Some multi-step iterative methods for solving nonlinear equations-I

1. Introduction

To find the root of nonlinear equation of the form \(f(x)=0,\) is the oldest and basic problem in numerical analysis. Newton's method is one of oldest and most powerful formula to approximate the root of nonlinear equations. It has second order convergence. Many modifications in Newton's method have been made to increase its convergence order using various techniques. Recently, many iterative methods with higher order convergence have been established using different techniques like Taylor's series, Adomain decomposition, homotopy, modified homotopy, decomposition, modified decomposition, interpolation, quadrature rules and their many modifications. see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and references therein. Chun [3] introduced some multi-step iterative methods using Adomain decomposition. These method requires higher order derivatives. Later on, Noor [4, 5] established some multi-step iterative methods that do not require higher order derivative using a different technique.

In this paper, some numerical methods based on decomposition technique are proposed for solving algebraic nonlinear equations. For this purpose, we write nonlinear equations as an equivalent system of equations and use technique introduced by Gejji and Jafari [9]. Recently, several iterative methods have been suggested by writing nonlinear equations as an equivalent system of equations and then using different techniques. In section 3, we give detailed proof regarding convergence of our newly established iterative methods. In the last section, numerical results are given to make comparison of these algorithm with some classical methods.

2. Iterative methods

Consider the nonlinear equation assume \(\alpha\) is a simple root of (1) and \(\gamma\) is an initial guess sufficiently close to \(\alpha\). We can write (1) as following coupled system; From Eq. (2), we have \[ x=\gamma -\frac{f(\gamma )}{f^{\prime }(\frac{x+\gamma }{2})}-\frac{g(x)}{% f^{\prime }(\frac{x+\gamma }{2})}. \] Let where and Now, we construct a sequence of higher-order iterative methods by applying modified decomposition technique introduced by Gejji and Jafari [8]. This technique consists a solution of equation (3) that can be written in the form of infinite series: \noindent and using Gejji and Jafari [8] technique, we decompose the nonlinear function \(N\) as; Thus from Eqs. (4), (7) and (8) we have \[ \sum\limits_{i=0}^{\infty }x_{i}=x_{0}+N(x_{0})+\sum_{i=1}^{\infty }\{N(\sum\limits_{j=0}^{i}x_{j})-N(\sum\limits_{j=0}^{i-1}x_{j})\}. \] So we have iteration scheme as \begin{eqnarray*} x_{0} &=&c \\ x_{1} &=&N(x_{0}) \\ x_{2} &=&N(x_{0}+x_{1})-N(x_{0}) \\ &&. \\ &&. \\ &&. \\ x_{n+1} &=&N(x_{0}+x_{1}+...+x_{n})-N(x_{0}+x_{1}+...+x_{n-1}),\text{ }% n=1,2,.. \end{eqnarray*} When \begin{eqnarray*} x &\approx &x_{0} \\ &=&c \\ &=&\gamma -\frac{f(\gamma )}{f^{\prime }(\frac{x+\gamma }{2})}. \end{eqnarray*} From above relation, we can write the algorithm as follows;
Algorithm 2.1: For any initial value \(x_{0},\) we compute the approximation solution \(x_{n+1}\), by the iteration scheme; \begin{eqnarray*} y_{n} &=&x_{n}-\frac{f(x_{n})}{f^{\prime }(x_{n})} \\ x_{n+1} &=&x_{n}-\frac{f(x_{n})}{f^{\prime }(\frac{x_{n}+y_{n}}{2})}. \end{eqnarray*} This algorithm has third order convergence and has been established by Frontini and Sormani [1]. When \begin{eqnarray*} x &\approx &x_{0}+x_{1} \\ &=&\gamma -\frac{f(\gamma )}{f^{\prime }(\frac{x+\gamma }{2})}-\frac{g(x_{0}) }{f^{\prime }(\frac{x+\gamma }{2})}. \end{eqnarray*} From Eq. (3), we see that \[ g(x_{0})=f(x_{0}), \] by substituting in above, we get \[ x=\gamma -\frac{f(\gamma )}{f^{\prime }(\frac{x+\gamma }{2})}-\frac{f(x_{0}) }{f^{\prime }(\frac{x_{0}+\gamma }{2})}, \] Using this, we suggest the following three step iterative methods for solving Eq. (1) as follows;
Algorithm 2.2: For any initial value \(x_{0},\) we compute the approximation solution \(x_{n+1}\), by the iteration scheme; \begin{eqnarray*} \text{ Predictor Steps;}\;\;\;\; y_{n} &=&x_{n}-\frac{f(x_{n})}{% f^{\prime }(x_{n})} \\ z_{n} &=&x_{n}-\frac{f(x_{n})}{f^{\prime }(\frac{x_{n}+y_{n}}{2})} \end{eqnarray*} \[ \text{Corrector Step; }\;\;\;\; x_{n+1}=z_{n}-\frac{f(z_{n})}{% f^{\prime }(\frac{x_{n}+z_{n}}{2})} \] When \begin{eqnarray*} x &\approx &x_{0}+x_{1}+x_{2} \\ &=&x_{0}+N(x_{0}+x_{1}) \\ &=&x_{0}-\frac{g(x_{0}+x_{1})}{f^{\prime }(\frac{x_{0}+x_{1}+\gamma }{2})} \\ &=&x_{0}-\frac{f(x_{0}+x_{1})+f(x_{0})}{f^{\prime }(\frac{x_{0}+x_{1}+\gamma }{2})} \end{eqnarray*} From this relation, we formulate four step iterative method for solving Eq. (1) as follows;
Algorithm 2.3: For any initial value \(x_{0},\) we compute the approximation solution \(x_{n+1}\), by the iteration scheme; \begin{eqnarray*} \text{ Predictor Steps;} \;\;\;\; y_{n} &=&x_{n}-\frac{f(x_{n})}{% f^{\prime }(x_{n})} \\ z_{n} &=&x_{n}-\frac{f(x_{n})}{f^{\prime }(\frac{x_{n}+y_{n}}{2})} \\ u_{n} &=&z_{n}-\frac{f(z_{n})}{f^{\prime }(\frac{x_{n}+z_{n}}{2})} \end{eqnarray*} \[ \text{Corrector Step;} \;\;\;\; x_{n+1}=z_{n}-\frac{f(u_{n})+f(z_{n})% }{f^{\prime }(\frac{x_{n}+u_{n}}{2})} \]

