Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 40E / b

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- Solution b

Solution

Subtracting the fixed point from the weighted Jacobi method gives

e^{(ν + 1)} = H_{ω} e^{(ν)}, H_{ω} = I - ω D^{- 1} A = I + \frac{ω}{2} A .

Put

x = π / (n + 1)

. Acting on the supplied sine eigenvector gives

(A v_{k})_{i} = sin ((i - 1) k x) - 2 sin (ik x) + sin ((i + 1) k x) = [- 2 + 2 cos (k x)] sin (ik x) .

Therefore

λ_{k} (A) = - 2 + 2 cos (k x) = - 4 sin^{2} \frac{k x}{2},

and the corresponding eigenvalues of the iteration matrix are

λ_{k} (ω) = 1 + \frac{ω}{2} λ_{k} (A) = 1 - ω + ω cos (k x) = 1 - 2 ω sin^{2} \frac{k x}{2} .

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