On the Rate of Convergence of Some New Modified Iterative Schemes ()

Renu Chugh, Sanjay Kumar

Department of Mathematics, M.D.U, Rohtak, India.

**DOI: **10.4236/ajcm.2013.34037
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Department of Mathematics, M.D.U, Rohtak, India.

In this article, following Bizare and Amriteimoori [1] and B. Parsad and R. Sahni [2], we modify Ishikawa, Agarwal *et al*., Noor, SP iterative schemes and compare the rate of convergence of Ishikawa, Agarwal *et al*., Noor, SP and new modified Ishikawa, Agarwal *et al.*, Noor, SP iterative schemes not only for particular fixed value of a_{n},b_{n},r_{n but also for varying the value of an,bn,rn. With the help of two numerical examples, we compare the converging step. }

Keywords

Metric Space; New Modified Ishikawa; New Modified Agarwal et al.; New Modified SP; New Modified Noor

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Chugh, R. and Kumar, S. (2013) On the Rate of Convergence of Some New Modified Iterative Schemes. *American Journal of Computational Mathematics*, **3**, 270-290. doi: 10.4236/ajcm.2013.34037.

1. Introduction

Let X be a complete metric space and T be self map, then is called set of fixed points of T. Now in literature, there are several iteration processes to find the fixed point of any equation. In complete metric space, Picard iteration process is defined as

which is used to approximate fixed points of mappings satisfying the condition

called Banach contraction condition.

In Ishikawa iteration process [3], is defined as

where are real sequences in [0,1].

Now for Agarwal et al. iteration [4], is defined as

where are real sequences in [0,1].

M. A. Noor defines [5] as

where are real sequences in [0,1].

For SP iteration [6] is defined as

where are real sequences in [0,1].

2. Preliminaries

In this paper following Bizare and Amriteimoori [1] and B. Parsad and R. Sahni [2] we prove the basic results in sequel. In [1] Bizare and Amriteimoori improved the picard iteration under following conditions:

1) Initial approximation is chosen in the interval [a, b], where function is defined.

2) Function has continuous derivative on (a, b).

3) for all

4) for all

**Definition 2.1 [1].** Let converges to α. If there exists an integer constant q and a real +ve constant C such that

q is called order and C is called constant of convergence

**Theorem 2.2([1,7]).** Let, if for and then sequence is of order q.

To improve the order of convergence of fixed iterative schemes, such that

. We determines from the following equation

which becomes

this is fixed point equation form .Now the assumption that yields to a system of linear equations which after solving [1] converted into upper triangular matrix which have nonzero diagonal entries. It means determinant is nonzero. So we determine uniquely.

Now the new Picard iteration becomes

(1)

where

(2)

Following Bhagwati Parsad and Ritu Shani [2] the new modified Ishikawa, Agarwal et al., Noor, SP iterations are defined as:

New modified Ishikawa iteration scheme

(3)

New modified Agarwal et al. iteration scheme

(4)

New modified Noor iteration scheme

(5)

New modified SP iteration scheme

(6)

where and are real sequences in [0, 1].

In this article, we compare the rate of convergence of new modified iterative schemes and simple iterative schemes with the help of the following examples

(7)

(8)

To find the fixed point we write and as

and

both equations has unique root in the interval (0,1) so we convert this in the fixed point form

and

and take respectively. Now we solve it by

For respective value of α, can be determined uniquely from system of linear equations as in

[5] for α = 0.5 and we have

(9)

After solving the system of linear equations we have

for second polynomial equation where α=0.35 and

System of linear equations become

(10)

Hence we get

3. Experiments

Now using the value of and iterative schemes we have following **Tables 1-24**, and**Figures 1-16**.

We take

Table 1. Simple Ishikawa for.

Table 2. Modified Ishikawa for.

Table 3. Simple Agarwal et al. for.

Table 4. Modified Agarwal et al. for.

Table 5. Simple SP for.

Table 6. Modified SP for.

Table 7. Simple Noor for.

Table 8. Modified Noor for.

(a)(b)(c)(d)

Figure 1. Graphical observations of simple Ishikawa iteration for. Here (a)-(d) show the graph for Table 1. The merging point with value 0.412391 is fixed point.

Figure 2. Graphical observations of new modified Ishikawa iteration for. Here (a)-(d) show the graph for Table 2. The merging point with value 0.412391 is fixed point.

Figure 3. Graphical observations of simple Agarwal et al. iteration for. Here (a)-(d) show the graph for Table 3. The merging point with value 0.412391 is fixed point.

Figure 4. Graphical observations of new modified Agarwal et al. iteration for. Here (a)-(d) show the graph for Table 4. The merging point with value 0.412391 is fixed point.

