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!
You start with x_{0}, and you know what f(x_{0}) and f'(x_{0}) is. You also know that the graph of f(x) is approximately a line passing through (x_{0},f(x_{0}) with slope f'(x_{0}). The first step is to find x_{1} which is the place where f(x) would equal zero IF f(x) was actually a linear function.
Most likely f(x_{1}) is not equal to zero, but since f(x) kind of looked like the line whose zero you calculated, you expect that x_{1} is closer to the zero of f(x) than x_{0} is. Then you just repeat the whole process with x_{1} in place of x_{0} to find x_{2} which is even closer, etc.