Concept

π‘₯π‘˜+1=π‘₯π‘˜βˆ’π‘“(π‘₯π‘˜)𝑓′(π‘₯π‘˜)

sometime we use 𝑔(π‘₯)

𝑔(π‘₯)=π‘₯βˆ’π‘“(π‘₯)𝑓′(π‘₯)

and

𝑔′(π‘₯)=𝑓(π‘₯)𝑓″(π‘₯)𝑓′(π‘₯)2

Examples

1𝐴

𝑓(π‘₯)=π΄βˆ’1π‘₯

then 𝑓(1𝐴)=0.

𝑔(π‘₯)=π‘₯βˆ’π‘“π‘“=π‘₯βˆ’(𝐴π‘₯2βˆ’π‘₯)=2π‘₯βˆ’π΄π‘₯2

√𝐴

𝑓(π‘₯)=π‘₯2βˆ’π΄

and

𝑔(π‘₯)=π‘₯βˆ’π‘₯2βˆ’π΄2π‘₯=12(π‘₯+𝐴π‘₯)

convergence rate

convergence rate for simple root of Newton’s method is quadratic (r = 2).
for multiple root(multiplicity π‘š>1), it has linear rate.

Variants