Abstract
Many real-life problems using mathematical modeling can be reduced to scalar and system of nonlinear equations. In this paper, we develop a family of three-step sixth-order method for solving nonlinear equations by employing weight functions in the second and third step of the scheme. Furthermore, we extend this family to the multidimensional case preserving the same order of convergence. Moreover, we have made numerical comparisons with the efficient methods of this domain to verify the suitability of our method.
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1 Introduction
It is well-known fact that a wide class of problems which arises in various branches of pure and applied sciences can be viewed in the general framework of the nonlinear equations and systems of nonlinear equations. Due to their importance, several iterative methods have been suggested and analyzed under certain conditions. Therefore, solving nonlinear equations and nonlinear systems efficiently and reliably has gained paramount importance in physics, engineering, operational research, and many other disciplines. This importance led to the development of many numerical techniques. However, most of them are iterative in nature, because analytic methods for such problems are almost unavailable. We can see several examples that show the applicability of these to real world problems; see [5, 9]. Narang et al. (2016) in [10] proposed fourth- and sixth-order methods for the nonlinear systems which were the extensions of earlier univariate schemes. Recently, researchers have proposed sixth-order iterative methods using weight functions and parameters (for example, see [1,2,3, 6,7,8, 13]). By getting motivation from the recent activities in this direction, we aim to propose a sixth-order family of Jarratt-type methods for solving scalar equations. Along with the perseverance of order of convergence, we then extend this family for the multidimensional case. The outline of the paper is as follows. In Sect. 2, a sixth-order scheme is presented along with their convergence analysis and numerical examples. In Sect. 3, we present extension of sixth-order scheme and their numerical examples. Section 4 and Section 5 are devoted to the efficiency of methods and concluding remarks.
2 Development of a sixth-order scheme for nonlinear scalar equation
We introduce a new sixth-order method for solving nonlinear equations.
2.1 Derivation of the scheme
For the development of our scheme, we use weight function approach. Our method is defined by the following three steps:
where \(P:{\mathbb {C}}\rightarrow {\mathbb {C}}\) and \(\ Q:{\mathbb {C}}\rightarrow {\mathbb {C}}\) are weight functions that are analytic in the neighborhood of 1 and \(u_{n}=\frac{f^{\prime }(x_{n})}{f^{\prime }(y_{n})}{,}\) \(v_{n}=\frac{f^{\prime }(y_{n})}{f^{\prime }(x_{n})}\). Theorem 2.1 demonstrates that the order of convergence reaches at six using particular conditions on these weight functions.
Theorem 2.1
Suppose \(f:D\subset {\mathbb {C}}\rightarrow {\mathbb {C}}\) be a sufficiently differentiable function in D containing a simple root \(\gamma \) of the equation f \((x)=0.\) Moreover, we suppose that an initial guess \(x_{0}\) is sufficiently close to \(\gamma \). Then, the family of iterative methods (1) attains order of convergence six using the following conditions on weight functions:
The error equation is given as
Proof Let us consider that \(e_{n} = x_{n}\) \(-\gamma \) be the error in the nth iteration. The Taylor’s series expansion of the function \(f(x_{n})\) and its first-order derivative \(f^{\prime }(x_{n})\) about \(x =\gamma \) with the assumption \(f^{\prime }(\gamma )\ne 0\) lead us to
where
for \(i=2,3,...\) and
Now
Using (2) and (3) in (4), we get
In view of the fact that \(f^{\prime }(y_{n})=\ f^{\prime }(x_{n})\mid _{e_{n}\longrightarrow y_{n}-\gamma },\) we obtain
where
With the help of (3) and (5), we obtain \(u_{n}=\frac{ f^{\prime }(x_{n})}{f^{\prime }(y_{n})}\) as
where
Next, we use Taylor’s series expansion of \(P(u_{n})\) about \(u_{n}=1\) up to fifth-order terms, as follows:
Therefore, we have
where
Moreover
where
Consequently, the second substep becomes
such that
Using the conditions on P and its derivatives as
(7) becomes
where
As \(f(z_{n})=\ f(x_{n})\mid _{e_{n}\longrightarrow z_{n}-\gamma },\) we obtain
where
With the help of (3) and (5), \(v_{n}=\frac{f^{\prime }(y_{n})}{f^{\prime }(x_{n})}\) is given by
where
Let us consider Taylor’s expansion for the weight function Q about \(v_{n}=1\) up to fifth-order terms as
Thus
where
Therefore, the final step takes the form
where
From (8), it is clear that for the following conditions on Q and on its derivatives:
our proposed scheme has the following error equation:
The error equation shows that the proposed scheme (1) approaches the sixth-order of convergence. \(\square \)
2.2 Particular cases of weight functions
Here are some particular cases of weight functions written as Case 1, Case 2, and Case 3.
