Table 17 System (31): Asymptotic error constants of the classical and generalized Newton methods
Soln | \(s_{i}(x)\) | \([\|\mu (x^{*})\|/2, \|\rho (x^{*})\|/2]\) | λ | \(\lambda _{N}/\lambda _{\mathrm{GN}}\) |
|---|---|---|---|---|
1 | \(x_{i}\) | [0.29,2.6] | 1.6 | |
\(x_{i}^{3}\) | [0.14,2.2] | 1.7 | 0.9 | |
\(\sinh x_{i}\) | [0.24,2.4] | 0.6 | 2.7 | |
\(e^{x_{i}}\) | [0.23,2.3] | 0.4 | 4 | |
2 | \(x_{i}\) | [0.29,2.6] | 1.0 | |
\(x_{i}^{3}\) | [0.14,2.2] | 1.5 | 0.7 | |
\(\sinh x_{i}\) | [0.24,2.4] | 0.6 | 1.7 | |
\(e^{x_{i}}\) | [0.34,3.0] | 1.5 | 0.7 | |
3 | \(x_{i}\) | [0,3.0] | 1.3 | |
\(x_{i}^{3}\) | [0,4.9] | 4.5 | 0.3 | |
\(\sinh x_{i}\) | [0,2.8] | 1.1 | 1.2 | |
\(e^{x_{i}}\) | [0,3.7] | 2.0 | 0.7 | |
4 | \(x_{i}\) | [0,3.0] | 1.3 | |
\(x_{i}^{3}\) | [0,4.9] | 0.4 | 3.3 | |
\(\sinh x_{i}\) | [0,2.8] | 1.1 | 1.2 | |
\(e^{x_{i}}\) | [0,2.5] | 1.0 | 1.3 | |
5 | \(x_{i}\) | − | 0.0 | |
\(x_{i}^{3}\) | − | − | − | |
\(\sinh x_{i}\) | − | 0.0 | − | |
\(e^{x_{i}}\) | [0,0.71] | 0.5 | 0.0 |