Both spectral sequences and persistent homology are tools in algebraic topology defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set $\mathbb{Z}$ of integer numbers. A recent work has shown the details of the relation between both concepts. Moreover, generalizations of both concepts have been proposed which originate from a different choice of the set of indices of the filtration, producing the new notions of multipersistence and spectral system. In this paper, we show that these notions are also related, generalizing results valid in the case of filtrations over $\mathbb{Z}$. By using this relation and some previous programs for computing spectral systems, we have developed a new module for the Kenzo system computing multipersistence. We also present a birth-death descriptor and a new invariant providing information on multifiltrations. This new invariant, in some cases, is able to provide more information than the rank invariant. We show some applications of our algorithms to spaces of infinite type via the effective homology technique, where the performance has also been improved by means of discrete vector fields.

频谱序列和持久同源性都是代数拓扑中的工具，它是根据在集合上建立索引的对象（例如拓扑空间或单纯复形）的过滤定义的 $\mathbb{ž}$整数。最近的工作显示了这两个概念之间关系的细节。此外，已经提出了这两种概念的概括，它们源自对过滤指数集的不同选择，从而产生了多重余辉和光谱系统的新概念。在本文中，我们证明了这些概念也是相关的，归纳了在以下情况下有效的过滤结果：$\mathbb{ž}$。通过使用这种关系和一些以前的用于计算光谱系统的程序，我们为Kenzo系统计算多持久性开发了一个新模块。我们还提供了一个生死描述符和一个新的不变式，提供有关多重过滤的信息。在某些情况下，这个新的不变式能够提供比等级不变式更多的信息。我们通过有效的同源性技术展示了我们的算法在无限类型空间上的一些应用，其中通过离散矢量场的性能也得到了改善。