Case 2.2
If we take the weight functions \(P\left( u\right) \) and \(Q\left( v\right) \) of the following form:
and
with
For \(b_{2}=4\)
Then, for \(u_{n}=\frac{f^{\prime }(x_{n})}{f^{\prime }(y_{n})}\) and \({v}_{n}{=}\frac{f^{\prime }(y_{n})}{f^{\prime }(x_{n})}\), we get a new sixth-order scheme, called as FS1
Case 2.3
When the weight functions \(P\left( u\right) \) and \(Q\left( v\right) \) are the rational function of the following form:
with
For \(b_{2}=2\)
Then, a new sixth-order scheme is given namely as FS2
Case 2.4
Next, we consider weight functions \(P\left( u\right) \) and \(Q\left( v\right) \) of the following form:
and
with
For \(b_{2}=3\)
Then, another sixth-order new scheme, namely FS3, is obtained as
2.3 Numerical results
Now, we want to verify the numerical results of our new schemes that are presented in the previous section. To demonstrate the suitability of our suggested schemes, we have considered some examples and compared the results of our schemes, namely, FS1, FS2, and FS3, with respect to the number of iterations n, absolute residual error of the corresponding function \( \mid f(x_{n})\mid ,\) error in two consecutive iterations \(\left| x_{n}-x_{n-1}\right| \), and computational order of convergence \(COC= \frac{\log \left[ f(x_{n+1})/f(x_{n})\right] }{\log \left[ f(x_{n})/f(x_{n-1})\right] }\). The previous methods for comparisons are considered as the sixth-order methods given by Behl et al. (2019) in [1] and Lee and Kim (2020) in [8] denoted by BS and LK. The numerical results are given in Tables 1 and 2.
Example 2.5
We choose a function from [4], which is
The function has two real and four complex roots. We take the real root \( \gamma =2\) and an initial guess \(x_{0}=2.5\).
Example 2.6
Consider the function
from [11]. The desired root for the function is \(\gamma =2.759+6.585i\). We take an initial guess \(x_{0}=3+7.4i\).
3 Extension of sixth-order method to the system of nonlinear equations
Now, we give an extension of our method to the system of nonlinear equations by preserving the order of convergence as in the case of scalar equations.
3.1 Derivation of the scheme
We use the weight function approach in the development of our scheme. Our method consists of three steps, which are given below. For the multidimensional case, the scheme (1) named as FS can be rewritten as
for the multivariate vector-valued function \(F:{\mathbb {D}}\subseteq {\mathbb {C}} ^{n}\rightarrow {\mathbb {C}}^{n}\) with \(n\in {\mathbb {N}}\)
and
Theorem 3.1
Let us suppose that \(F:{\mathbb {D}}\subseteq {\mathbb {C}} ^{n}\rightarrow {\mathbb {C}}^{n}\) with \(n\in N\) be a sufficiently Frechet differentiable function in \({\mathbb {D}}\) containing simple root \(\Upsilon .\) In addition, that convergence is guaranteed if we consider that initial guess \( X^{\left( 0\right) }\) is close to the root \(\Upsilon .\) Then, the numerical scheme (14) has sixth-order convergence for the following conditions on weight functions:
where \(P,Q:{\mathbb {C}}^{n\times n}\rightarrow {\mathbb {C}}^{n\times n}\) are matrix functions, sufficiently Frechet differentiable in a neighborhood of I (I is \( n\times n\) identity matrix).
