Ball Comparison for Some Efficient Fourth Order Iterative Methods Under Weak Conditions

We provide a ball comparison between some 4-order methods to solve nonlinear equations involving Banach space valued operators. We only use hypotheses on the first derivative, as compared to the earlier works where they considered conditions reaching up to 5-order derivative, although these derivatives do not appear in the methods. Hence, we expand the applicability of them. Numerical experiments are used to compare the radii of convergence of these methods.


Introduction
Let E 1 , E 2 be Banach spaces and D ⊂ E 1 be a nonempty and open set.Set LB(E 1 , E 2 ) = {M : E 1 → E 2 }, bounded and linear operators.A plethora of works from numerous disciplines can be phrased in the following way: using mathematical modelling, where λ : D → E 2 is a continuously differentiable operator in the Fréchet sense.Introducing better iterative methods for approximating a solution s * of expression ( 1) is a very challenging and difficult task in general.Notice that this task is extremely important, since exact solutions of Equation (1) are available in some occasions.
We are motivated by four iterative methods given as Mathematics 2019, 7, 89; doi:10.3390/math7010089www.mdpi.com/journal/mathematicsand    y j =x j − H j λ (x j ) −1 λ(x j ) x n+1 =z j − 3I − H j λ (x j ) −1 [x j , z j ; λ] λ (x j ) −1 λ(z j ), (5) where H 0 j = H 0 (x j ), x 0 , y 0 ∈ D are initial points, H(x) = 2I + H 0 (x), H j = H(X j ) ∈ LB(E 1 , E 1 ), A j = λ (x j ) −1 λ (y j ), z j = x j +y j 2 , B j = λ (y j ) −1 λ (x j ), and [•, •; λ] : D × D → LB(E 1 , E 1 ) is a first order divided difference.These methods specialize to the corresponding ones (when E 1 = E 2 = R i , i is a natural number) studied by Nedzhibov [1], Hueso et al. [2], Junjua et al. [3], and Behl et al. [4], respectively.The 4-order convergence of them was established by Taylor series and conditions on the derivatives up to order five.Even though these derivatives of higher-order do not appear in the methods (2)- (5).Hence, the usage of methods (2)-( 5) is very restricted.Let us start with a simple problem.Set E 1 = E 2 = R and D = [− 5  2 , 3  2 ].We suggest a function λ : A → R as Then, s * = 1 is a zero of the above function and we have Then, the third-order derivative of function λ (x) is not bounded on D. The methods (2)-( 5) cannot be applicable to such problems or their special cases that require the hypotheses on the third or higher-order derivatives of λ.Moreover, these works do not give a radius of convergence, estimations on x j − s * , or knowledge about the location of s * .The novelty of our work is that we provide this information, but requiring only the derivative of order one, for these methods.This expands the scope of utilization of them and similar methods.It is vital to note that the local convergence results are very fruitful, since they give insight into the difficult operational task for choosing the starting points/guesses.
Otherwise with the earlier approaches: (i) We use the Taylor series and high order derivative, (ii) we do not have any clue for the choice of the starting point x 0 , (iii) we have no estimate in advance about the number of iterations needed to obtain a predetermined accuracy, and (iv) we have no knowledge of the uniqueness of the solution.
The work lays out as follows: We give the convergence of these iterative schemes (2)-( 5) with some main theorems in Section 2. Some numerical problems are discussed in the Section 3. The final conclusions are summarized in Section 4.
Theorem 1.Under the conditions (A) sequence {x j } starting at x 0 ∈ S(s * , R) − {s * } converges to s * , {x j } ⊂ S(x, R) so that with ψ 1 and ψ 2 functions considered previously and R is given in (9).Moreover, s * is a unique solution in the set D 1 .
Hence, we arrived at the next Theorem.
Theorem 2. Under the conditions (A) , the conclusions of Theorem 1 hold for method (3).
Proof.Next, we deal with method (4) in the similar way.Let ϕ 0 , ϕ, ϕ 1 , ρ 0 , ρ 1 , ρ, ψ 1 , R 1 , and ψ1 , be as in the case of method (3).We consider functions ψ 2 and ψ2 on I 1 as The minimal zero of ψ2 (t) = 0 is denoted by R 2 in (0, ρ), and set Notice again that from the second substep of method (4), we have so The rest follows as in Theorem 1.
Hence, we arrived at the next following Theorem.
Theorem 3.Under the conditions (A) , conclusions of Theorem 1 hold for scheme (4).
Then, using the estimates and Here, recalling that z 0 = x 0 +y 0 2 , we also used the estimates 41) and (42).
Hence, we arrived at the next following Theorem.Theorem 4.Under the conditions (C), the conclusions of Theorem 1 hold for method (5).

