Consider the problem $minimize$ $f (x) = x^4-1$. solve this problem using newton's method. start from $x_0=4$ and perform three iterations. prove that the iterates converge to the solution. what is the rate of convergence? can you explain this?

F (x) = x⁴-1

f' (x) = 4x³

Newton’s Method: x[n+1]=x[n]-f (x[n]) / f' (x[n]); x[n+1]=x[n] - (x[n]⁴-1) / 4x[n]³

x₁=3.00390625

x₂=2.26215 ...

x₃=1.7182 ...

X'=X - (X⁴-1) / 4X³=X-X/4+1/4X³ is a symbolic way of writing the recursive formula, where X' represents the next iteration.

When X'≈X, - X/4+1/4X³≈0; so X/4≈1/4X³; X≈1/X³, so X⁴≈1 and X⁴-1≈0. But this is f (x) ≈0. Hence Newton’s Method converges to a solution.

The rate of change is x[n+1]-x[n] = - (x[n]⁴-1) / 4x[n]³=x[n]/4-1/4x[n]³ or symbolically - X/4+1/4X³.

Note that the method converges to one solution. A different x₀ will possibly converge to the solution x=-1.