Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 40C / c

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- Solution c

Solution

Because

A x^{*} = b

, the error

e^{(k)} = x^{*} - x^{(k)}

obeys

e^{(k + 1)} = ((1 + β) I - α A) e^{(k)} - β e^{(k - 1)} .

Thus the heavy-ball error propagation matrix is

(e^{(k + 1)} e^{(k)}) = M (e^{(k)} e^{(k - 1)}), M = ((1 + β) I - α A I - β I 0) .

If

A = diag (λ_{1}, \dots, λ_{n})

, permute the coordinates from

(e_{1}^{(k)}, \dots, e_{n}^{(k)}, e_{1}^{(k - 1)}, \dots, e_{n}^{(k - 1)})

to

(e_{1}^{(k)}, e_{1}^{(k - 1)}, \dots, e_{n}^{(k)}, e_{n}^{(k - 1)}) .

For the corresponding permutation matrix

P

,

PM P^{T} = diag (M_{1}, \dots, M_{n}), M_{i} = (1 + β - α λ_{i} 1 - β 0) .

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