Family of Schröder’s Method for Multiple Zeros

: Here, we suggest a high-order optimal variant/modiﬁcation of Schröder’s method for obtaining the multiple zeros of nonlinear uni-variate functions. Based on quadratically convergent Schröder’s method, we derive the new family of fourth -order multi-point methods having optimal convergence order. Additionally, we discuss the theoretical convergence order and the properties of the new scheme. The main ﬁnding of the present work is that one can develop several new and some classical existing methods by adjusting one of the parameters. Numerical results are given to illustrate the execution of our multi-point methods. We observed that our schemes are equally competent to other existing methods.


Introduction
There are several issues from chemistry, physics, applied mathematics, scientific computing, economics and engineering that can be transformed to It is almost inaccessible to find the solution by an analytical approach. Therefore, we worry about the iterative schemes to find the multiple solution r m with known multiplicity m > 1 of the uni-variate function in Equation (1).
Construction of the iterative schemes [1][2][3] (that do not essentially require the value of second-order derivative of Θ) having optimal convergence order [4] is the foremost and most difficult issue in the region of numerical analysis. It is also interesting with practical perspective because sometimes it is very difficult to compute the second-order derivative.
Some high-order variants and modifications of Newton's method that can solve multiple roots are suggested and investigated [5][6][7][8][9][10][11]. Out of them, only two iterative methods by Li et al. [6,7], Sharma and Sharma [8], and Zhou et al. [9] have optimal fourth-order convergence. We have very limited fourth-or high-order schemes until now that are capable of solving multiple roots. Khattri and Abbasbandy [12] developed the following scheme for obtaining optimal families of fourth-order methods for simple zeros of uni-variate functions where α j ∈ R. Zhou et al. [9] presented an optimal scheme of fourth-order that needs one value of Θ and two of derivative Θ at each step and is defined as: where m denotes the multiplicity of the zero.
No doubt, the above-mentioned scheme given by Zhou et al. [9] is a more general form of the fourth-order iterative schemes. This scheme increases the order of convergence of methods from linear to quartic and gives an optimal family of fourth-order methods. However, the proposed family of methods is dependent upon the weight function. Thus, whenever someone wants to develop a new method from this family, he/she has to construct a new weight function that employs the hypotheses mentioned in [9]. It is not easily accessible to obtain such a weight function. Therefore, this is the main drawback of this scheme.
We are keen to construct a new general class of Schröder's method where there is no need for the weight function(s) to develop new methods having optimal fourth-order convergence. For this, we extend the idea of Khattri and Abbasbandy [12] for the multiple zeros. The principle benefit of the scheme is that we have two free disposable parameters. Therefore, one can obtain several new and some existing optimal families by adjusting one of the parameters, unlike the weight function(s) in [9]. Our iterative techniques are equally competent to the existing schemes.

Development of the Fourth-Order Methods
Here, we suggest a new optimal class of Schröder's method. For this, we consider a Schröder's method [13] that is defined as follows: The principle advantages of this method are that it does not depend upon the multiplicity of the required zero along and it has second-order convergence. The simplest way to obtain this method is to adopt the classical Newton's method to the function η(x) = Θ(x) Θ (x) that has a simple zero corresponding to the multiple root of function Θ.
We assume the following Newton-like method for multiple roots To abolish second-order derivative from the scheme in Equation (4), expand the function Θ (y l ) Θ (x l ) about the point x = x l by Taylor's series expansion, which leads us to further yielding Using Equation (7) in the Schröder's method in Equation (4), we obtain The method in Equation (8) has the following error equation Due to Kung-Traub conjecture [4], Equation (8) is a non optimal scheme because of linear convergence order. Additionally, it uses three functional evaluations at each step. To increase the convergence order of Equation (8), we consider the following iterative scheme where γ, β, α 0 , α 1 , α 2 ∈ R are free variables. These variables are responsible for obtaining optimal fourth-order convergence. Theorem 1 illustrates the restrictions on the parameters, so the above scheme in Equation (9) reaches at optimal convergence order.
Theorem 1. Let Θ : A ⊆ C → C be a holomorphic function in the region A that encloses the desired multiple zero of Θ(x), (say x = r m ) with known multiplicity m ≥ 1. Then, the family in Equation (9) has fourth-order of convergence, when where u = m m+2 m and α 1 , α 2 ∈ R are free variables.
Proof. We assume that x = r m is a multiple zero of multiplicity m of Θ(x). Now, expand Θ(x l ) and Θ (x l ) about a point x = r m with the help of the Taylor's series expansion. Then, we yield and , k = 1, 2, 3, 4, . . . are the asymptotic error constants.
By adopting Equations (11) and (12), we obtain Furthermore, we have and Using Equations (14) and (15) in the scheme in Equation (9), we have where For attaining fourth-order convergence of the scheme in Equation (9), the coefficients of e l , e 2 l , and e 3 l should be zero simultaneously. From Equation (16), we obtain the following values of α, β, α 0 involving two free disposable parameters α 1 and α 2 where u = m m+2 m .
By inserting Equation (17) into Equation (16), we yield This confirms that the new scheme in Equation (14) attains the optimal convergence order by just consuming three functional evaluations at each step. Hence, it completes the proof.

Particular Forms
Since we have two free disposable parameters, namely α 1 and α 2 , we can simply develop several new and some existing classical iterative functions by adjusting one of the free parameters in the scheme in Equation (9).

