Theory and Modern Applications
Table 2 Pointwise absolute errors in the maximum norm and computed rates of convergence versus the number of sub-intervals N with \(s_{0} = 0.2, 1.0\) for each in Example 4.1. We set \(\alpha _{1} = 0.4\), \(\alpha _{2} = 1.7\), and m is fixed at 10
N | Newton’s method | Halley’s method | ||||||
|---|---|---|---|---|---|---|---|---|
\(s_{0}=0.2\) | \(s_{0}=1.0\) | \(s_{0}=0.2\) | \(s_{0}=1.0\) | |||||
Max. error | Rate | Max. error | Rate | Max. error | Rate | Max. error | Rate | |
10 | 1.17E−02 | – | 1.17E−02 | – | 1.02E−03 | – | 1.02E−03 | – |
20 | 5.63E−03 | 1.059 | 5.63E−03 | 1.059 | 1.83E−04 | 2.486 | 1.83E−04 | 2.486 |
40 | 1.78E−03 | 1.665 | 1.78E−03 | 1.665 | 3.63E−05 | 2.333 | 3.63E−05 | 2.332 |
80 | 4.92E−04 | 1.855 | 4.92E−04 | 1.854 | 5.47E−06 | 2.730 | 5.47E−06 | 2.730 |
160 | 1.29E−04 | 1.931 | 1.29E−04 | 1.931 | 7.51E−07 | 2.866 | 7.51E−07 | 2.866 |
320 | 3.30E−05 | 1.966 | 3.30E−05 | 1.966 | 9.88E−08 | 2.925 | 9.88E−08 | 2.925 |