In this paper, we introduce the notion of outer generalized inverses, with predefined range and null space, of tensors with rational function entries equipped with the Einstein product over an arbitrary field, of characteristic zero, with or without involution. We assume that the involved tensor entries are rational functions of unassigned variables or rational expressions of functional entries. The research investigates the replacements in two stages. The lower-stage replacements assume replacements of unknown variables by constant values from the field. The higher-order stage assumes replacements of functional entries by unknown variables. This approach enables the calculation on tensors over meromorphic functions to be simplified by analogous calculations on matrices whose elements are rational expressions of variables. In general, the derived algorithms permit symbolic computation of various generalized inverses which belong to the class of outer generalized inverses, with prescribed range and null space, over an arbitrary field of characteristic zero. More precisely, we focus on a few algorithms for symbolic computation of outer inverses of matrices whose entries are elements of a field of characteristic zero or a field of meromorphic functions in one complex variable over a connected open subset of \({\mathbb {C}}\). Illustrative numerical results validate the theoretical results.

在本文中，我们介绍具有预定义范围和零空间的外部广义逆的概念，这些张量具有有理函数项，在任意域上具有爱因斯坦乘积，特征域为零，​​有或没有对合。我们假设所涉及的张量条目是未分配变量的有理函数或功能项的有理表达式。该研究分两个阶段研究替代品。低级替换假设使用字段中的常数替换未知变量。高阶阶段假定功能项被未知变量替代。通过对元素是变量的有理表达式的矩阵进行类似计算，该方法可以简化对亚纯函数的张量的计算。一般来说，推导的算法允许符号化计算各种广义逆，这些广义逆属于外部广义逆的类，具有规定范围和零空间，位于特征零的任意字段上。更精确地讲，我们集中于几种用于矩阵外部逆的符号计算的算法，这些矩阵的项是特征零域或亚纯函数域的一个复杂变量在所连接的开放子集上的元素。\（{\ mathbb {C}} \）。说明性的数值结果验证了理论结果。