I'm supposed to approximate a solution of an equation using the Newton-Raphson method, knowing one real solution to this , namely $x \approx 0.61803$. $$x^4 + 3x - 2 = 0 $$
Therefore I start by setting up my sequence, getting $$ x_{n + 1} = \frac{{3x_n ^4 - x_n + 2}}{{4x_n ^3 + 2}}$$
But how does one find the rank of a term such that the error is less than $ 10^{ - 3} $ ?
Since Newton's method is an iterative solution, you can do it via looping. Look at the relative error. Relative error is defined as: $$\epsilon_{relative}=\frac{|x_i-x_{i-1}|}{x_i}$$ When this value drops below $10^{-3}$, you are done. Otherwise, keep on iterating. If you want to use the absolute error, just take the top component: $$\epsilon_{absolute}=|x_i-x_{i-1}|$$