Numerical Applications
We test the theoretical results on many examples.In addition, we use five examples and out of them: The first one is a counter example where the earlier results are applicable; the next three are real life problems, e.g., a chemical engineering problem, an electron trajectory in the air gap among two parallel surfaces problem, and integral equation of Hammerstein problem, which are displayed in Examples 1-5.The last one compares favorably (5) to the other three methods.Moreover, the solution to corresponding problem are also listed in the corresponding example which is correct up to 20 significant digits.However, the desired roots are available up to several number of significant digits (minimum one thousand), but due to the page restriction only 30 significant digits are displayed.
We compare the four methods namely (2)-( 5), denoted by N M, HM, J M, and BM, respectively on the basis of radii of convergence ball and the approximated computational order of convergence , j = 2, 3, 4, ... (for the details please see Cordero and Torregrosa [5]) (ACOC).We have included the radii of ball convergence in the following Tables 1-6 except, the Table 4 that belongs to the values of abscissas t j and weights w j .We use the Mathematica 9 programming package with multiple precision arithmetic for computing work.
We choose in all examples H 0 (x) = 0 and In addition, we choose the following stopping criteria (i) x j+1 − x j < and (ii) λ(x j ) < , where = 10 −250 .
But, λ (x) is unbounded on Ω at x = 0.The solution of this problem is s * = 1 π .The results in Nedzhibov [1], Hueso et al. [2], Junjua et al. [3], and Behl et al. [4] cannot be utilized.In particular, conditions on the 5th derivative of λ or may be even higher are considered there to obtain the convergence of these methods.But, we need conditions on λ according to our results.In additon, we can choose The distinct radius of convergence, number of iterations n, and COC (ρ) are mentioned in Table 1.The distinct radius of convergence, number of iterations n, and COC (ρ) are mentioned in Table 2.  39) is violated with these choices of ϕ i .This is the reason that R is zero in the method BM.Therefore, our results hold only, if x 0 = s * .
Example 3.An electron trajectory in the air gap among two parallel surfaces is formulated given as where e, m, x 0 , v 0 , and E 0 sin(ωt + α) are the charge, the mass of the electron at rest, the position, velocity of the electron at time t 0 , and the RF electric field among two surfaces, respectively.For particular values of these parameters, the following simpler expression is provided: The solution of function f 3 is s * ≈ −0.309093271541794952741986808924.Moreover, we have The distinct radius of convergence, number of iterations n, and COC (ρ) are mentioned in Table 3.
where the kernel U is We phrase (47) by using the Gauss-Legendre quadrature formula with 1 0 φ(t)dt 10 ∑ k=1 w k φ(t k ), where t k and w k are the abscissas and weights respectively.Denoting the approximations of x(t i ) with x i (i = 1, 2, 3, ..., 10), then we yield the following 8 × 8 system of nonlinear equations The values of t k and w k can be easily obtained from Gauss-Legendre quadrature formula when k = 8 mentioned in Table 4.The distinct radius of convergence, number of iterations n, and COC (ρ) are mentioned in Table 5.  39) is violated with these choices of ϕ i .This is the reason that R is zero in the method BM.Therefore, our results hold only, if x 0 = s * .
Example 5. We consider a boundary value problem from [8], which is defined as follows: We assume the following partition on [0, 1] , where x j+1 = x j + h, h = 1 j .

Table 1 .
Comparison on the basis of different radius of convergence for Example 1.Equation (39) is violated with these choices of ϕ i .This is the reason that R is zero in the method BM.Therefore, our results hold only, if x 0 = s * .

Table 2 .
Comparison on the basis of different radius of convergence for Example 2.

Table 3 .
Comparison on the basis of different radius of convergence for Example 3.
Equation (39) is violated with these choices of ϕ i .This is the reason that R is zero in the method BM.Therefore, our results hold only, if x 0 = s * .Example 4. Considering mixed Hammerstein integral equation Ortega and Rheinbolt [8], as x

Table 5 .
Comparison on the basis of different radius of convergence for Example 4.