Several New Third-Order and Fourth-Order Iterative Methods for Solving Nonlinear Equations

Table 1

Functions and their roots.

𝑓(𝑥)	𝛼
𝑓1(𝑥)=[sin(𝑥)]^2+𝑥	𝛼1=0
𝑓2(𝑥)=[sin(𝑥)]^2−𝑥^2+1	𝛼2≈1.404491648215341226035086817786
𝑓3(𝑥)=𝑒^−𝑥+sin(𝑥)−1	𝛼3≈2.076831274533112613070044244750
𝑓4(𝑥)=𝑥^2+sin(𝑥)+𝑥	𝛼4=0
𝑓5(𝑥)=sin[2cos(𝑥)]−1−𝑥^2+𝑒^sin(𝑥3)	𝛼5≈1.306175201846827825014842909066
𝑓6(𝑥)=𝑥^6−10𝑥^3+𝑥^2−𝑥+3	𝛼6≈0.658604847118140436763860014710
𝑓7(𝑥)=𝑥^4−𝑥^3+11𝑥−7	𝛼7≈0.803511199110777688978137660293
𝑓8(𝑥)=𝑥^3−cos(𝑥)+2	𝛼8≈−1.172577964753970012673332714868
𝑓9(𝑥)=√𝑥−cos(𝑥)	𝛼9≈0.641714370872882658398565300316
𝑓10(𝑥)=log(𝑥)−𝑥^3+2sin(𝑥)	𝛼10≈1.297997743280371847164479238286