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For $f(x)=x^TAx/2-b^Tx$ with $A$ positive definite and residual $r=b-Ax$, exact line search along $r$ uses
$$
t=\frac{r^Tr}{r^TAr}.
$$
The objective error contracts by
$$
1-\frac{(r^Tr)^2}{(r^TA^{-1}r)(r^TAr)}
\leq 1-\frac{\lambda_{\min}(A)}{\lambda_{\max}(A)}.
$$


Exact line search for a positive-definite quadratic

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

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 40A / b / iii / Solution

Exact line search for a positive-definite quadratic

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Exact line search for a positive-definite quadratic

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