Fixed point iteration does not give me expected value

Question

I have this function $$x^4 + 3x - 1 = 0$$ I want to find the root inside $[0,2]$ using the Fixed point iteration. Starting from $X = 1$ and using: $x = \sqrt[3]{\dfrac{1 - 3x}{x}}$ I should get the root $0.33$ but I am getting the root $-1.54$ How is that possible?

Answer 1

Why do you think you should get the root $0.33$, which is really approximately $0.32941$? The derivative of your expression is well greater than $1$ in absolute value at the root, so any error will be magnified about $-28$ times in each iteration. The root is a repelling fixed point of your iteration. You need to find an iteration where the derivative of the right hand side is less than $1$ in absolute value at the root. Cube roots are good when the root is large, not so when the root is small.

If you use $x_{i+1}=\frac 13(1-x_i^4)$ you converge nicely. Taking a high power of a number smaller than $1$ has a small derivative.