发表机构
Universitat Politècnica de València（瓦伦西亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过引入标量范数型加速器，将Hyperpower逆矩阵近似方法的收敛阶提高50%至六阶，在保持四阶计算成本的同时提升效率，经数值实验及图像复原应用验证了其有效性。
AI 中文摘要
在本手稿中，首次使用标量范数型加速器来提高求解矩阵方程$X^{-1}-A=0$的已知迭代方法的收敛阶，同时减少矩阵-矩阵乘积的数量。具体而言，所提方案将收敛阶提高了50%，以矩阵-矩阵乘法计算量计，在达到四阶Hyperpower方法计算成本的同时，实现了六阶收敛。主要结果中，不仅证明了收敛性及其阶数，还证明了其效率与稳定性。该新技术即便在包含多达$10^6$个元素的大型矩阵上，与现有的四阶、六阶及八阶方案相比也能提供良好的数值结果，且有望开辟一类具有良好性能与可扩展性的新方法。最后，通过将其应用于数字图像复原，展示了该方法与现有方法相比的性能表现。
英文摘要
In this manuscript, scalar norm-type accelerators are used for first time in order to increase the order of convergence of known iterative methods for solving the matrix equation $X^{-1}-A=0$, while decreasing the number of matrix-matrix products. Specifically, our proposed scheme upgrades the order of convergence in a $50\%$, achieving the sixth-order of convergence with the computational cost of fourth-order Hyperpower method, in terms of matrix-matrix multiplications. In the main results, not only the convergence and its order is proven, but also its efficiency and stability. This new technique provides good numerical results, even compared with existing fourth-, sixth- and eighth-order schemes in large matrices of up to $10^6$ entries, but also promises the opening of a new kind of procedures with good performance and scalability. Finally, an application to digital image restoration is made to show the behavior of the method, compared with existing ones.
Comments14 pages
