In this research article, we interrogate two new modifications in inverse Weierstrass iterative method for estimating all roots of non-linear equation simultaneously. These modifications enables us to accelerate the convergence order of inverse Weierstrass method from 2 to 3. Convergence analysis proves that the orders of convergence of the two newly constructed inverse methods are 3. Using computer algebra system Mathematica, we find the lower bound of the convergence order and verify it theoretically. Dynamical planes of the inverse simultaneous methods and classical iterative methods are generated using MATLAB (R2011b), to present the global convergence properties of inverse simultaneous iterative methods as compared to classical methods. Some non-linear models are taken from Physics, Chemistry and engineering to demonstrate the performance and efficiency of the newly constructed methods. Computational CPU time, and residual graphs of the methods are provided to present the dominance behavior of our newly constructed methods as compared to existing inverse and classical simultaneous iterative methods in the literature.
A large number of physical and theoretical problems arise in various fields of mathematical, physical and engineering sciences which can be formulated as a non-linear equation:
The most primitive and popular iterative technique for approximating single root of
In the year 2016, Nedzhibov et al. [
In the last few years, lot of work has been done on those numerical iterative methods which approximate single root at a time. Besides these methods in literature, there is another class of derivative free iterative schemes which approximate all roots of
Among derivative free simultaneous methods, Weierstrass-Dochive method (abbreviated as WDK) is the most attractive method given by:
where
is Weierstrass’ Correction,
G.H Nedzibove presented two new modifications of
First modification (abbreviated as INHB):
where
Second modification (abbreviated as INHH):
The main aim of this research article is to accelerate the convergence order of
Here, we propose the following methods by replacing
and
where
In this section, we prove third order convergence of the methods IWKM1 and IWKM2.
Let
where norm in
Using Theorem 1, we have:
then there exists,
and C
where
Thus,
Using
Thus, from last inequality, the convergence order of
For a fixed point
Thus, we get:
Using the expression
If we assume all error are of the same order, i.e.,
Hence, from
Consider
and the first components of
and so on.
The lower bound of the convergence is obtained until the first non-zero element of row is found. The Mathematica codes are given for each of the considered methods as:
To provoke the basins of attraction of iterative schemes NM, INM, WDK, INHB, INHH, IWKM1, IWKM2 for the roots of non-linear equation, we execute the real and imaginary parts of the starting approximations represented as two axes over a mesh of
The basins of attraction of single root finding iterative schemes NM and INM are shown in
NM | INM | WDK | INHB | INHH | IWKM1 | IWKM2 |
---|---|---|---|---|---|---|
1.00205 | 0.08826 | 0.12388 | 0.09124 | 0.12212 | 0.04352 | 0.03521 |
Some non-linear models from engineering and applied sciences are considered to illustrate the performance and efficiency of WDK, INHB, INHH, IWKM1 and IWKM2. All calculations are done with 64 digits floating point arithmetic. The following stopping criteria are used to terminate the computer program:
where
Method | CO | CPU | n | ||||
---|---|---|---|---|---|---|---|
WDK | 2 | 0.188 | 8 | 2.5e-13 | 2.1e-13 | 5.1e-9 | 1.5e-9 |
INHB | 2 | 0.172 | 8 | 0.009 | 0.009 | 7.2e-17 | 3.3e-18 |
INHH | 2 | 0.140 | 8 | 4.0e-12 | 3.9e-12 | 3.4e-14 | 6.8e-12 |
IWKM1 | 3 | 0.141 | 8 | 0.0 | 0.0 | 5.8e-40 | 6.8e-13 |
IWKM2 | 3 | 0.125 | 8 | 8.2e-25 | 6.1e-25 | 8.2e-23 | 6.7e-13 |
Method | CO | CPU | n | ||||
---|---|---|---|---|---|---|---|
WDK | 2 | 0.031 | 7 | 0.002 | 0.002 | 0.0 | 5.8e-26 |
INHB | 2 | 0.031 | 7 | 0.002 | 0.002 | 4.9e-26 | 0.0 |
INHH | 2 | 0.047 | 7 | 0.003 | 0.003 | 4.3e-24 | 3.3e-21 |
IWKM1 | 3 | 0.016 | 7 | 2.5e-4 | 2.5e-4 | 3.8e-25 | 1.0e-28 |
IWKM2 | 3 | 0.015 | 7 | 5.5e-4 | 5.6e-4 | 0.0 | 4.9e-25 |
Method | CO | CPU | n | |||
---|---|---|---|---|---|---|
WDK | 2 | 0.017 | 4 | 7.2 | 7.4 | 9.8 |
INHB | 2 | 0.013 | 4 | 5.2 | 5.2 | 0.05 |
INHH | 2 | 0.016 | 4 | 6.4e-18 | 7.8e-18 | 1.3e-12 |
IWKM1 | 3 | 0.011 | 4 | 1.5e-18 | 5.0e-17 | 0.8e-20 |
IWKM2 | 3 | 0.010 | 4 | 8.0e-20 | 8.0e-20 | 3.2e-65 |
As described in [
is the fractional conversion of nitrogen, hydrogen feed at 250 atm and 227k.
The exact roots of
The initial calculated values of
Problem of beam positioning [
The exact roots of
The initial calculated values of
In Predator-Prey model, predation rate is denoted by
where r is number of aphids as preys [
Taking k = 30 (aphids eaten rate), a = 20 (number of aphids) and
The exact roots of
The initial estimates for
Here, we have developed two new inverse simultaneous methods of order three for determining all the roots of non-linear equations simultaneously. It must be pointed out that so far there exists an inverse simultaneous iterative scheme of order two only in the literature. We have made here comparison with the methods INHB, INHH and with classical Weierstrass-Dochive method WDK all of order two. The dynamical behavior/basins of attractions of iterative methods IWKM1, IWKM2 are also discussed here to show the global convergence behavior. Single root finding methods may have divergence region. From