Efficient iterative scheme for solving non-linear equations with engineering applications

Figures & data

Figure 1. (a–e), shows the strange fixed point of iterative methods S1–S5, respectively.

Figure 2. Shows zone of the stability region of iterative method S1 for $\alpha =1.$

Figure 3. (a–d) depicts the dynamical planes corresponding to the  stable behaviour of iterative method S1 for various values of $\alpha =-2i,10,20,000\pm 20,000i$ respectively.

Figure 4. (a,b) depicts the dynamical planes corresponding to the unstable behaviour of iterative method S1 for various values of $\alpha =200\pm 200i$.

Figure 6. (a–l) show basins of attraction for methods S1–S5 and MM1–MM7, respectively, for function $f_1\left(x\right)$.

Figure 7. (a–l) show basin of attraction for methods S1–S5 and MM1–MM7, respectively, for function $f_2\left(x\right)$.

Figure 8. (a–l) show basins of attraction for methods S1–S5 and MM1–MM7, respectively, for function $f_3\left(x\right)$.

Figure 9. (a–l) show basins of attraction for methods S1–S5 and MM1–MM7, respectively, for function $f_4\left(x\right)$.

Table 1. Elapsed time in seconds.

Method	S1	S2	S3	S4	S5
$f_3\left(x\right)=-{\displaystyle \frac{9.8}{800}}{x}^2-0.4x+10+{\displaystyle \frac{400}{19.6}}$	0.0714	0.0192	0.03231	0.0714	0.0153
$f_{3\ast }\left(x\right)=e^{x^2}-\log \left(x\right)$	0.0184	0.0177	0.03191	0.0124	0.0191
$f_5\left(x\right)=x^4+4x^3-24x^2+16x+16$	3.1714	4.3152	5.73231	3.2714	2.3152
$f_6\left(x\right)=x^4-7.79075x^3+14.7445x^2+2.511x-1.674$	3.9124	4.0193	3.83191	2.5124	3.0153
MM1	MM2	MM3	MM4	MM5	MM6	MM7
3.1422	4.3231	7.1293	8.1422	4.3231	0.0714	0.0152
1.2382	1.4319	2.1609	3.2482	3.4719	1.1124	4.2193
4.1422	5.3231	5.1293	5.1422	4.3231	5.4714	5.9152
7.2382	7.4819	6.1659	5.2382	5.4319	8.1124	5.0193

Table 2. Numerical comparison of S1–S5 and MM1–MM7.

Method	$\left|x_3-\zeta \right|$	$\left|f\left(x_3\right)\right|$	CPU	$\rho$
$f_1\left(x\right)=c-{\displaystyle \frac{\pi {\left({\displaystyle \frac{h}{\cos \left(x\right)}}\right)}^{2}F}{0.5\pi D^2\left(1+\sin \left(x\right)-0.5\cos \left(x\right)\right)}}$
S1	0.3e−157	3.1e−322	0.0130	7.9
S2	4.1e−150	0.8e−319	0.0124	8.0
S3	3.3e−160	0.4e−320	0.0113	8.0
S4	6.4e−153	9.8e−325	0.0182	8.0
S5	0.5e−161	2.8e−331	0.0171	7.9
MM1	9.3e−118	0.3e−144	0.0274	7.9
MM2	1.8e−120	6.7e−140	0.0361	8.0
MM3	8.3e−137	7.2e−145	0.0252	8.01
MM4	7.1e−140	9.9e−150	0.0328	7.89
MM5	4.6e−125	9.2e−180	0.0179	8.9
MM6	6.2e−117	8.3e−175	0.0327	8.1
MM7	5.5e−116	3.3e−140	0.0251	8.0

Table 3. Numerical comparison of S1–S5 and MM1–MM7.

Method	$\left|x_3-\zeta \right|$	$\left|f\left(x_3\right)\right|$	CPU	$\rho$
$f_2\left(x\right)=v\left(t\right)-\sqrt{2gx}\times \tanh \left({\displaystyle \frac{t}{2l}}\sqrt{2gx}\right)$
S1	7.3e−182	9.1e−224	0.0134	8.0
S2	3.5e−185	5.0e−220	0.0151	8.0
S3	0.3e−190	0.4e−235	0.0140	8.0
S4	0.4e−180	0.8e−231	0.0123	7.0
S5	6.5e−181	8.1e−219	0.0112	6.9
MM1	8.3e−130	3.7e−170	0.0274	7.9
MM2	9.3e−133	2.6e−170	0.0365	8.0
MM3	4.9e−135	4.2e−171	0.0255	8.0
MM4	8.1e−140	5.9e−165	0.0267	8.0
MM5	0.4e−141	9.2e−117	0.0196	7.9
MM6	2.2e−137	8.3e−155	0.0227	7.8
MM7	3.5e−138	1.2e−173	0.0238	8.1

Table 4. Numerical comparison of S1–S5 and MM1–MM7.

