 Research
 Open Access
On a new semilocal convergence analysis for the Jarratt method
 Ioannis K Argyros^{1},
 Yeol Je Cho^{2}Email author and
 Sanjay Kumar Khattri^{3}
https://doi.org/10.1186/1029242X2013194
© Argyros et al.; licensee Springer 2013
 Received: 29 November 2012
 Accepted: 7 April 2013
 Published: 22 April 2013
Abstract
We develop a new semilocal convergence analysis for the Jarratt method. Through our new idea of recurrent functions, we develop new sufficient convergence conditions and tighter error bounds. Numerical examples are also provided in this study.
MSC:65H10, 65G99, 65J15, 47H17, 49M15.
Keywords
 Jarratt method
 Newtontype methods
 Banach space
 Fréchetderivative
 majorizing sequence
 recurrent functions
1 Introduction
where F is a Fréchetdifferentiable operator defined on a convex subset $\mathcal{D}$ of a Banach space $\mathcal{X}$ with values in a Banach space $\mathcal{Y}$.
A large number of problems in applied mathematics and also in engineering are solved by finding the solutions of certain equations. For example, dynamic systems are mathematically modeled by difference or differential equations and their solutions usually represent the states of the systems. For the sake of simplicity, assume that a timeinvariant system is driven by the equation $\dot{x}=Q(x)$ for some suitable operator Q, where x is the state. Then the equilibrium states are determined by solving equation (1.1). Similar equations are used in the case of discrete systems. The unknowns of engineering equations can be functions (difference, differential and integral equations), vectors (systems of linear or nonlinear algebraic equations) or real or complex numbers (single algebraic equations with single unknowns). Except in special cases, the most commonly used solution methods are iterative  when starting from one or several initial approximations, a sequence is constructed that converges to a solution of the equation. Iteration methods are also applied for solving optimization problems. In such cases, the iteration sequences converge to an optimal solution of the problem at hand. Since all of these methods have the same recursive structure, they can be introduced and discussed in a general framework.
for each $n\ge 0$. The fourth order of (JM) is the same as that of a twostep Newton method [1]. But the computational cost is less than that of Newton’s method. In each step, we save one evaluation of the derivative and the computation of one inverse.
Here, we use our new idea of recurrent functions in order to provide new sufficient convergence conditions, which can be weaker than before [4]. Using this approach, the error bounds and the example on the distances are improved (see Example 3.5 and Remarks 3.6). This new idea can be used on other iterative methods [1].
2 Semilocal convergence analysis of (JM)
We present our Theorem 2.1 in [4] in an affine invariant form since ${F}^{\prime}{({x}_{0})}^{1}F$ can be used for F in the original proof of Theorem 2.1.
 (1)The scalar sequences $\{{v}_{n}\}$ and $\{{w}_{n}\}$ given by$\begin{array}{l}{w}_{n}={v}_{n}{g}^{\mathrm{\prime}}{({v}_{n})}^{1}g({v}_{n}),\\ {b}_{n}={g}^{\mathrm{\prime}}{({v}_{n})}^{1}({g}^{\mathrm{\prime}}({v}_{n}+\frac{2}{3}({w}_{n}{v}_{n})){g}^{\mathrm{\prime}}({v}_{n})),\\ {v}_{n+1}={w}_{n}\frac{3}{4}{b}_{n}(1\frac{3}{2}{b}_{n})({w}_{n}{v}_{n})\end{array}\}$(2.11)
 (2)The sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ generated by (JM) are well defined, remain in $\overline{U}({x}_{0},{v}^{\star})$ for all $n\ge 0$ and converge to a unique solution ${x}^{\star}\in \overline{U}({x}_{0},{v}^{\star})$ of the equation $F(x)=0$, which is the unique solution of the equation $F(x)=0$ in $U({x}_{0},{v}^{\star \star})$. Moreover, the following estimates hold for all $n\ge 0$:
Remarks 2.2 The bounds of Theorem 2.1 can be improved under the same hypotheses and computational cost in two cases as follows.
for all $x\in \mathcal{D}$. We can find upper bounds on the norms $\parallel {F}^{\prime}{(x)}^{1}{F}^{\prime}({x}_{0})\parallel $ using ${M}_{0}$, which is actually needed, and not K used in [4].
Note also that if ${M}_{0}=K$, then ${\overline{v}}_{n}={v}_{n}$, ${\overline{w}}_{n}={w}_{n}$.
Case 2. In view of the upper bound for $\parallel F({x}_{n+1})\parallel $ obtained in Theorem 2.1 in [4] and (2.21), $\{{t}_{n}\}$, $\{{s}_{n}\}$ given in (3.9) and (3.10) are also even more precise majorizing sequences for $\{{x}_{n}\}$ and $\{{y}_{n}\}$. Therefore, if they converge under certain conditions (see Lemma 3.2), then we can produce a new semilocal convergence theorem for (JM) with sufficient convergence conditions or bounds that can be better than the ones of Theorem 2.1 (see also Theorem 3.4 and Example 3.5).
Similar favorable comparisons (due to (2.20)) can be made with other iterative methods of the fourth order [1, 11].
3 Semilocal convergence analysis of (JM) using recurrent functions
We show the semilocal convergence of (JM) using recurrent functions. First, we need the following definition.
and the polynomial ${h}_{1}$ has a unique positive root ${\varphi}_{{h}_{1}}$.
If ${\varphi}_{1}=1$, then assume that (3.3) holds as a strict inequality. From now on (3.1)(3.3) constitute the (C) conditions.
We can show the following result on the majorizing sequences for (JM).
This completes the induction for (3.4)(3.9). It follows that the sequences $\{{s}_{n}\}$ and $\{{t}_{n}\}$ are nondecreasing, bounded from above by ${t}^{\star \star}$ given in a closed form by (3.6) and converge to their unique least upper bound ${t}^{\star}\in [0,{t}^{\star \star}]$. This completes the proof. □
which completes the induction for (3.38). This completes the proof. □
then the solution ${x}^{\star}$ is unique in $U(x,R)$.
The scalar sequences $\mathbf{\{}{\mathit{s}}_{\mathit{n}}\mathbf{\}}$ and $\mathbf{\{}{\mathit{t}}_{\mathit{n}}\mathbf{\}}$ are given by equation ( 3.5 ) in Lemma 3.2
n  ${\mathit{t}}_{\mathit{n}}$  ${\mathit{s}}_{\mathit{n}}$  ${\mathit{s}}_{\mathit{n}}\mathbf{}{\mathit{t}}_{\mathit{n}}$  $\mathit{\varphi}\mathbf{(}{\mathit{s}}_{\mathit{n}}\mathbf{}{\mathit{t}}_{\mathit{n}}\mathbf{)}$ 

