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Exact line search for a positive-definite quadratic

For

f (x) = x^{T} A x /2 - b^{T} x

with

A

positive definite and residual

r = b - A x

, exact line search along

r

uses

t = \frac{r ^{T} r}{r ^{T} A r} .

The objective error contracts by

1 - \frac{( r ^{T} r ) ^{2}}{( r ^{T} A ^{- 1} r ) ( r ^{T} A r )} \leq 1 - \frac{λ _{m i n} ( A )}{λ _{m a x} ( A )} .

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