Mathematical Problems in Engineering

Research Article

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

Table 2

Interesting choices of 𝛽 , 𝐺 ( 0 ) , and 𝑄 ( 4 ) ( 0 ) in (2.13), which provide very efficient optimal root solvers.
Method 𝛽 𝐺 ( 0 ) 𝑄 ( 4 ) ( 0 ) Error equation
1 4 3 2 7 4 4 −219 𝑒 𝑛 + 1 = 1 𝑐 2 6 4 2 ( 1 1 𝑐 2 2 − 3 𝑐 3 ) ( − 2 7 𝑐 2 3 + 8 8 𝑐 2 𝑐 4 ) 𝑒 8 𝑛 + 𝑂 ( 𝑒 9 𝑛 )
2 4 3 0 −120 𝑒 𝑛 + 1 1 = − 1 2 ( 𝑐 2 2 ( 1 1 𝑐 2 2 − 3 𝑐 3 ) ( 9 𝑐 2 𝑐 3 − 4 𝑐 4 ) ) 𝑒 8 𝑛 + 𝑂 ( 𝑒 9 𝑛 )
3 − 1 2 −1 −54 𝑒 𝑛 + 1 = 1 4 𝑐 2 𝑐 3 ( 9 𝑐 2 2 𝑐 3 − 2 𝑐 2 3 − 4 𝑐 2 𝑐 4 ) 𝑒 8 𝑛 + 𝑂 ( 𝑒 9 𝑛 )
4 − 1 2 0 0 𝑒 𝑛 + 1 = 1 4 𝑐 2 2 𝑐 3 ( 9 𝑐 2 ( 𝑐 2 2 + 𝑐 3 ) − 4 𝑐 4 ) 𝑒 8 𝑛 + 𝑂 ( 𝑒 9 𝑛 )
50 −2 −96 𝑒 𝑛 + 1 1 = − 4 ( 𝑐 2 ( 𝑐 2 2 − 𝑐 3 ) ( 1 7 𝑐 2 2 𝑐 3 − 4 𝑐 2 3 − 4 𝑐 2 𝑐 4 ) ) 𝑒 8 𝑛 + 𝑂 ( 𝑒 9 𝑛 )