Figure 5. Graphical observations of simple SP iteration for. Here Here (a)-(d) show the graph for Table 5. The merging point with value 0.412391 is fixed point.

Figure 6. Graphical observations of new modified SP iteration for. Here (a)-(d) show the graph for Table 6. The merging point with value 0.412391 is fixed point.

Figure 7. Graphical observations of simple Noor iteration for. Here (a)-(d) show the graph for Table 7. The merging point with value 0 .412391 is fixed point.

Figure 8. Graphical observations of new modified Noor iteration for. Here (a)-(d) show the graph for Table 8. The merging point with value 0.412391 is fixed point.

Table 9. Simple Ishikawa for.

Table 10. Modified Ishikawa for.

Table 11. Simple Agarwal et al. for.

Table 12. Modified Aggarwal et al. for.

Table 13. Simple SP for.

Table 14. Modified SP for.

Table 15. Simple Noor for.

Table 16. Modified Noor for.

Figure 9. Graphical observations for simple Ishikawa iteration for. Here (a)-(d) show the graph for Table 9. The merging point with value 0.33981 is fixed point.

Figure 10. Graphical observations for new modified Ishikawa iteration for. Here (a)-(d) show the graph for Table 10. The merging point with value 0.33981 is fixed point.

Figure 11. Graphical observations for simple Agarwal iteration for. Here (a)-(b) show the graph for Table 11. The merging point with value 0.33981 is fixed point.

Figure 12. Graphical observations for simple Agarwal iteration for. Here (a)-(b) show the graph for Table 12. The merging point with value 0.33981 is fixed point.

Figure 13. Graphical observations for simple SP iteration for. Here (a)-(b) show the graph for Table 13. The merging point with value 0.33981 is fixed point.

Figure 14. Graphical observations for new modified SP iteration for. Here (a)-(b) show the graph for Table 14. The merging point with value 0.33981 is fixed point.

Figure 15. Graphical observations for simple Noor iteration for. Here (a)-(b) show the graph for Table 15. The merging point with value 0.33981 is fixed point.

Figure 16. Graphical observations for new modified Noor iteration for. Here(a)-(b) show the graph for Table 16. The merging point with value 0.33981 is fixed point.

4. Observations

Table 17. Simple-Ishikawa for.

Table 18. Modified-Ishikawa for.

Table 19. Simple-Ishikawa for.

Table 20. Modified-Ishikawa for.

Table 21. Simple-Agarwal for.

Table 22. Modified-Agarwal for.

Table 23. Simple-Agarwal for.

Table 24. Modified-Agarwal for.

We have noted the converging step of different iterations in tabular form and compare the conversing step for different value of a, b, c. Now by comparative analysis we noted that

1) For, simple Ishikawa do not converge for, and, but new modified Ishikawa converges for all values of a and b converges faster than Ishikawa iteration for corresponding values of a, b. Also it converges at lesser step as a and b both approaches one but not so in case of simple Ishikawa as observe from **Tables 17 **and**18**. Similarly if we compare the both iterations for as observed from **Tables 19 **and **20** that as we increase values of a and b simultaneously than converging step decreases for both iterations but modified Ishikawa iteration converges at lesser step for.

2) As observed from **Tables 21** and **22** for simple Agarwal et al. do not converge for all values of a and b it converges for

, , , , ,

but modified new Agarwal et al. iteration converges at lesser step for all values of a, b. For both iterations converge for all values of a and b but modified iteration converges faster than simple iteration 3) The simple SP iteration converges at lesser step for when a = 1/2, b = 1/2, c = 1/2 as we increases a and b the step of convergence increases. But do not converge if a and b approaches to one whereas modified new SP iteration converges for all values of a, b, c and at lesser step than simple SP. For both iteration converges for all values of a, b and c but modified new SP converge faster than Simple SP iteration.

4) For simple Noor and modified new Noor iteration result is same as for SP and modified new SP iterations.

5. Conclusion

By the observation formed from the program and graph drawn in C^{++} and Mathematica for and polynomial, we conclude that the modified Ishikawa, Agarwal et al., SP, Noor are faster than simple Ishikawa, Agarwal et al., SP, Noor; but if we compare modified Ishikawa, Agarwal et al., SP, Noor with decreasing order of rate of convergence of modified Agarwal et al., SP, Noor, Ishikawa, modified new Agarwal et al. have consistent rate of convergence. The graphs drawn are based on data formed from C^{++} program and plot the data in mathematica to show the fixed point.

NOTES

Conflicts of Interest

The authors declare no conflicts of interest.

[1] |
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[2] | B. Prasad and R. Sahni, “Convergence of Some General Iterative Schemes,” International Journal of Mathematical Analysis, Vol. 5, No. 25, 2011, pp. 1237-1242. |

[3] |
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[5] |
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[6] |
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[7] |
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