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 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.