# 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}

**2013**:194

https://doi.org/10.1186/1029-242X-2013-194

© 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

## 1 Introduction

where *F* is a Fréchet-differentiable 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 time-invariant 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 two-step 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.

**Theorem 2.1**

*Let*$F:\mathcal{D}\subseteq \mathcal{X}\to \mathcal{Y}$

*be thrice differentiable*.

*Assume that there exist*${x}_{0}\in \mathcal{D}$, $L\ge 0$, $M\ge 0$, $N\ge 0$

*and*$\eta \ge 0$

*such that*

*and*

*where*${v}^{\star}$

*and*${v}^{\star \star}$

*are the zeros of functions*

*given by*

*Then the following hold*:

- (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)

*for each*$n\ge 0$

*are non*-

*decreasing and converge to their common limit*${v}^{\star}$,

*so that*

- (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*$n\ge 0$:

*where*

**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.

**Definition 3.1**Let $L\ge 0$, ${M}_{0}>0$, $M>0$, $N\ge 0$ and $\eta >0$ be given constants. Define the polynomials on $[0,+\mathrm{\infty})$ for some $\alpha >0$ by

*g*, ${g}_{1}$ have unique positive roots denoted by ${\varphi}_{{f}_{1}}$, ${\varphi}_{g}$ and ${\varphi}_{{g}_{1}}$ (given in an explicit form), respectively, by the Descartes rule of signs. Moreover, assume

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).

**Lemma 3.2**

*Under the*(C)

*conditions*,

*choose*

*Then the scalar sequences*$\{{s}_{n}\}$, $\{{t}_{n}\}$

*given by*

*are non*-

*decreasing*,

*bounded from above by*

*and converge to their unique least upper bound*${t}^{\star}\in [0,{t}^{\star \star}]$.

*Moreover*,

*the following estimate holds*:

*where*

*Proof*We show, using induction on

*k*, that

*α*. Moreover, the estimates (3.7) and (3.9) hold for $n=0$ by (3.5), the choice of ${\varphi}_{0}$ and (3.4). Let us assume (3.7)-(3.9) hold for all $k\le n$. We have in turn by the induction hypotheses:

*g*and its unique positive root ${\varphi}_{g}\in [0,1)$ are given in Definition 3.1. The estimate (3.11) is true if

This completes the induction for (3.4)-(3.9). It follows that the sequences $\{{s}_{n}\}$ and $\{{t}_{n}\}$ are non-decreasing, 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. □

*Under the hypotheses of Lemma*3.2,

*further assume*

*where*

*Fix*

*Define the parameters*${p}_{0}$,

*p*

*by*

*and a function*${g}_{3}$

*on*$[1,1/q)$

*by*

*Moreover*,

*assume*

*Then the following estimates hold for all*$k\ge 0$:

*Proof*We show

which completes the induction for (3.38). This completes the proof. □

**Theorem 3.4**

*Under the hypotheses*(3.1)-(3.5)

*and*(3.23),

*further assume that the hypotheses of Lemma*3.2

*hold and*

*Then the sequences*$\{{x}_{n}\}$

*and*$\{{y}_{n}\}$

*generated by*(JM)

*are well defined*,

*remain in*$\overline{U}(x,{t}^{\star})$

*for all*$n\ge 0$

*and converge to a unique solution*${x}^{\star}$

*of the equation*$F(x)=0$

*in*$\overline{U}(x,{t}^{\star})$.

*Moreover*,

*the following estimates hold*:

*Furthermore*,

*under the hypotheses of Proposition*3.3,

*the estimates*(3.33)

*also hold*.

*Finally*,

*if*$R\ge {t}^{\star}$

*such that*

*and*

*then the solution* ${x}^{\star}$ *is unique in* $U(x,R)$.

**Example 3.5**Let $\mathcal{X}=\mathcal{Y}={\mathbb{R}}^{2}$, $\mathcal{D}={[1,3]}^{2}$, ${x}_{0}={(2,2)}^{T}$ and define a function

*F*on $\mathcal{D}$ by

**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**

From Table 1, we observe the following:

⧫ The sequences $\{{t}_{n}\}$ and $\{{s}_{n}\}$ are non-decreasing.

⧫ 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**

**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 $F(x)=0$.

**Remarks 3.6**[1, 6, 7]

- (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)).

*δ*seems to be $\delta ={\delta}_{3}$. Let

- (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 $\eta >0$.

*α*and

*β*, we have

Note that the *p*-Jarratt-type 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 third-order 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: 2012-0008170).

## Authors’ Affiliations

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