For either Barzilai-Borwein rule, the worst asymptotic gradient root factor on strongly convex quadratics is exactly (κ(H)-1)/(κ(H)+1), and the same constant governs local nonlinear convergence under strict differentiability.
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The Sharp Worst-Case Asymptotic Rate of the Barzilai--Borwein Method in $\mathbb R^d$ and Hilbert Spaces
For either Barzilai-Borwein rule, the worst asymptotic gradient root factor on strongly convex quadratics is exactly (κ(H)-1)/(κ(H)+1), and the same constant governs local nonlinear convergence under strict differentiability.