This work presents a novel matrix-based method for constructing an approximation Hessian using only function evaluations. The method requires less computational power than interpolation-based methods and is easy to implement in matrix-based programming languages such as MATLAB. As only function evaluations are required, the method is suitable for use in derivative-free algorithms. For reasonably structured sample sets, the method is proven to create an order-$1$ accurate approximation of the full Hessian. Under more specialized structures, the method is proved to yield order-$2$ accuracy. The undetermined case, where the number of sample points is less than required for full interpolation, is studied and error bounds are developed for the resulting partial Hessians.

翻译：本文提出了一种新颖的基于矩阵的方法，仅利用函数评估构建近似海森矩阵。该方法所需计算量低于基于插值的方法，且易于在MATLAB等矩阵编程语言中实现。由于仅需函数评估，该方法适用于无导数优化算法。对于具有合理结构化的样本集，该方法被证明能构建出全海森矩阵的一阶精确近似；在更特殊的结构条件下，可达到二阶精度。本文还研究了样本点数不足全插值需求的不确定情形，并推导了由此产生的部分海森矩阵的误差界。