Proof Let us consider that \(E_{n}\) = \(X^{(n)}-\Upsilon \) be the error in the \( n^{th}\) iteration. The Taylor’s series expansion of the function \(F(X^{(n)})\) and \(F^{\prime }(X^{(n)})\) with the assumption \( \mid F^{\prime }(\Upsilon )\mid \) \(\ne 0\) leads us to
where
for \(i=2,3,...\) and
Now, for the first substep
Applying Taylor’s series to (17), we get
where
Also, \(F^{\prime }(Y^{(n)})\) is given by
where
Next, for the Taylor’s series expansion of the function \( U^{(n)}=(F^{\prime }(Y^{(n)}))^{-1}F^{\prime }(X^{(n)})\)
where
Moreover, \(P(U^{(n)})\) is given by
where
Let us now consider the second substep
as
where
Taking the conditions
(19) becomes
where
Similarly, \(F(Z^{(n)})\) is given as
where
Also, applying Taylor’s series to \(V^{(n)}=(F^{\prime }(X^{(n)}))^{-1}F^{\prime }(Y^{(n)})\), we get
where
Similarly, \(Q(V^{(n)})\) is given as
where
Finally, Taylor’s expansion of the last step gives
where
It is apparent that taking the following conditions on the weight function Q:
we obtain the following error equation from (21):
This asymptotic error constant reveals that the proposed scheme (14) reaches at sixth-order convergence. It completes the proof. \(\square \)
Next, we take some special cases of our proposed scheme (14), which are as follows:
Case 1 When the weight functions P(U) and Q(V) are polynomial functions of the following form:
with
for \(b_{2}=4\)
Then, we get a sixth-order scheme, named as FS4 which is given below
Case 2 If we take the weight functions P(U) and Q(V) of the following form:
with
for \(b_{2}=2\)
Then, we obtain the following sixth-order scheme called as FS5:
Case 3 If we take weight functions P(U) and Q(V) of the following form:
and
with
for \(b_{2}=3\)
Then, we obtain the sixth-order scheme called as FS6
3.2 Numerical results
Now, we want to verify the numerical results of our iterative method. For this purpose, we consider some examples and compare the results of our scheme, namely, FS4, FS5, and FS6, with respect to number of iterations n, absolute residual error of the corresponding function in \(\left\| {F}\left( X^{(n)}\right) \right\| \), and absolute error in two consecutive iterations \(\left\| {X}^{(n)}{-X} ^{(n-1)}\right\| \) that are given in Tables 3, 4, 5. For the sake of comparison, we consider the sixth-order methods given by Behl and Argyros (2020) [2], Kansal et al. (2021) [6], Lee and Kim (2020) [8], and Behl et al. (2019) [1], namely, BA, KC, LK, and BS, respectively.
Example 3.2
We take a \(3\times 3\) system \(F_{1}\left( X\right) \) of nonlinear equations from [6], such that
where
The exact solution for the system is \(\Upsilon =\left( 0.6982886,0.6285243,0.3425642\right) \). We choose an initial guess as \( X^{(0)}=\left( 1,1,1\right) \)
Example 3.3
Let us take a substance that is under observation in a bounded domain \( \Omega \in {\mathbb {R}} ^{2}\) with continuous boundary \(\partial \Omega \). The two-dimensional nonlinear diffusion–reaction equation for the concentration \(w\left( x,t\right) \) of the substance in a bounded domain is represented by an initial-boundary value problem [8]
where \(w=g\) on the boundary.
Here, \(\Delta \) is Laplacian operator, a is positive constant, and \(d>0\) is diffusion coefficient. Let us observe the concentration of substance in a unit square region, such that \(\Omega =\left[ 0,1\right] \times \left[ 0,1 \right] \) and let we take \(d=1\), \(a=1\). To get the steady state solution, Eq. (26) is converted in the following form:
with Dirichlet boundary conditions
We use central-divided difference formula by taking step-length \(h=1/4\) between the space components of the unit square region, and then, we discretize Eq. (27) into a system of nonlinear equations. This system consists of 25 nodes. Among these 25 nodes, 16 are boundary nodes and 9 nodes represent the interior nodal variables. We solve the system for the interior nodal variables say \(x_{1},x_{2},...x_{9.}\) The desired solution to the problem is
We take \(X^{\left( 0\right) }=\left( 1,1,1,1,1,1,1,1,1\right) ^{T}\) as an initial guess (Table 6).
Example 3.4
Consider the Van der Waals equation of state from [3]
in an interval [0, 2]. Let the boundary conditions are
We consider the following partition for the interval [0, 2]:
and assume that
The central finite-difference formula for the first and second-order derivative is given as
By substituting central-difference formula in Eq. (28), we obtain \(\left( m-1\right) \times \left( m-1\right) \) system of nonlinear equations of the following form:
We take \(\beta =\frac{1}{2}\) and \(X^{\left( 0\right) }=(-1,-2,-3,-4,-5,-6,-7,-8,-9)^{T}\), \(k=1,2,...,m-1.\) We consider \(m=10\) and solve the system of nine nonlinear equations.
4 Efficiency of the methods
Consider the efficiency index [12] (EI), EI \(=\) \(p^{\frac{1}{d}}\), where p represents the order and d represents the total number of functional evaluations. Moreover, the computational efficiency index (CE) [4] is characterized as \(CE=p^{\frac{1}{(d+op)}}\) where op is the operations cost per cycle. We have made comparisons of our scheme FS for EI and CE with the sixth-order methods given in Sect. 3, namely, BA, KC, LK, and BS.
5 Conclusion
We developed a new sixth-order scheme for the univariate as well as for the multidimensional case. The numerical results of our scheme compared with those of existing families of Jarratt-type methods show that our scheme performs better than the existing ones.
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Yaseen, S., Zafar, F. A new sixth-order Jarratt-type iterative method for systems of nonlinear equations. Arab. J. Math. 11, 585–599 (2022). https://doi.org/10.1007/s40065-022-00380-2
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DOI: https://doi.org/10.1007/s40065-022-00380-2