3. Convergence Analysis

In this section, we discuss in detail the convergence analysis of Algorithm 2.2 and 2.3 established above.

Theorem 3.1. For an open interval \(I \subset \mathbb{R}\) let \(f:I \rightarrow\mathbb{R}\) and \(\alpha \in I\) be simple zero of \(f(x)=0.\) If \(f\) is differentiable and \(x_{0}\) is sufficiently close to \(\alpha\) then three-step iterative method defined by Algorithm 2.2 has fourth order convergence.

Proof. Expanding \(f(x)\) by Taylor's series about \(\alpha ,\) we have

where \(c_{k}=\frac{1}{k!}\frac{f^{(k)}(\alpha )}{f^{^{\prime }}(\alpha )}\), and \(e_{n}=x_{n}-\alpha .\) \[ y_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime }(x_{n})} \] From above equations,we get Now, expanding \(f^{\prime }(\frac{x_{n}+y_{n}}{2})\) by Taylor's series about \(\alpha\), Dividing Eq. (9) by Eq. (12) Now \[ z_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime }(\frac{x_{n}+y_{n}}{2})} \] Putting values in above, we get Expanding \(f(z_{n})\) and \(f^{\prime }(\frac{x_{n}+z_{n}}{2})\) by Taylor's series about \(\alpha,\) Dividing (15) by Eq. (16), we have \begin{align*} \frac{f(z_{n})}{f^{\prime }(\frac{x_{n}+z_{n}}{2})} &=(c_{2}^{2}-\frac{1}{4}% c_{3})e_{n}^{3}+(4c_{2}c_{3}-4c_{2}^{3}-\frac{1}{2}c_{4})e_{n}^{4}+(\frac{27% }{8}c_{3}^{2}-\frac{73}{4}c_{3}c_{2}^{2}\\&+\frac{5}{2}c_{2}c_{4}+ 10c_{2}^{4}-\frac{11}{16}c_{5})e_{n}^{5}+... \end{align*} Corrector step of Algorithm 2.2 is \[ x_{n+1}=z_{n}-\frac{f(z_{n})}{f^{\prime }(\frac{x_{n}+z_{n}}{2})} \] By substitution and simplification, we have Hence Algorithm 2.2 has fourth order convergence.

