One popular research area in mathematics is to find the solution α=(α1,α2,…,αn)t of the system of nonlinear equation F(X)=0 , where F(X)=(f1(x),f2(x),…,fn(x))t , and X=(x1,x2,…xn)t∈Rn . This type of problems occurs in many applied sciences like engineering, physics, biology and chemistry. Many researchers developed iterative methods for solving this kind of systems using different techniques. The most popular iterative method for solving system of nonlinear equations is the well-known Newton’s method which has second order of convergence [1]. To improve the order of convergence and increase the accuracy of the solution obtained, many researchers tried to improve Newton’s method. Some authors used different forms and modifications based on Adomian decomposition technique for solving systems of nonlinear equations, see for instance [2–6]. Another way to improve some schemes for systems of nonlinear equation is by using homotopy analysis method and homotopy perturbation method, see for example [7,8]. Grau-Sánchez et al. [9] used the harmonic mean of the derivative to improve an iterative scheme for solving systems of nonlinear equations. By applying some quadrature formulas, some researchers implement their techniques to solve systems of nonlinear equations, for instance [10–12]. Also, some derivative-free schemes for systems of nonlinear equations were proposed, see for example [13–15] and the references therein. One of the well-known modifications of Newton method is Jarratt method of order four. Cordero et al. [16] extended Jarratt method to solve systems of nonlinear equations preserving the same order of convergence. Many variants of Jarratt type methods have been developed, see for example [17–19] and the references therein. Many other different orders of convergence schemes for nonlinear modules can be found in the literature, see for example [20,21] and the references therein.

Some techniques to improve the order of convergence of the iterative schemes for systems of nonlinear equations have been proposed, for instance, see [22,23]. In general, obtaining a higher-order iterative method is not the only important thing; as the computational and the time cost are crucial issue also. So, establishing a high order iterative method based on low computational and time cost is very important.

In this paper, we develop a new multi-step scheme of arbitrary odd order for nonlinear equations. The proposed method can be used in the multidimensional case preserving the same order. The convergence analysis of the new scheme is discussed. Several examples are given to show the efficiency of the generalized method and its comparison with other iterative schemes of the same order. To confirm the applicability of the new technique, we apply the new technique to some real-life problems.

In this section we will derive the proposed technique for nonlinear modules. We begin by writing the function f(x) as:

(1) f(x)=∫xnx⁡f′(t)dt+f(xn).

As we want f(x)=0 , and by using midpoint quadrature formula and writing the equation as an iterative scheme, one gets

(2) xn+1=xn−f(xn)f′(xn+xn+12).

Now, to write the iterative scheme (2) in explicit form, replace f′(xn+xn+12) by f′(xn+xn+1∗2) where xn+1∗ is the Newton step. So, scheme (2) becomes:

(3) {yn=xn−12f(xn)f′(xn),xn+1=xn−f(xn)f′(yn).

The iterative method (3) was proposed by Frontini and Sormani [10]. The multidimensional case of scheme (3) was discussed by Homeier [24] and can be written as:

(4) {Yn=Xn−12F′(Xn)−1F(Xn),Xn+1=Xn−F′(Yn)−1F(Xn),

where F′(Xn)−1 and F′(Yn)−1 are the inverse of the first Fréchet derivative of F(Xn) and F(Yn) respectively. Scheme (4) is of third-order of convergence, and requires at each iteration the evaluation of one function, two Fréchet derivatives and two matrix inversions. In order to increase the convergence order and the computational efficiency of scheme (4) Sharma et al. [20] proposed a new scheme of the fifth-order of convergence by adding one step to scheme (4):

(5) {Yn=Xn−12F′(Xn)−1F(Xn),Wn=Xn−F′(Yn)−1F(Xn),Xn+1=Wn−(2F′(Yn)−1−F′(Xn)−1)F(Wn).

Per iteration, scheme (5) requires the evaluations of two functions, two Fréchet derivatives and two matrix inversions.

