Newton Raphson method and Fixed Point Iteration method ?

[atomic_light11]

Hi everyone, I has been learning numerical method recently, i am very wonder how fixed point iteration method and Newton raphson method works (a more insight explanation rather than mathematical proof ) thanks!

 

[Office_Shredder]

Intuitively it works as follows:

You start with x₀, and you know what f(x₀) and f'(x₀) is. You also know that the graph of f(x) is approximately a line passing through (x₀,f(x₀) with slope f'(x₀). The first step is to find x₁ which is the place where f(x) would equal zero IF f(x) was actually a linear function.

Most likely f(x₁) is not equal to zero, but since f(x) kind of looked like the line whose zero you calculated, you expect that x₁ is closer to the zero of f(x) than x₀ is. Then you just repeat the whole process with x₁ in place of x₀ to find x₂ which is even closer, etc.