Theory and Modern Applications
Table 6 Pointwise absolute errors in the maximum norm, computed rates of convergence obtained by the proposed schemes, number of iterations k, and CPU time in second executed with various fractional orders versus the number of sub-intervals N in Example 4.2 as we set \(s_{0}=0.2\), \(\mathit{Tol}=10^{-10}\) for both shooting techniques
N | Newton’s method \((\alpha _{1}, \alpha _{2})\) | Halley’s method \((\alpha _{1}, \alpha _{2})\) | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
(0.9,1.1) | (0.7,1.3) | (0.5,1.5) | (0.3,1.7) | (0.1,1.9) | (0.9,1.1) | (0.7,1.3) | (0.5,1.5) | (0.3,1.7) | (0.1,1.9) | ||
64 | Max. error | 1.25E − 04 | 1.10E − 05 | 4.15E − 05 | 4.97E − 05 | 5.10E − 05 | 3.63E − 06 | 9.39E − 08 | 6.56E − 07 | 7.63E − 07 | 6.59E − 07 |
Rate | – | – | – | – | – | – | – | – | – | – | |
k | 5 | 5 | 5 | 5 | 5 | 6 | 5 | 5 | 5 | 5 | |
CPU time (sec) | 0.930 | 0.871 | 0.879 | 0.888 | 0.892 | 2.233 | 1.320 | 4.985 | 1.342 | 1.331 | |
128 | Max. error | 2.71E − 05 | 5.55E − 06 | 1.14E − 05 | 1.29E − 05 | 1.32E − 05 | 4.00E − 07 | 3.39E − 08 | 9.36E − 08 | 1.02E − 07 | 8.86E − 08 |
Rate | 2.201 | 9.891 | 1.861 | 1.942 | 1.955 | 3.181 | 1.471 | 2.808 | 2.899 | 2.896 | |
k | 5 | 5 | 5 | 5 | 5 | 6 | 5 | 5 | 5 | 5 | |
CPU time (sec) | 3.347 | 3.335 | 3.332 | 3.406 | 3.411 | 6.076 | 5.011 | 19.757 | 5.046 | 5.075 | |
256 | Max. error | 5.80E − 06 | 1.89E − 06 | 3.00E − 06 | 3.30E − 06 | 3.35E − 06 | 4.36E − 08 | 6.32E − 09 | 1.25E − 08 | 1.33E − 08 | 1.16E − 08 |
Rate | 2.226 | 1.555 | 1.928 | 1.970 | 1.974 | 3.197 | 2.421 | 2.905 | 2.944 | 2.935 | |
k | 5 | 5 | 5 | 5 | 5 | 6 | 5 | 5 | 5 | 5 | |
CPU time (sec) | 13.195 | 13.212 | 13.037 | 13.417 | 13.467 | 23.836 | 19.863 | 77.831 | 19.576 | 19.782 | |
512 | Max. error | 1.22E − 06 | 5.61E − 07 | 7.71E − 07 | 8.34E − 07 | 8.47E − 07 | 4.72E − 09 | 1.04E − 09 | 1.62E − 09 | 1.70E − 09 | 1.49E − 09 |
Rate | 2.249 | 1.752 | 1.962 | 1.984 | 1.983 | 3.209 | 2.604 | 2.950 | 2.966 | 2.955 | |
k | 5 | 5 | 5 | 5 | 5 | 6 | 5 | 5 | 5 | 5 | |
CPU time (sec) | 51.750 | 52.685 | 51.734 | 53.075 | 55.703 | 94.335 | 78.871 | 77.349 | 78.020 | 78.308 | |
1024 | Max. error | 2.52E − 07 | 1.56E − 07 | 1.96E − 07 | 2.10E − 07 | 2.14E − 07 | 5.06E − 10 | 1.52E − 10 | 2.06E − 10 | 2.16E − 10 | 1.91E − 10 |
Rate | 2.273 | 1.848 | 1.979 | 1.991 | 1.988 | 3.221 | 2.779 | 2.973 | 2.978 | 2.965 | |
k | 5 | 5 | 5 | 5 | 5 | 6 | 5 | 5 | 5 | 5 | |
CPU time (sec) | 204.322 | 203.996 | 206.051 | 209.157 | 210.729 | 372.048 | 310.732 | 306.249 | 305.918 | 309.241 | |
2048 | Max. error | 5.12E − 08 | 4.17E − 08 | 4.93E − 08 | 5.27E − 08 | 5.37E − 08 | 5.39E − 11 | 2.12E − 11 | 2.58E − 11 | 2.75E − 11 | 2.44E − 11 |
Rate | 2.301 | 1.902 | 1.988 | 1.994 | 1.991 | 3.232 | 2.839 | 2.999 | 2.973 | 2.972 | |
k | 5 | 5 | 5 | 5 | 5 | 6 | 5 | 5 | 5 | 5 | |
CPU time (sec) | 818.055 | 806.967 | 823.096 | 830.832 | 836.200 | 1452.813 | 1231.030 | 1223.399 | 1224.377 | 1227.764 | |