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Table 5 Comparison of optimal 3rd order methods

From: On iterative techniques for estimating all roots of nonlinear equation and its system with application in differential equation

Method n CPU ρ \(\Vert \mathbf{x}^{(k+1)}-\mathbf{x}^{(k)} \Vert \) \(\Vert \mathbf{Fx}^{(k)} \Vert \)
Solving a system of nonlinear equations \(\mathbf{F}_{3}(\mathbf{X}) \) and \(\mathbf{F}_{4}(\mathbf{X})\)
\(\mathbf{F}_{3}(\mathbf{X})\); \((\mathbf{1.5},\mathbf{1.5})^{T}\); \(\mathbf{X}_{0}=(\mathbf{0.8},\mathbf{1},\mathbf{0.8})^{T}\), \(\alpha =\frac{-1}{100{,}000}\)
QQ1 3 0.015 3.3 1.7e−31 5.1e−32
EE1 3 0.016 3.0 1.2e−17 1.1e−17
EE2 3 0.017 2.9 2.3e−16 2.1e−17
EE3 3 0.017 3.0 2.5e−17 2.1e−17
\(\mathbf{F}_{4}(\mathbf{X})\); \(\mathbf{X}_{0}=(\mathbf{0.5},\mathbf{0.5},\mathbf{0.5})^{T}\), \(\alpha =\frac{-1}{100{,}000}\)
QQ1 3 0.013 3.0 9.9e−11 1.7e−21
EE1 3 0.031 2.7 1.1e−7 4.0e−18
EE2 3 0.016 3.0 2.3e−16 8.1e−18
EE3 3 0.016 2.9 9.0e−11 2.3e−17