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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 41C / d / Solution

Since

r^{(k)} = b - A x^{(k)} = - \nabla f (x^{(k)})

, exact line search minimizes

f (x^{(k)} + α r^{(k)})

. Differentiating with respect to

α

gives

- (r^{(k)})^{T} r^{(k)} + α (r^{(k)})^{T} A r^{(k)} = 0,

so

α^{(k)} = \frac{( r ^{(k)} ) ^{T} r ^{(k)}}{( r ^{(k)} ) ^{T} A r ^{(k)}} .

Moreover,

r^{(k + 1)} = r^{(k)} - α^{(k)} A r^{(k)},

and substitution of the displayed step size yields

(r^{(k)})^{T} r^{(k + 1)} = 0 .

Thus consecutive residuals are orthogonal.

Solved by gpt-5.6-sol high.

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