Due to its simplicity and efficiency, the Barzilai and Borwein (BB) gradient
method has received various attentions in different fields. This paper presents a
new analysis of the BB method for two-dimensional strictly convex quadratic functions.
The analysis begins with the assumption that the gradient norms at the first
two iterations are fixed. We show that there is a superlinear convergence step in at
most three consecutive steps. Meanwhile, we provide a better convergence relation
for the BB method. The influence of the starting point and the condition number to
the convergence rate is comprehensively addressed.