Theory and Modern Applications
\(\alpha _{2}\) | N = 64 | N = 128 | N = 256 | N = 512 | N = 1024 | N = 2048 | |
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Finite difference method [22] | |||||||
1.3 | error | 6.557E−002 | 4.240E−002 | 2.509E−002 | 1.387E−002 | 7.337E−003 | 3.782E−003 |
rate | 0.629 | 0.757 | 0.855 | 0.919 | 0.956 | 0.977 | |
1.5 | error | 2.271E−002 | 1.265E−002 | 6.749E−003 | 3.509E−003 | 1.795E−003 | 9.104E−004 |
rate | 0.844 | 0.906 | 0.944 | 0.967 | 0.980 | 0.988 | |
1.7 | error | 1.110E−002 | 5.968E−003 | 3.140E−003 | 1.628E−003 | 8.356E−004 | 4.262E−004 |
rate | 0.895 | 0.927 | 0.948 | 0.962 | 0.971 | 0.978 | |
1.9 | error | 5.650E−003 | 2.978E−003 | 1.557E−003 | 8.090E−004 | 4.184E−004 | 2.156E−004 |
rate | 0.017 | 0.050 | 0.084 | 0.117 | 0.168 | 0.255 | |
Newton’s method with linear explicit | |||||||
1.3 | error | 5.35E−03 | 2.55E−03 | 1.14E−03 | 4.95E−04 | 2.09E−04 | 8.68E−05 |
rate | 0.935 | 1.068 | 1.157 | 1.211 | 1.244 | 1.265 | |
1.5 | error | 5.35E−03 | 2.55E−03 | 1.14E−03 | 4.95E−04 | 2.09E−04 | 8.68E−05 |
rate | 0.935 | 1.068 | 1.157 | 1.211 | 1.244 | 1.265 | |
1.7 | error | 4.19E−04 | 1.39E−04 | 4.51E−05 | 1.45E−05 | 4.62E−06 | 1.46E−06 |
rate | 1.562 | 1.594 | 1.618 | 1.636 | 1.650 | 1.660 | |
1.9 | error | 1.00E−04 | 2.85E−05 | 8.07E−06 | 2.27E−06 | 6.35E−07 | 1.77E−07 |
rate | 1.798 | 1.810 | 1.821 | 1.830 | 1.837 | 1.844 | |
Newton’s method | |||||||
1.3 | error | 4.04E−03 | 6.03E−04 | 4.95E−05 | 8.12E−06 | 5.88E−06 | 2.17E−06 |
rate | 2.238 | 2.745 | 3.607 | 2.606 | 0.465 | 1.438 | |
1.5 | error | 2.41E−04 | 8.57E−05 | 2.50E−05 | 6.74E−06 | 1.74E−06 | 4.42E−07 |
rate | 0.329 | 1.489 | 1.775 | 1.893 | 1.949 | 1.980 | |
1.7 | error | 1.53E−04 | 4.05E−05 | 1.03E−05 | 2.58E−06 | 6.40E−07 | 1.58E−07 |
rate | 1.779 | 1.919 | 1.975 | 1.998 | 2.010 | 2.017 | |
1.9 | error | 9.08E−05 | 2.29E−05 | 5.74E−06 | 1.43E−06 | 3.58E−07 | 8.92E−08 |
rate | 1.941 | 1.984 | 1.997 | 2.001 | 2.003 | 2.004 | |
Halley’s method | |||||||
1.3 | error | 1.00E−03 | 9.19E−05 | 6.67E−06 | 3.25E−07 | 1.60E−08 | 8.44E−09 |
rate | 3.056 | 3.456 | 3.782 | 4.358 | 4.348 | 0.920 | |
1.5 | error | 8.24E−06 | 2.48E−06 | 8.70E−07 | 3.02E−07 | 1.05E−07 | 3.64E−08 |
rate | 2.111 | 1.732 | 1.511 | 1.524 | 1.526 | 1.525 | |
1.7 | error | 1.26E−05 | 3.60E−06 | 1.07E−06 | 3.26E−07 | 9.97E−08 | 3.06E−08 |
rate | 1.862 | 1.802 | 1.746 | 1.719 | 1.707 | 1.703 | |
1.9 | error | 4.87E−06 | 1.06E−06 | 2.53E−07 | 6.42E−08 | 1.68E−08 | 4.44E−09 |
rate | 2.365 | 2.200 | 2.063 | 1.980 | 1.937 | 1.916 |