Structured matrices with symbolic sizes appear frequently in the literature, especially in the description of algorithms for linear algebra. Recent work has treated these symbolic structured matrices themselves as computational objects, showing how to add matrices with blocks of different symbolic sizes in a general way while avoiding a combinatorial explosion of cases. The present article introduces the concept of hybrid intervals, in which points may have negative multiplicity. Various operations on hybrid intervals have compact and elegant formulations that do not require cases to handle different orders of the end points. This makes them useful to represent symbolic block matrix structures and to express arithmetic on symbolic block matrices compactly. We use these ideas to formulate symbolic block matrix addition and multiplication in a compact and uniform way.

翻译：文献中常出现具有符号大小的结构化矩阵，尤其是在线性代数算法的描述中。近期研究将此类符号结构化矩阵本身视为计算对象，展示了如何在避免情形组合爆炸的前提下，以通用方式处理具有不同符号大小分块的矩阵加法。本文引入混合区间概念，其中点可具有负重数。混合区间上的各类运算具有紧凑而优雅的公式化表达，无需针对端点不同顺序分情形处理。这使得混合区间能有效表征符号分块矩阵结构，并紧凑表达符号分块矩阵的算术运算。我们运用这些思想，以紧凑统一的方式构建了符号分块矩阵的加法与乘法运算。