Find a suitable rearrangement such that the sequence

Question

Question:

Consider the equation

cos (Πx) - x³ = 0.

(a) Use MATLAB to plot a graph on any interval you consider appropriate that shows all roots of this equation. Provide argunimits that there are rm other roots.

(b) Give a reason why the iterative scheme

x_n+1 = [cos(Πx_n)^1/3,

is not suitable for determining a sequence that converges to the largest positive root starting from x_o = 0.5.

(c) Find a suitable rearrangement such that the sequence starting with x_o = 0.5 converges to the largest positve root. Show that your rearrangement satisfies the criteria of Theorem.

(d) Determine an approximation to the largest positive root up to 10^-6 accuracy.

(e) Explain how you can estimate the number of iterations needed.

(f) Argue why the estimated number of iterations needed does or does tot differ from the actual number of iterations needed in part (d).