Theorem 3.2. Let \(f:I\subset \mathbb{R}\rightarrow\mathbb{R}\) for an open interval \(I\) and \(\alpha \in I\) be simple zero of \(f(x)=0.\) If \(f\) is differentiable and \(x_{0}\) is sufficiently close to \(\alpha\) then three-step iterative method defined by Algorithm 2.3 has fifth order convergence and satisfies the error equation \[ x_{n+1}=\alpha +(-\frac{1}{4}c_{3}c_{2}^{2}+c_{2}^{4})e_{n}^{5}+O(e_{n}^{6}),% \text{ and }e_{n}=x_{n}-\alpha. \]

Proof. From Eq. (17), we have \[ u_{n}=\alpha +(4c_{2}c_{3}-4c_{2}^{3}-\frac{1}{2}c_{4})e_{n}^{4}+(\frac{27}{8% }c_{3}^{2}-\frac{73}{4}c_{3}c_{2}^{2}+\frac{5}{2}c_{2}c_{4}+10c_{2}^{4}-% \frac{11}{16}c_{5})e_{n}^{5}+..., \] Expanding \(f(u_{n})\) and \(f^{\prime }(\frac{x_{n}+u_{n}}{2})\) by Taylor's series about \(\alpha ,\)

Now \[ x_{n+1}=z_{n}-\frac{f(u_{n})+f(z_{n})}{f^{\prime }(\frac{x_{n}+u_{n}}{2})}, \] by substituting values and simplifying, we have \[ x_{n+1}=\alpha +(-\frac{1}{4}c_{3}c_{2}^{2}+c_{2}^{4})e_{n}^{5}+O(e_{n}^{6}). \] Hence Algorithm 2.3 has fifth order convergence.

