  In this paper, we combine the $m$th-order Taylor expansion of the objective
function with cubic Hermite interpolation conditions. Then, we derive a series
of modified secant equations with higher accuracy in approximation of the
Hessian matrix of the objective function. A modified Wolfe line search is also
developed. It overcomes the weakness of the typical constraint which is imposed
on modified secant equations and to keep the curvature condition met.
Therefore, based on the modified secant equation and Wolfe line search, an
improved spectral conjugate gradient algorithm is proposed. Under some mild
assumptions, the algorithm is showed to be globally convergent for general
nonconvex functions. Numerical results are also reported for verifying the
effectiveness.
