 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-1/18c/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

ρ (w) = w^{2} - w, σ (w) = \frac{5 w ^{2} + 8 w - 1}{12} .

Expansion at zero gives

ρ (e^{z}) - z σ (e^{z}) = - \frac{1}{24} z^{4} + O (z^{5}) .

Part (a) therefore shows that the method has order exactly three.

It is consistent because its order is at least one. The roots of

ρ (w) = w (w - 1)

are

0

and

1

; both lie in the closed unit disk, and the only unit-modulus root is simple. Thus the method is zero-stable. By the Dahlquist equivalence theorem, consistency and zero stability imply convergence. Hence the method is

third-order and convergent .

 New to topics? Read the docs here!