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Theory and Modern Applications

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