4. Applications

In this section, we present some numerical examples to examine the validity and efficieny of our newly devolped algorithms namely, Algorithm 2.2 and Algorithm 2.3. We also provide comparison of these algorithms with Newton's method(NM), Abbasbandy's method (AM), Householder's method(HHM), Halley's method (HM), Chun's method (CM) [3] and Noor's method(NR)[4]. We use \(\epsilon =10^{-25}\). We use the following stopping criteria; \vskip-1cm \begin{eqnarray*} (i)\text{}\;\;\;\; |x_{n}-x_{n-1}| &<&\in \\ (ii)\text{}\;\;\;\;|f(x_{n})| &<&\in \end{eqnarray*} We consider the following nonlinear scalar equations for comparison: \begin{eqnarray*} f_{1}(x) &=&\sin ^{2}x-x^{2}+1=0 \\ f_{2}(x) &=&x^{2}-e^{x}-3x+2=0 \\ f_{3}(x) &=&\cos x-x=0 \\ f_{4}(x) &=&(x-1)^{3}-1=0 \\ f_{5}(x) &=&x^{3}-10=0 \\ f_{6}(x) &=&e^{x^{2}+7x-30}-1=0. \end{eqnarray*} Comparison Table \begin{array}{ccccc} \text{Examples} & & \text{Iterations} & x_{n} & f(x_{n}) \\ f_{1},\text{ }x_{0}=1 & & & & \\ NM & & 7 & 1.40449164831534122635086 & -1.05e^{-50} \\ AM & & 5 & 1.40449164831534122635086 & -5.82e^{-54} \\ HM & & 4 & 1.40449164831534122635086 & -5.45e^{-63} \\ CM & & 5 & 1.40449164831534122635086 & -2.01e^{-64} \\ HHM & & 6 & 1.40449164821534122603508 & 1.82e^{-25} \\ NR & & 5 & 1.40449164821534122603508 & 9.23e^{-26} \\ \text{Alg. }2.2 & & 3 & 1.40449164821534122603508 & 3.40e^{-25} \\ \text{Alg. }2.3 & & 3 & 1.40449164821534122603509 & 2.32e^{-44} \\ f_{2},\text{ }x_{0}=2 & & & & \\ NM & & 6 & 0.257530285439860760455367 & 2.94e^{-55} \\ AM & & 5 & 0.257530285439860760455367 & 1.03e^{-63} \\ HM & & 5 & 0.257530285439860760455367 & 0 \\ CM & & 4 & 0.257530285439860760455367 & 1.05e^{-62} \\ HHM & & 5 & 0.257530285439860760455367 & -6.01e^{-25} \\ NR & & 5 & 0.257530285439860760455367 & 1.08e^{-23} \\ \text{Alg. }2.2 & & 3 & 0.257530285439860934975933 & 1.72e^{-69} \\ \text{Alg}.2.3 & & 2 & 0.257530285439860934975933 & 1.01e^{-29} \\ \end{array} \begin{array}{ccccc} \text{Examples} & & \text{Iterations} & x_{n} & f(x_{n}) \\ f_{3},\text{ }x_{0}=1.7 & & & & \\ NM & & 5 & 0.739085133215160641655372 & -2.04e^{-31} \\ AM & & 4 & 0.739085133215160641655372 & -7.13e^{-47} \\ HM & & 4 & 0.739085133215160641655372 & -5.04e^{-58} \\ CM & & 4 & 0.739085133215160641655372 & 0 \\ HHM & & 4 & 0.739085133215160641655372 & 3.91e^{-17} \\ NR & & 4 & 0.739085133215160645628855 & 6.66e^{-18} \\ \text{Alg. }2.2 & & 3 & 0.739085133215160641655312 & 3.21e^{-50} \\ \text{Alg.}2.3 & & 3 & 0.739085133215160641655312 & 1.21e^{-100} \\ f_{4},\text{ }x_{0}=1.5 & & & & \\ NM & & 8 & 2.0000000000000000000001234 & 2.05e^{-42} \\ AM & & 5 & 2 & 0 \\ HM & & 5 & 2 & 0 \\ CM & & 5 & 2 & 0 \\ HHM & & 7 & 2.000000000000000000000828 & 2.47e^{-22} \\ NR & & 5 & 2.000000000000000000000302 & 1.12e^{-24} \\ \text{Alg. }2.2 & & 4 & 2.000000000000000000000007 & 1.83e^{-44} \\ \text{Alg. }2.3 & & 4 & 2.000000000000000000000000 & 5.07e^{-27}\\ f_{5},\text{ }x_{0}=1.5 & & & & \\ N.M. & & 7 & 2.154434690031883721759235 & 2.05e^{-54} \\ A.M. & & 5 & 2.154434690031883721759235 & -5.01e^{-63} \\ H.M. & & 5 & 2.154434690031883721759235 & -5.03e^{-62} \\ C.M. & & 5 & 2.154434690031883721759235 & -5.01e^{-64} \\ H.H.M. & & 6 & 2.154434690031883721759293 & 7.86e^{-27} \\ NR.M. & & 4 & 2.154434690031883721759292 & 8.98e^{-24} \\ \text{Alg. }2.2 & & 3 & 2.154434690031883721759294 & 2.00e^{-27} \\ \text{Alg. }2.3 & & 3 & 2.154434690031883721759293 & 1.33e^{-50} \\ f_{6,}\text{ }x_{0}=3.5 & & & & \\ N.M. & & 13 & 3.000000000000000000000000 & 1.52e^{-48} \\ A.M. & & 7 & 3.000000000000000000000000 & -4.32e^{-48} \\ H.M. & & 8 & 3.000000000000000000000000 & 2.01e^{-62} \\ C.M. & & 8 & 3.000000000000000000000000 & 2.05e^{-62} \\ H.H.M. & & 12 & 3.000000000000000000000000 & 5.47e^{-25} \\ NR.M. & & 6 & 3.00000010285594873990664 & 1.34e^{-06} \\ \text{Alg. }2.2 & & 4 & 3.00000000000000000000000 & 4.76e^{-30} \\ \text{Alg. }2.3 & & 4 & 3.00000000000000000000000 & 1.99e^{-102}% \end{array}

5. Conclusion

We have established two new algorithms of fourth and fifth order convergence by using modified decomposition technique to coupled system. We have discussed convergence analysis of these newly established algorithms. Computational comparison of these algorithms has been made with some well known methods for solving nonlinear equations. From numerical table, we see that our new methods converge fast to the true solution of equations and are comparable with some classical methods.

Competing Interests

The author(s) do not have any competing interests in the manuscript.