Find a solution of $x^3 - x-1=0$ in $[1,2]$

Question

Using Banach contraction theorem to find a solution of $x^3 - x-1=0$ in [$1,2$], by approximation.

Can anyone give an example of Banach contraction, so I can complete my lecture note to find a solution of $x^3 - x-1=0$ in $[1,2]$?

Answer 1

It is also known as the Banach fixed-point theorem. You want to find a function $f$ so that $f(x)=x$ at the root you are seeking and $|f'(x)| \lt q \lt 1$ over the interval $[1,2]$. Then if you start at any $x_0 \in [1,2]$ and iterate $x_{i+1}=f(x_i)$ you will converge to the root you seek. The derivative condition insures that you get closer and closer with each iteration. A choice which fails is $f(x)=x^3-1$ because the derivative is too large. You need to find another $f(x)$ that doesn't change so much with $x$. You should present your $f(x)$, then show the result at each iteration until you have converged to acceptable accuracy.

Answer 2

Hint: the equation $x^3-x-1=0$ is equivalent to $$x^3 = x+ 1$$ or $$x = \sqrt[3]{x+1}.$$ Now try Ross' answer.