Now, to derive the new scheme for solving systems of nonlinear equations, we start by composing scheme (3) to additional Newton step, that is:

(6) {yn=xn−12f(xn)f′(xn),wn=xn−f(xn)f′(yn),xn+1=wn−f(wn)f′(wn).

Now, to reduce number of functional evaluations at each iteration, we will use divided difference approximation to write the derivative f′(wn) using some already computed functions from the previous steps. To do that, one can write

(7) f′(yn)=f(yn)−f(xn)yn−xn=f[yn,xn]≈f′(xn),

in the same manner, we have

(8) f′(yn)=f(yn)−f(wn)yn−wn=f[yn,wn]≈f′(wn),

by adding (7) and (8), one easily can conclude that

(9) f′(wn)≈2f′(yn)−f′(xn).

Now, if we substitute (9) in (6), then we will have a new scheme for solving nonlinear equations:

(10) {yn=xn−12f(xn)f′(xn),wn=xn−f(xn)f′(yn),xn+1=wn−f(wn)2f′(yn)−f′(xn).

To generalize scheme (10) to the multidimensional case to solve systems of nonlinear modules, the scheme becomes:

(11) {Yn=Xn−12F′(Xn)−1F(Xn),Wn=Xn−F′(Yn)−1F(Xn),Xn+1=Wn−(2F′(Yn)−F′(Xn))−1F(Wn).

Scheme (11) requires at each iteration the evaluation of two functions, two Fréchet derivatives and three matrix inversions. The proposed scheme is of fifth-order of convergence as we will see in the next section.

If we repeat using the same idea of the derivation of scheme (11), we can write a general scheme for solving system of nonlinear equations, and this is the main motivation of our work. The general scheme can be written as:

(12) {X1,n=X0,n−12F′(X0,n)−1F(X0,n),X2,n=X0,n−F′(X1,n)−1F(X0,n),X3,n=X2,n−(2F′(X1,n)−F′(X0,n))−1F(X2,n),⋮Xq,n=Xq−1,n−(2F′(X1,n)−F′(X0,n))−1F(Xq−1,n),⋮Xm,n=Xm−1,n−(2F′(X1,n)−F′(X0,n))−1F(Xm−1,n).

Per iteration, scheme (12) requires the evaluation of m−1 functions, two Fréchet derivatives and three matrix inversions. We will prove in the next section that scheme (12) is of order 2m−1 for any integer m≥3 .

We will discuss in this section the order of convergence of the proposed schemes (11) and (12). Assume for the next theorems that Cj=1j!F′(α)−1F(j)(α) , j≥2 , and en=Xn−α .

Theorem 1 Let α be the solution of the system F(X)=0 where F:D⊆Rn→Rn be a sufficiently differentiable function on a neighborhood D of α . Suppose that F′(X) is continuous and nonsingular in α . If X0∈D is an initial approximation which is close enough to α , then the sequence {Xn}n≥0 obtained by scheme (11) converges to the root α , and the order of convergence equals 5 , with asymptotic equation ek+1=18(4C22−3C3)(4C22−C3)ek5 .

Proof. By using the Taylor expansion of F(Xn) we can write F(Xn)=F′(α)en+C2en2+C3en3+C3en3+C4en4+…) . Now, we use the following Mathematica code to show the convergence order of scheme (11) for m=3 :

In[1]:= F[e_]:= dF[α](e+C2e2+C3e3+C4e4);

In[2]:= y=e-Series[12(F′[e])−1F[e],{e,0,2}];

In[3]:= w=e-Series[(F′[y])−1F[e],{e,0,2}];

In[4]:= en+1=w−(2F′[y]−F′[e])−1F[w]// //FullSimplify

Out[4]:= (2C24−2C22C3+3C328)e5+O[e6].

the code shows that we have en+1=(2C24−2C22C3+3C328)en5 , which can be written as:

en+1=18(4C22−3C3)(4C22−C3)en5 .

By this, we show that scheme (11) is at least of fifth-order of convergence.

