 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-1/40c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

A = M - N

with

M

A x = b

is equivalent to

x = M^{- 1} N x + M^{- 1} b,

x^{(k + 1)} = H x^{(k)} + M^{- 1} b, H = M^{- 1} N .

For the Jacobi method, write

A = D + L + U

with

D

L, U

strictly lower and upper triangular, and take

M = D, N = - (L + U) .

Thus

x^{(k + 1)} = - D^{- 1} (L + U) x^{(k)} + D^{- 1} b .

The iteration converges for every initial vector exactly when the spectral radius of its iteration matrix satisfies

ρ (- D^{- 1} (L + U)) < 1.

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