Special cases:
Equation (18) is a new fourth-order optimal family for multiple roots. (18):

Special cases of the scheme in Equation
which is another new fourth-order optimal technique.
, the family in Equation (18) provides the following new fourth-order optimal scheme , the scheme in Equation (18) further yields This is a well-known fourth-order optimal technique developed by Zhou et al. [9]. Case 2. For α 1 = 1, the general class in Equation (9) leads us to another new optimal fourth-order scheme: Sub special case of the family in Equation (22): (i) For α 2 = 0, the family in Equation (22) gives a well-known fourth-order method developed by Li et al. [7], which is defined as follows: It is obvious to observe that each step of the schemes in Equations (18)- (20) and (22) consumr only three functional evaluations each iteration, viz. one value of Θ(x) and two of Θ (x). For checking the efficiency of the new technique in Equation (9), we adopt the efficiency index a 1 ρ (for more details, please see [14]). Thus, we have a = 4 and ρ = 3 that provide us E ∼ = 1.587, which is far better than third-order and modified Newton's technique E ∼ = 1.442 and E ∼ = 1.414, respectively.

Numerical Results
We verified the potency of the new optimal techniques. We computee them with the iterative expressions (35) and (37) described by Sharifi et al. in [15], called by (SH1) and (SH2), respectively. In addition, we considered method (11) from Zhou et al. [9] (defined by ZS) for comparison with our methods. Further, we compared them with expressions (69) and (75) mentioned by Li et al. [6], called by (LS1) and (LS2), respectively. Finally, we computed them with expression (1) depicted by Sharma and Sharma [8], known as (SAS).
In Tables 1-6, we display the values of iterates (x l ), the absolute residual errors | f (x l )| and the absolute errors |x l+1 − x l |. We adopted the the following formula [16], for the computational order of convergence (COC) In the above COC, we need the exact zero r m . However, there are several practical situations where the exact solution is not accessible. Therefore, Ezquerro and Hernández, [17] suggested the following technique whereě l = x l − x l−1 and there is no need of an exact zero. During the current numerical experiments with programming language Mathematica (Version-9), all computations were done with 300 digits of mantissa, which minimizes round-off errors.
In addition, the initial guesses of each Θ i are mentioned in the corresponding table. Further, (b 1 ± b 2 ) express as (b 1 × 10 ±b 2 ).

Example 1. Study of multi-factor effect
The mathematical expression of the trajectory of an electron that moves between two parallel sheets along with air gap is defined as follows: where m and e stand for the mass and charge of an electron at the rest position, respectively. In addition, v 0 and x 0 are known as the velocity and position of an electron at the initial timing t 0 , respectively. Finally, E 0 sin(ωt + α) denotes a RF electric field among two sheets. With the particular values of these parameters, Equation (18) leads us to The above function has a solution x = 1.570796326794896619231322 of multiplicity 3 and numerical results of Equation (25) are depicted in Table 1.

Example 2. Eigen value problem:
We considered a matrix of order 8 × 8, defined as follows: We have the following characteristic equation of the above matrix: Equation (26) has a zero at x = 4 of multiplicity m = 3. The computational results on Equation (26) are mentioned in Table 2. Table 2. Convergence study of distinct iterative functions on Θ 2 (x).

Example 3. Chemical engineering problem:
We next assumed a fourth-order polynomial expression from [18,19]. That expression outlines the fraction conversion problem of nitrogen and hydrogen when they are transformed into ammonia (such fractions are known as fractional conversion). Adopting 250 atm and 500 • C leads to Equation (27) has two real and two complex zeros. However, we chose complex zero ξ = 3.9485424455620457727 + 0.3161235708970163733i to illustrate the application on complex roots. Table 3. Convergence study of distinct iterative functions on Θ 3 (x).

Example 4. (Continuous stirred tank reactor (CSTR)):
We assumed a problem of continuous stirred tank reactor (CSTR). We observed a reaction scheme that develops in the chemical reactor (see [20] for more information), which is defined as follows: where the components T and W 1 are fed at the amount of q-Q and Q, respectively, to the chemical reactor. The above model was studied in detail by Douglas [21] to find a good and simple system that can control feedback problem. Finally, he transferred the above model to the following mathematical expression: where K H denotes for the gaining proportional controller. The control system in Equation (29) Table 4. Example 5. We considered an academic problem, which is given as follows: Equation (31) has a zero x = 0 of multiplicity 20. Table 5. Convergence study of distinct iterative functions on Θ 5 (x). Example 6. We picked another academic problem from Zeng [22]: In the above function, we have four multiple zeros x = 1, 2, 3 and 4 and multiplicity of the corresponding zero is 20, 15, 10 and 5, respectively. All multiple zero are quite close to each other. We chose x = 1 multiple zero of multiplicity 20 for the computational point of view and the results are depicted in Table 6 . Table 6. Convergence study of distinct iterative functions on Θ 6 (x).
Methods l x l |Θ(x l )| |x l+1 − x l | ρ derivatives. The principle benefit of our family is that there is no need of weight functions to further construct new optimal fourth-order methods. We also show that we can easily obtain several new and some existing classical techniques by adjusting one of the parameters from α 1 and/or α 2 . We conclude from numerical experimentation that our methods have at least equal performance as compared to ZS, LS1, LS2, SAS and also demonstrate better behavior than SH1 and SH2 in Examples 1, 2, and 4-6.
In Example 3, our method in Equation (20) illustrates the better performance as compared to all other considered methods and the results can be seen in Table 3.