Now, we want to discuss the order of convergence of the generalized scheme given by (12).

Theorem 2 Let α be the solution of the system F(X)=0 where F:D⊆Rn→Rn be a sufficiently differentiable function on a neighborhood D of α . Suppose that F′(X) is continuous and nonsingular in α . If X0∈D is an initial approximation which is close enough to α , then the sequence {Xn}n≥0 obtained by scheme (12) converges to the root α , and the order of convergence equals 2m−1 , for any integer m≥3 , with asymptotic equation of the form ek+1=12m(4C22−3C3)m−2(4C22−C3)ek2m−1 .

Proof. We will use the mathematical induction to prove the convergence order of scheme (12).

Firstly, we will prove that scheme (12) is convergent for m=3 , and the convergence order satisfies 2(3)−1=5 . Note that for m=3 , scheme (12) reduces to scheme (11) which we have been proved that it has the fifth-order of convergence in the previous theorem. Now, to complete the proof using the mathematical induction, suppose that scheme (12) is true and converges for all m≤r for some positive 3$]]>r>3 and satisfy the given asymptotic equation. We need to show that the scheme converges for m=r+1 , and satisfy the given asymptotic equation. To do so, consider the following code of Mathematica:

In[1]:= F[e_]:= dF[α](e+C2e2+C3e3+C4e4);

In[2]:= y=e-Series[12(F′[e])−1F[e],{e,0,6}];

In[3]:= w=e-Series[(F′[y])−1F[e],{e,0,6}];

In[4]:= x[m_]:=x[m] =12m(4C22−3C3)m−2(4C22−C3)en2m−1// //FullSimplify

In[5]:= en+1=x[r]−(2F′[y]−F′[e])−1F[x[r]]// //FullSimplify

Out[5]:= 12r+1(4C22−3C3)r−2(4C22−C3)en2r−1

Hence, this shows that for m=r+1 , we have ek+1=12r+1(4C22−3C3)r−1(4C22−C3)ek2r+1 .

In this section, we compare the efficiency index of our proposed method with other methods in the literature. Commonly in the literature, the efficiency index EI=p1d is used, where p is the order of convergence of the iterative scheme, and d is the number of functions needed to be found per iteration in the iterative scheme. Another common index that can be used in the comparison between iterative scheme is the computational efficiency index CEI=p1d+op , where op is the number of operations per iteration in the iterative scheme. The evaluation of any scalar function is considered as an operation.

To find the number of functions required to be found per iteration in an iterative scheme, the following rules applied: Any computation of F(X) needs n evaluations of scalar functions. Any computation of the Jacobian F′(X) needs n2 evaluations of scalar functions. Also, the floating points for obtaining the LU factorization are 23n3 , and to solve the triangular system we need n2 floating points operations. Finally, n2 operations required to find a matrix-vector multiplication, and n3 operations needed to find a matrix-matrix multiplication.

We compare the efficiency index and the computational efficiency index for the proposed method (PM) (11) to the following iterative schemes:

• The third-order Frontini-Sormani method (FS) [10] given by (4).

• The fifth order scheme (CHMT) proposed by Cordero et al. [23], given by

(13) {Yn=Xn−F′(Xn)−1F(Xn),Wn=Xn−2(F′(Yn)+F′(XN))−1F(Xn),Xn+1=Wn−F′(Yn)−1F(Wn).

• The fifth order scheme (MMK) proposed by Waseem et al. [4], which is given by:

(14) {Yn=Xn−F′(Xn)−1F(Xn),Wn=Yn−F′(Xn)−1F(Yn),Zn=Wn−F′(Xn)−1F(Wn),Xn+1=Zn−F′(Xn)−1F(Zn).

• The fifth-order iterative scheme (SG) presented by Sharma et al. [20], which is defined by scheme (5).

