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A framework model of multi-step quasi-Newton methods developed which utilizes values of the objective function. The model is constructed using iteration genereted data from the m+1 most recent iterates/gradient evaluations. It hosts double free parameters which introduce a certain degree of flexibility. This permits the interpolating polynomials to exploit available computed function values which are otherwise discarded and left unused. Two new algorithms are derived for those function values incorporated in the update of the inverse Hessian approximation at each iteration to accelerate convergence. The idea of incorporating function values configure quasi-Newton methods, but the presentation constitutes a new approach for such algorithms. Several earlier works only include function values data in the update of the Hessian approximation numerically to improve the convergence of Secant-like methods. The methods are a useful tool for solving nonlinear problems arising in engineering, physics, machine learning, decision science, approximations techniques to Bayesian Regressors and a variety of numerical analysis applications.

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