0 




1 




2 




3 




4 




5 




6 




From Table 1, we observe the following:
⧫ The sequences $\{{t}_{n}\}$ and $\{{s}_{n}\}$ are nondecreasing.
⧫ The sequences $\{{t}_{n}\}$ and $\{{s}_{n}\}$ are bounded from above by ${t}^{\star \star}$.
⧫ The estimate (3.7) holds.
The scalar sequences $\mathbf{\{}{\mathit{w}}_{\mathit{n}}\mathbf{\}}$ and $\mathbf{\{}{\mathit{v}}_{\mathit{n}}\mathbf{\}}$ are given by equation ( 2.11 ) in Theorem 2.1
n  ${\mathit{w}}_{\mathit{n}}$  ${\mathit{v}}_{\mathit{n}}$  ${\mathit{w}}_{\mathit{n}}\mathbf{}{\mathit{v}}_{\mathit{n}}$ 

0 



1 



2 



3 



4 



5 



6 



7 



8 



9 



To validate the estimate ( 3.33 ) of Proposition 3.3
n  ${\mathit{s}}_{\mathit{n}}\mathbf{}{\mathit{t}}_{\mathit{n}}$  ${\mathbf{(}\mathit{t}}_{\mathit{n}\mathbf{+}\mathbf{1}}\mathbf{}{\mathit{s}}_{\mathit{n}}\mathbf{)}$  $\frac{\mathit{p}}{{\mathit{q}}^{\mathbf{2}}}\sqrt{{\mathbf{(}\mathit{q}\mathit{\eta}\mathbf{)}}^{{\mathbf{4}}^{\mathit{k}\mathbf{+}\mathbf{1}}}}$  $\frac{\mathbf{1}}{\mathit{q}}{\mathbf{(}\mathit{q}\mathit{\eta}\mathbf{)}}^{{\mathbf{4}}^{\mathit{k}}}$ 

0 




1 




2 




3 




4 




5 




6 




7 




8 




9 




10 




In Table 3, we observe that the estimates (3.33) are also true. Hence the conclusions of Proposition 3.3 also hold for the equation $F(x)=0$.
 (1)The condition (3.32) can be replaced by a stronger, but easier to check$\frac{2\eta}{2\delta}\le {p}_{0},$(3.44)
for $\delta \in I$ (see (3.13) and (3.21)).
 (2)
The ratio of convergence ‘qη’ given in Proposition 3.3 can be smaller than ‘$\sqrt[3]{5}\theta $’ given in Theorem 2.1 for q close to $\sqrt[3]{b}$ and M, N, L not all zero and $\eta >0$.
Note that the pJarratttype method ($p\in [0,1]$) given in [8] uses (2.1)(2.5), but the sufficient convergence conditions are different from the ones given in the study and guarantees only thirdorder convergence (not fourth obtained here) in the case of the Jarratt method (for $p=2/3$).
4 Conclusions
We developed a semilocal convergence analysis, using recurrent functions, for the Jarratt method to approximate a locally unique solution of a nonlinear equation in a Banach space. A numerical example and some favorable comparisons with previous works are also reported.
Declarations
Acknowledgements
The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 20120008170).
Authors’ Affiliations
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