A comparison of the number of functional evaluations of the selected iterative schemes is illustrated in Tab. 1. Also, the computational efficiency indices of the selected schemes are compared (for n=2,3,4,5,10,20,50 ), see Fig. 1. Note that the proposed scheme does not attain the best efficiency in this comparison, especially for small n . We will see in the next two sections that this issue does not affect the scheme negatively when applied to some numerical tests.

Fig. 2 illustrates the efficiency indices for the selected methods. Note that CHMT, SG and our proposed method have the same efficiency indices. However, this does not guarantee that they have the same behavior, accuracy and computational time cost.

The concept of basins of attraction is a method to show how different starting points affect the behavior of the function. In this way, we can compare different root-finding schemes depending on the convergence area of the basins of attraction. In this sense, the iterative scheme is better if it has a larger area of convergence. Here, we mean by the area of convergence, the number of convergent points to a root α of f(x) in a selected range.

To check the stability and the area of convergence of our proposed method, we select the case m=3 of scheme (12). We denote the proposed method by PM 5 . For comparison, we compare PM 5 with the following schemes of the same order of convergence: The scheme CHMT given by Cordero et al. (13), the scheme SG proposed by Sharma and Gupta (5), and the scheme MMK presented by Waseem et al. (14). We choose three test examples to visualize the basins of attraction. All examples are polynomials with roots of multiplicity one. The test polynomials are

• P1(z)=z3−z , with roots z=0 , ±1 .

• P2(z)=z4−1 , with roots z=±i , ±1 .

• P3(z)=z5+2z−1 , with roots z=−0.945068±0.854518i , 0.486389 , 0.701874±0.879697i

A 4×4 region is centered at the origin to cover all the zeros of the selected polynomials. The step size selected is 0.01 ; thus, 401×401=160801 points in a uniform grid are selected as initial point for the iterative schemes to generate the basins of attraction. The exact roots were assigned as black dots on the graph. If the scheme needs less number of iterations to converge to a specific root, then the region of that roots appears darker. The convergence criterion selected is a tolerance of 10−3 with a maximum of 100 iterations. All calculations have been performed on Intel Xeon CPU-E5-2690 0@2.90 GHz with 32 GB RAM, using Microsoft Windows 10, 64 bit based on X64-based processor. Mathematica 9 has been used to generate all graphs and computations. The dynamics of the four test problems are shown in Figs. 3–5 respectively.

Basins of attraction of PM 5 shows that the proposed method is comparable to other methods of the same order, with an area of convergence which is larger or the same as the areas of convergence of the other methods used in the comparison.

In this part, we consider some numerical problems to clarify the computational efficiency and convergence behavior of the proposed scheme. All calculations have been performed using 4000 significant digits on Mathematics 9. For comparisons, we find the number of iterations n needed to satisfy the stopping criterion Xn−Xn−1+F(Xn)<10−150 for each selected method. Also, we use the approximated computational order of convergence for each iterative scheme, which can be found by

ACOC≈ln((Xn+1−xn)/(Xn−Xn−1))ln((Xn−Xn−1)/(Xn−1−Xn−2)).

Finally, we compare for the selected schemes the distance between two consecutive iterations Xn−Xn−1 and the value of F(Xn) for n=1,2,3 .

To be consistent in the comparison, we compare the proposed scheme PM 5 defined by scheme (12) for m=3 , to the original method which we derived from, that is, FS scheme given by (4). Also, we use the following fifth-order iterative schemes in the comparison: CHMT 5 method defined by scheme (13), MMK method defined by (14), and SG method defined by (5). To test the efficiency of the extension of our proposed scheme to higher orders schemes, we compare the proposed scheme PM 7 of seventh-order given by scheme (12) for m=4 , to the extension of CHMT 5 to the seventh-order scheme CHMT 7 given by:

(15) {Yn=Xn−F′(Xn)−1F(Xn),Wn=Xn−2(F′(Yn)+F′(XN))−1F(Xn),Zn=Wn−F′(Yn)−1F(Wn),Xn+1=Zn−F′(Yn)−1F(Zn).