 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ii/paper-1/38c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

With

D = diag (a_{11}, \dots, a_{nn})

x^{(k + 1)} = x^{(k)} + ω D^{- 1} (b - A x^{(k)}) = (I - ω D^{- 1} A) x^{(k)} + ω D^{- 1} b .

Each new component uses only the previous iterate's components;

ω = 1

is ordinary Jacobi iteration. Errors evolve by

e^{(k + 1)} = (I - ω D^{- 1} A) e^{(k)}

 New to topics? Read the docs here!