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On a new semilocal convergence analysis for the Jarratt method
Journal of Inequalities and Applications volume 2013, Article number: 194 (2013)
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.
In this study, we are concerned with the problem of approximating a locally unique solution of the equation
where F is a Fréchet-differentiable operator defined on a convex subset of a Banach space with values in a Banach space .
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 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.
Many authors have developed high order methods for generating a sequence approximating . A survey of such results can be found in [, and the references there] (see also [2–11]). The natural generalization of the Newton method is to apply a multipoint scheme. Suppose that we know the analytic expressions of , and at a recurrent step for each . In order to increase the order of convergence and to avoid the computation of the second Fréchet-derivative, we can add one more evaluation of or , where and are real constants that are independent of and , whereas is generated by a Newton-step. A two-point scheme for functions of one variable was found and developed by Ostrowski . Following this idea, we provide a semilocal as well as a local convergence analysis for a fourth-order inverse free Jarratt-type method (JM) [1, 4] given by
for each . The fourth order of (JM) is the same as that of a two-step Newton method . 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 . 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 .
2 Semilocal convergence analysis of (JM)
We present our Theorem 2.1 in  in an affine invariant form since can be used for F in the original proof of Theorem 2.1.
Theorem 2.1 Let be thrice differentiable. Assume that there exist , , , and such that
for each ,
where and are the zeros of functions
Then the following hold:
The scalar sequences and given by(2.11)
for each are non-decreasing and converge to their common limit , so that
The sequences and generated by (JM) are well defined, remain in for all and converge to a unique solution of the equation , which is the unique solution of the equation in . Moreover, the following estimates hold for all :(2.13)
Remarks 2.2 The bounds of Theorem 2.1 can be improved under the same hypotheses and computational cost in two cases as follows.
Case 1. Define a function by
In view of (2.2), there exists such that
for all . We can find upper bounds on the norms using , which is actually needed, and not K used in .
It follows from (2.21) and the Banach lemma on invertible operators  that exists and
We can use (2.21) instead of the less precise one used in :
This suggests that more precise scalar majorizing sequences , can be used and they are defined as follows for initial iterates , :
A simple induction argument shows that, if , then
Note also that if , then , .
Case 2. In view of the upper bound for obtained in Theorem 2.1 in  and (2.21), , given in (3.9) and (3.10) are also even more precise majorizing sequences for and . 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).
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 , , , and be given constants. Define the polynomials on for some by
Moreover, define a scalar by
The polynomials , g, have unique positive roots denoted by , and (given in an explicit form), respectively, by the Descartes rule of signs. Moreover, assume
Under the conditions (3.1), (3.2), respectively,
and the polynomial has a unique positive root .
Set . Furthermore, assume
If , 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 , given by
are non-decreasing, bounded from above by
and converge to their unique least upper bound . Moreover, the following estimate holds:
Proof We show, using induction on k, that
The estimate (3.8) holds for by the choice of α. Moreover, the estimates (3.7) and (3.9) hold for by (3.5), the choice of and (3.4). Let us assume (3.7)-(3.9) hold for all . We have in turn by the induction hypotheses:
Hence, instead of (3.9), we can show
The estimate (3.8) can be written as
So, we can show, instead of (3.8),
The estimate (3.11) motivates us to define polynomials on (for ) by
or, since for , define the polynomials on by
We need a relationship between two consecutive polynomials :
where g and its unique positive root are given in Definition 3.1. The estimate (3.11) is true if
since by (3.14) we have
But (3.16) is true by the definition of and (3.4). Define
Then we also have
This completes the induction for (3.8). The estimate (3.10) is true if
The estimate (3.20) motivates us to define polynomials on by
We need a relationship between two consecutive polynomials :
where and the unique positive root are given in Definition 3.1. The estimate (3.20) is true if
But (3.24) is true by the definition of and (3.4). Define a function on by
Then we have
This completes the induction for (3.4)-(3.9). It follows that the sequences and are non-decreasing, bounded from above by given in a closed form by (3.6) and converge to their unique least upper bound . This completes the proof. □
Under the hypotheses of Lemma 3.2, further assume
Define the parameters , p by
and a function on by
Then the following estimates hold for all :
Proof We show
If the estimate (3.34) holds, then we have
which implies the second equation in (3.33). We have the estimate
that is, we have
Instead of showing (3.34), we can show
By the hypothesis (3.32), we have
We also have
We get in turn
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 and generated by (JM) are well defined, remain in for all and converge to a unique solution of the equation in . Moreover, the following estimates hold:
Furthermore, under the hypotheses of Proposition 3.3, the estimates (3.33) also hold. Finally, if such that
then the solution is unique in .
Example 3.5 Let , , and define a function F on by
Using (2.1)-(2.7), we obtain
Hence the conclusions of Theorem 2.1 hold for the equation . Considering the hypotheses of Theorem 3.4, from Lemma 3.2, we have
and, from Definition 3.1, we get
Consequently, from the definition of (see Definition 3.1), we obtain
and from the definition of (see Definition 3.1), we obtain
From the equation (3.6),
From Table 1, we observe the following:
⧫ The sequences and are non-decreasing.
⧫ The sequences and are bounded from above by .
⧫ The estimate (3.7) holds.
Let us now compare the bounds between Theorems 2.1 and 3.4. From equation (2.10), we get
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 .
The condition (3.32) can be replaced by a stronger, but easier to check(3.44)
for (see (3.13) and (3.21)).
The best possible choice for δ seems to be . Let
In this case, (3.44) is written as
The ratio of convergence ‘qη’ given in Proposition 3.3 can be smaller than ‘’ given in Theorem 2.1 for q close to and M, N, L not all zero and .
Set and . Note that and . By comparing α and β, we have
Case 1. If or and , then we have
Case 2. If and , then we have
Case 3. If , then we have
Note that the p-Jarratt-type method () given in  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 ).
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.
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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).
The authors declare that they have no competing interests.
All authors jointly worked on the results and they read and approved the final manuscript.