Theory and Modern Applications
Table 4 Comparison of optimal 3rd order methods
Method | n | CPU | ρ | \(\Vert \mathbf{x}^{(k+1)}-\mathbf{x}^{(k)} \Vert \) | \(\Vert \mathbf{Fx}^{(k)} \Vert \) |
|---|---|---|---|---|---|
Solving a system of nonlinear equations \(\mathbf{F}_{1}(\mathbf{X}) \) and \(\mathbf{F}_{2}(\mathbf{X})\) | |||||
\(\mathbf{F}_{1}(\mathbf{X})\); \((\mathbf{1.5},\mathbf{1.5})^{T}\); \(\mathbf{X}_{0}=(\mathbf{0.6},\mathbf{1.4})^{T}\), \(\alpha =\frac{-1}{100{,}000}\) | |||||
QQ1 | 3 | 0.012 | 3.0 | 7.9e−17 | 9.7e−17 |
EE1 | 3 | 0.031 | 2.9 | 3.8e−9 | 1.4e−9 |
EE2 | 3 | 0.015 | 3.0 | 5.9e−17 | 2.3e−16 |
EE3 | 3 | 0.562 | 3.0 | 2.1e−15 | 2.1e−15 |
\(\mathbf{F}_{2}(\mathbf{X})\); \(\mathbf{X}_{0}=(\mathbf{1.5},\mathbf{1.0})^{T}\), \(\alpha =\frac{-1}{100{,}000}\) | |||||
QQ1 | 4 | 0.017 | 3.1 | 1.2e−21 | 0.1e−21 |
EE1 | 4 | 0.016 | 2.9 | 8.3e−17 | 1.1e−18 |
EE2 | 4 | 0.016 | 3.0 | 1.8e−16 | 1.0e−17 |
EE3 | 4 | 0.016 | 3.0 | 5.3e−17 | 5.3e−17 |