Mathematical Problems in Engineering

Research Article

A General Three-Step Class of Optimal Iterations for Nonlinear Equations

Table 1

Typical forms of the weight functions satisfying (2.5).
Weight functionForm 1Form 2Form 3
𝐺 ( 𝑡 ) 1 + 𝑡 2 1 + 𝜇 𝑡 3 , 𝜇 ≠ 0 ( 1 + 𝜇 𝑡 2 ) 1 / 𝜇 , 𝜇 ≠ 0
𝐻 ( 𝜏 ) 9 6 𝜏 9 6 𝜏 + 𝜇 𝜏 2 𝜏 ( 9 / 6 ) + 𝜇 𝜏
𝑄 ( 𝛾 ) − ( 9 + 1 8 𝛽 ) 6 𝛾 3 − ( 9 + 1 8 𝛽 ) 6 𝛾 3 + 𝜇 𝛾 4 − ( 9 + 1 8 𝛽 ) 6 𝛾 3 + 𝜇 𝛾 4 + 𝜇 𝛾 5
𝑡 = 𝑓 ( 𝑧 ) / 𝑓 ( 𝑦 ) , 𝜏 = 𝑓 ( 𝑧 ) / 𝑓 ( 𝑥 ) , 𝛾 = 𝑓 ( 𝑦 ) / 𝑓 ( 𝑥 ) (without the index n), and 𝛽 ∈ ℝ .