Author Contributions
Conceptualization, A.C. and J.R.T.; methodology, J.G.M.; software, A.R.-C.; validation, A.C., J.G.M. and J.R.T.; formal analysis, J.R.T.; investigation, A.C.; resources, A.R.-C.; writing—original draft preparation, A.R.-C.; writing—review and editing, A.C. and J.R.T.; visualization, J.G.M.; supervision, A.C. and J.R.T. All authors have read and agreed to the published version of the manuscript.
Figure 1.
Efficiency index IO for systems of size to .
Figure 1.
Efficiency index IO for systems of size to .
Figure 2.
Efficiency index IO for systems of size to .
Figure 2.
Efficiency index IO for systems of size to .
Figure 3.
Eigenvalues associated to the fixed points. (a) . (b) .
Figure 3.
Eigenvalues associated to the fixed points. (a) . (b) .
Figure 4.
Parameter line of in . (a) . (b) .
Figure 4.
Parameter line of in . (a) . (b) .
Figure 5.
Parameter line of in .
Figure 5.
Parameter line of in .
Figure 6.
Feigenbaum diagrams of , for , from different critical points. (a) and . (b) a detail. (c) a detail.
Figure 6.
Feigenbaum diagrams of , for , from different critical points. (a) and . (b) a detail. (c) a detail.
Figure 7.
Strange attractors of for in blue quadruple-period cascade. (a) . (b) .
Figure 7.
Strange attractors of for in blue quadruple-period cascade. (a) . (b) .
Figure 8.
Details Strange attractors of . (a) a detail. (b) a detail.
Figure 8.
Details Strange attractors of . (a) a detail. (b) a detail.
Figure 9.
Feigenbaum diagrams of , for , from different critical points. (a) and . (b) a detail. (c) a detail.
Figure 9.
Feigenbaum diagrams of , for , from different critical points. (a) and . (b) a detail. (c) a detail.
Figure 10.
Strange attractors of for in blue quadruple-period cascade. (a) . (b) a detail. (c) into the detail.
Figure 10.
Strange attractors of for in blue quadruple-period cascade. (a) . (b) a detail. (c) into the detail.
Figure 11.
Dynamical planes for some stable values of parameter . (a) . (b) . (c) . (d) .
Figure 11.
Dynamical planes for some stable values of parameter . (a) . (b) . (c) . (d) .
Figure 12.
Unstable dynamical planes of on . (a) . (b) .
Figure 12.
Unstable dynamical planes of on . (a) . (b) .
Figure 13.
Periodic orbits for parameter for different initial values. (a) . (b) . (c) .
Figure 13.
Periodic orbits for parameter for different initial values. (a) . (b) . (c) .
Table 1.
Computational cost (products/quotients) of proposed and comparison methods.
Table 1.
Computational cost (products/quotients) of proposed and comparison methods.
Method | Complexity C |
---|
Newton | |
| |
| |
| |
| |
| |
| |
ACTV | |
Table 2.
Results for function , using as seed .
Table 2.
Results for function , using as seed .
Iterative Method | k | | | | | Cpu-Time |
---|
| 4 | 6.0038 | | 0 | | 3.5022 |
| 4 | 6.0001 | | 0 | | 3.5053 |
| 4 | 5.883 | | | | 3.8991 |
| 4 | 5.874 | | | | 3.4959 |
Newton | 9 | 2.0000 | | | | 1.2855 |
| 4 | 6.0011 | | 0 | | 1.6573 |
| 4 | 6.0011 | | 0 | | 1.5505 |
| 4 | 6.004 | | 0 | | 1.7605 |
| 4 | 6.0008 | | 0 | | 1.5495 |
| 4 | 6.0101 | | 0 | | 3.6422 |
| 4 | 6.0000 | | 0 | | 3.5589 |
| 4 | 6.159 | | | | 3.9764 |
| 4 | 6.0026 | | 0 | | 1.6411 |
Table 3.
Results for function , with initial estimation .
Table 3.
Results for function , with initial estimation .
Iterative Method | k | | | | | Cpu-Time |
---|
| 4 | 6.0096 | | 0 | | 6.4105 |
| 4 | 6.0214 | | 0 | | 6.1329 |
| 4 | 6.0161 | | 0 | | 6.2416 |
| 4 | 6.0162 | | 0 | | 6.0655 |
Newton | 8 | 2.0004 | | | | 1.4579 |
| 4 | 5.995 | | 0 | | 2.4032 |
| 4 | 6.0076 | | 0 | | 2.1022 |
| 4 | 6.007 | | 0 | | 2.3387 |
| 4 | 5.9824 | | 0 | | 2.2433 |
| 4 | 6.0048 | | 0 | | 5.7749 |
| 4 | 5.9613 | | 0 | | 6.0901 |
| 4 | 5.9973 | | | | 7.0308 |
| 4 | 6.0019 | | 0 | | 2.5955 |
Table 4.
Results for function and initial guess .
Table 4.
Results for function and initial guess .
Iterative Method | k | | | | | Cpu-Time |
---|
| 10 | 5.9096 | | | | 8.2697 |
| 79 | 6.0503 | | 0 | | 65.9107 |
| - | - | - | - | - | - |
| - | - | - | - | - | - |
Newton | - | - | - | - | - | - |
| - | - | - | - | - | - |
| - | - | - | - | - | - |
| 6 | 5.9119 | | | | 2.4022 |
| - | - | - | - | - | - |
| - | - | - | - | - | - |
| - | - | - | - | - | - |
| 26 | 5.994 | | 0 | | 26.0787 |
| - | - | - | - | - | - |
Table 5.
Results for function and initial approximation .
Table 5.
Results for function and initial approximation .
Iterative Method | k | | | | | Cpu-Time |
---|
| 4 | 6.0917 | | 0 | | 9.0684 |
| 4 | 7.0212 | | 0 | | 9.1124 |
| 4 | 4.7204 | | | | 8.6135 |
| 4 | 4.3741 | | | | 8.8727 |
Newton | 9 | 2.0083 | | | | 2.0165 |
| 4 | 6.3232 | | 0 | | 2.5118 |
| 4 | 6.2666 | | 0 | | 2.4156 |
| 4 | 6.3173 | | 0 | | 2.4766 |
| 4 | 6.2481 | | 0 | | 2.3827 |
| 4 | 5.7907 | | | | 8.5868 |
| 4 | 7.0301 | | 0 | | 8.7981 |
| 4 | 4.726 | | | | 8.9312 |
| 4 | 6.3144 | | 0 | | 2.5340 |
Table 6.
Results for function using as initial estimation .
Table 6.
Results for function using as initial estimation .
Iterative Method | k | | | | | Cpu-Time |
---|
| 5 | 6.000 | | 0 | | 427.4885 |
| 5 | 5.9965 | | 0 | | 411.3637 |
| 7 | 6.0011 | | 0 | | 625.0273 |
| 16 | 6.000 | | 0 | | 1367.1656 |
Newton | 11 | 2.000 | | | | 24.0160 |
| 5 | 5.9999 | | 0 | | 42.2095 |
| 5 | 6.000 | | 0 | | 35.5499 |
| 5 | 5.9999 | | 0 | | 45.1993 |
| 5 | 6.000 | | 0 | | 40.3692 |
| 5 | 5.9991 | | 0 | | 482.4615 |
| 5 | 5.9627 | | 0 | | 598.1063 |
| 5 | 5.9723 | | | | 593.7798 |
| 5 | 5.9999 | | 0 | | 59.5617 |