Theory and Modern Applications
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 | 2.26E − 05 | 6.92E − 07 | 6.70E − 06 | 8.99E − 06 | 9.95E − 06 | 7.65E − 07 | 5.55E − 08 | 1.33E − 07 | 1.84E − 07 | 1.95E − 07 |
Rate | – | – | – | – | – | – | – | – | – | – | |
k | 20 | 10 | 9 | 8 | 6 | 22 | 10 | 9 | 8 | 7 | |
CPU time (sec) | 3.969 | 1.227 | 1.427 | 1.230 | 0.941 | 5.149 | 2.296 | 2.055 | 1.896 | 1.680 | |
128 | Max. error | 5.04E − 06 | 6.51E − 07 | 1.88E − 06 | 2.35E − 06 | 2.57E − 06 | 8.54E − 08 | 2.93E − 09 | 1.95E − 08 | 2.43E − 08 | 2.55E − 08 |
Rate | 2.166 | 8.900 | 1.833 | 1.937 | 1.954 | 3.164 | 4.244 | 2.767 | 2.920 | 2.933 | |
k | 20 | 10 | 9 | 8 | 7 | 22 | 11 | 9 | 8 | 7 | |
CPU time (sec) | 3.969 | 4.636 | 5.262 | 4.543 | 4.066 | 18.971 | 9.424 | 7.705 | 6.906 | 6.112 | |
256 | Max. error | 1.10E − 06 | 2.50E − 07 | 4.99E − 07 | 6.00E − 07 | 6.54E − 07 | 9.41E − 09 | 8.67E − 10 | 2.64E − 09 | 3.11E − 09 | 3.27E − 09 |
Rate | 2.192 | 1.381 | 1.914 | 1.968 | 1.974 | 3.181 | 1.754 | 2.888 | 2.962 | 2.964 | |
k | 20 | 10 | 9 | 8 | 7 | 22 | 11 | 9 | 8 | 7 | |
CPU time (sec) | 15.179 | 17.853 | 20.053 | 17.621 | 15.541 | 73.376 | 33.136 | 29.642 | 26.633 | 23.345 | |
512 | Max. error | 2.38E − 07 | 7.81E − 08 | 1.29E − 07 | 1.52E − 07 | 1.65E − 07 | 1.03E − 09 | 1.72E − 10 | 3.43E − 10 | 3.94E − 10 | 4.14E − 10 |
Rate | 2.211 | 1.679 | 1.955 | 1.983 | 1.984 | 3.194 | 2.331 | 2.944 | 2.981 | 2.979 | |
k | 20 | 10 | 9 | 8 | 7 | 22 | 10 | 9 | 8 | 7 | |
CPU time (sec) | 60.494 | 69.994 | 80.021 | 69.867 | 62.629 | 289.130 | 131.007 | 116.497 | 106.946 | 92.665 | |
1024 | Max. error | 5.08E − 08 | 2.23E − 08 | 3.27E − 08 | 3.82E − 08 | 4.17E − 08 | 1.12E − 10 | 2.71E − 11 | 4.37E − 11 | 4.96E − 11 | 5.23E − 11 |
Rate | 2.228 | 1.810 | 1.975 | 1.990 | 1.989 | 3.205 | 2.669 | 2.971 | 2.990 | 2.986 | |
k | 20 | 10 | 9 | 8 | 7 | 22 | 10 | 9 | 8 | 7 | |
CPU time (sec) | 239.810 | 275.430 | 313.605 | 276.502 | 244.386 | 1132.549 | 515.751 | 461.774 | 414.636 | 363.701 | |
2048 | Max. error | 1.07E − 08 | 6.05E − 09 | 8.26E − 09 | 9.59E − 09 | 1.05E − 08 | 1.20E − 11 | 3.90E − 12 | 5.50E − 12 | 6.25E − 12 | 6.58E − 12 |
Rate | 2.246 | 1.880 | 1.986 | 1.994 | 1.991 | 3.215 | 2.798 | 2.990 | 2.990 | 2.991 | |
k | 20 | 10 | 9 | 8 | 7 | 22 | 11 | 9 | 8 | 7 | |
CPU time (sec) | 961.302 | 1099.115 | 1235.384 | 1096.546 | 963.071 | 4517.411 | 2247.280 | 1837.934 | 1634.163 | 1436.284 |