Theory and Modern Applications
N | \(\alpha _{2} = 1.1\) | \(\alpha _{2} = 1.3\) | \(\alpha _{2} = 1.5\) | \(\alpha _{2} = 1.7\) | \(\alpha _{2} = 1.9\) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Max. error | Rate | Time | Max. error | Rate | Time | Max. error | Rate | CPU time | Max. error | Rate | Time | Max. error | Rate | Time | |
Modified integral discretization scheme [21] | |||||||||||||||
64 | 9.56E−03 | – | 0.028 | 5.14E−03 | – | 0.023 | 3.91E−03 | – | 0.102 | 3.76E−03 | – | 0.024 | 4.07E−03 | – | 0.023 |
128 | 3.89E−03 | 1.298 | 0.048 | 1.60E−03 | 1.686 | 0.048 | 1.05E−03 | 1.895 | 0.038 | 9.44E−04 | 1.995 | 0.049 | 1.02E−03 | 1.996 | 0.049 |
256 | 1.65E−03 | 1.236 | 0.162 | 5.20E−04 | 1.618 | 0.163 | 2.87E−04 | 1.872 | 0.100 | 2.37E−04 | 1.997 | 0.163 | 2.55E−04 | 1.998 | 0.163 |
512 | 7.22E−04 | 1.193 | 0.566 | 1.78E−04 | 1.548 | 0.578 | 8.01E−05 | 1.840 | 0.328 | 5.92E−05 | 1.998 | 0.583 | 6.39E−05 | 1.999 | 0.586 |
1024 | 3.22E−04 | 1.165 | 1.220 | 6.36E−05 | 1.484 | 1.221 | 2.30E−05 | 1.803 | 0.738 | 1.51E−05 | 1.971 | 1.239 | 1.60E−05 | 1.999 | 1.228 |
2048 | 1.45E−04 | 1.148 | 3.304 | 2.36E−05 | 1.432 | 3.338 | 6.77E−06 | 1.762 | 2.735 | 3.90E−06 | 1.955 | 3.363 | 4.00E−06 | 2.000 | 3.364 |
Newton’s method | |||||||||||||||
64 | 4.42E−03 | – | 0.026 | 1.31E−03 | – | 0.012 | 1.14E−03 | – | 0.152 | 1.27E−03 | – | 0.012 | 1.42E−03 | – | 0.012 |
128 | 1.89E−03 | 1.225 | 0.012 | 3.83E−04 | 1.775 | 0.012 | 2.95E−04 | 1.955 | 0.013 | 3.17E−04 | 1.999 | 0.011 | 3.55E−04 | 2.000 | 0.011 |
256 | 8.10E−04 | 1.224 | 0.036 | 1.27E−04 | 1.588 | 0.023 | 7.62E−05 | 1.955 | 0.031 | 7.92E−05 | 2.002 | 0.023 | 8.87E−05 | 2.000 | 0.022 |
512 | 3.48E−04 | 1.219 | 0.047 | 6.31E−05 | 1.012 | 0.045 | 1.97E−05 | 1.949 | 0.049 | 1.98E−05 | 2.004 | 0.045 | 2.22E−05 | 2.000 | 0.045 |
1024 | 1.50E−04 | 1.213 | 0.109 | 2.88E−05 | 1.130 | 0.103 | 5.15E−06 | 1.938 | 0.113 | 4.92E−06 | 2.006 | 0.150 | 5.54E−06 | 2.000 | 0.103 |
2048 | 6.91E−05 | 1.120 | 0.268 | 1.26E−05 | 1.191 | 0.263 | 1.36E−06 | 1.922 | 0.287 | 1.22E−06 | 2.008 | 0.307 | 1.38E−06 | 2.001 | 0.281 |