Method	$\left|x_3-\zeta \right|$	$\left|f\left(x_3\right)\right|$	CPU	$\rho$
$f_3\left(x\right)=-{\displaystyle \frac{9.8}{800}}{x}^2-0.4x+10+{\displaystyle \frac{400}{19.6}}$
S1	5.3e−61	2.1e−495	0.0151	7.9
S2	4.5e−71	5.8e−577	0.0141	8.0
S3	6.3e−73	4.4e−592	0.0153	8.0
S4	2.4e−78	6.8e−636	0.0164	8.2
S5	2.5e−64	2.1e−522	0.0152	8.1
MM1	5.3e−16	3.5e−67	0.0474	7.0
MM2	1.3e−37	2.7e−231	0.0169	8.0
MM3	4.3e−39	2.2e−202	0.0158	8.0
MM4	4.1e−42	4.9e−259	0.0169	6.7
MM5	4.4e−31	1.2e−417	0.0164	7.9
MM6	1.2e−17	6.3e−474	0.0166	7.8
MM7	1.5e−66	3.2e−540	0.0155	7.9

Table 5. Numerical comparison of S1–S5 and MM1–MM7.

Method	$\left|x_3-\zeta \right|$	$\left|f\left(x_3\right)\right|$	CPU	$\rho$
$f_4\left(x\right)=-100\ast {10}^{-9}{x}^3+9.3100\ast x^2-18.6200\ast x+9.3100$
S1	4.7e−130	1.5e−722	0.0213	8.1
S2	5.0e−135	2.1e−735	0.0252	8.0
S3	2.1e−136	1.2e−711	0.0251	7.9
S4	1.2e−137	4.6e−773	0.0242	7.9
S5	2.9e−138	1.5e−647	0.0262	8.0
MM1	1.6e−88	3.0e−334	0.0333	7.9
MM2	4.9e−81	2.5e−335	0.0315	7.7
MM3	9.0e−75	1.4e−216	0.0264	6.9
MM4	1.0e−50	5.0e−154	0.0376	7.5
MM5	7.1e−49	1.7e−187	0.0284	7.9
MM6	3.4e−55	3.1e−197	0.0375	7.7
MM7	3.7e−96	6.8e−563	0.0367	8.1

Table 6. Numerical comparison of S1–S5 and MM1–MM7.

Method	$\left|x_3-\zeta \right|$	$\left|f\left(x_3\right)\right|$	CPU	$\rho$
$f_5\left(x\right)=x^4+4x^3-24x^2+16x+16$
S1	4.7e−91	1.5e−721	0.0151	7.9
S2	5.0e−101	2.1e−802	0.0152	8.0
S3	2.1e−103	1.2e−821	0.0153	8.1
S4	1.2e−109	4.6e−873	0.0151	7.9
S5	2.9e−94	1.5e−747	0.0163	8.0
MM1	1.6e−23	3.0e−90	0.0314	7.9
MM2	4.9 −35	2.5 e−325	0.0165	8.0
MM3	9.0e−30	1.4e−116	0.0165	7.9
MM4	1.0e−59	5.0e−354	0.0167	8.0
MM5	7.1e−49	1.7e−287	0.0168	7.9
MM6	3.4e−25	3.1e−97	0.0319	7.7
MM7	3.7e−96	6.8e−763	0.0166	7.8

Table 7. Numerical comparison of S1–S5 and MM1–MM7.

Method	$\left|x_3-\zeta \right|$	$\left|f\left(x_3\right)\right|$	CPU	$\rho$
$f_6\left(x\right)=x^4-7.79075x^3+14.7445x^2+2.511x-1.674$
S1	4.8e−16	1.3e−120	0.0162	8.0
S2	7.4e−29	9.4e−224	0.0161	7.9
S3	5.7 e−26	1.5e−200	0.0160	8.0
S4	3.5 e−29	1.6e−226	0.0161	8.1
S5	4.0e−20	5.5e−154	0.0153	7.9
MM1	2.4e−6	1.5e−21	0.0314	7.9
MM2	1.0e−22	8.7e−131	0.0165	8.0
MM3	3.9 e−16	7.5e−162	0.0167	7.0
MM4	1.1 e−19	6.4e−113	0.0158	7.9
MM5	5.1e−5	5.8e−124	0.0163	7.5
MM6	8.4e−8	1.4e−127	0.0164	7.5
MM7	3.3e−25	2.6e−194	0.0158	8.0