  We propose a family of spectral gradient methods, whose stepsize is
determined by a convex combination of the long Barzilai-Borwein (BB) stepsize
and the short BB stepsize. Each member of the family is shown to share certain
quasi-Newton property in the sense of least squares. The family also includes
some other gradient methods as its special cases. We prove that the family of
methods is $R$-superlinearly convergent for two-dimensional strictly convex
quadratics. Moreover, the family is $R$-linearly convergent in the
any-dimensional case. Numerical results of the family with different settings
are presented, which demonstrate that the proposed family is promising.
