On the Rate of Convergence of Some New Modified Iterative Schemes ()
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, andFigures 1-16.
We take 
Table 3. Simple Agarwal et al. for
.
Table 4. Modified Agarwal et al. for
.
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 11. Simple Agarwal et al. for
.
Table 12. Modified Aggarwal et al. 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
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 and18. 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