In the present note we consider a type of matrices stemming in the context of the numerical approximation of distributed order fractional differential equations (FDEs): from one side they could look standard, since they are, real, symmetric and positive definite. On the other hand they present specific difficulties which prevent the successful use of classical tools. In particular the associated matrix-sequence, with respect to the matrix-size, is ill-conditioned and it is such that a generating function does not exists, but we face the problem of dealing with a sequence of generating functions with an intricate expression. Nevertheless, we obtain a real interval where the smallest eigenvalue belongs, showing also its asymptotic behavior. We observe that the new bounds improve those already present in the literature and give a more accurate spectral information, which are in fact used in the design of fast numerical algorithms for the associated large linear systems, approximating the given distributed order FDEs. Very satisfactory numerical results are presented and critically discussed, while a section with conclusions and open problems ends the current note.

翻译：在本说明中,我们考虑了在分布式分数差方程(FDEs)数字近似范围内产生的一种矩阵:从一边看,它们可以看标准,因为它们是真实的、对称的和肯定的。另一方面,它们提出了阻碍成功使用古典工具的具体困难。特别是,相关的矩阵序列在矩阵大小方面条件不完善,因此产生功能不存在,但我们面临处理以复杂表达方式生成函数的顺序的问题。然而,我们得到了最小的精精度值属于真实的间隔,也显示了其无光度行为。我们注意到,新的界限改善了文献中已经存在的界限,提供了更准确的光谱信息,实际上用于设计相关的大线性系统快速数字算法,与给的分布式FDE相匹配。提出了非常令人满意的数字结果,并进行了批判性的讨论,同时有结论和公开问题的一节结束了本说明。