Many iterative algorithms in optimization, computational geometry, computer algebra, and other areas of computer science require repeated computation of some algebraic expression whose input changes slightly from one iteration to the next. Although efficient data structures have been proposed for maintaining the solution of such algebraic expressions under low-rank updates, most of these results are only analyzed under exact arithmetic (real-RAM model and finite fields) which may not accurately reflect the complexity guarantees of real computers. In this paper, we analyze the stability and bit complexity of such data structures for expressions that involve the inversion, multiplication, addition, and subtraction of matrices under the word-RAM model. We show that the bit complexity only increases linearly in the number of matrix operations in the expression. In addition, we consider the bit complexity of maintaining the determinant of a matrix expression. We show that the required bit complexity depends on the logarithm of the condition number of matrices instead of the logarithm of their determinant. We also discuss rank maintenance and its connections to determinant maintenance. Our results have wide applications ranging from computational geometry (e.g., computing the volume of a polytope) to optimization (e.g., solving linear programs using the simplex algorithm).

翻译：许多优化、计算几何、计算机代数及计算机科学其他领域的迭代算法需要重复计算某个代数表达式，其输入在相邻迭代间仅发生微小变化。尽管已有针对低秩更新下维护此类代数表达式解的高效数据结构，但多数分析仅基于精确算术（实RAM模型和有限域），可能无法准确反映真实计算机的复杂度保证。本文在字RAM模型下，分析了涉及矩阵求逆、乘法、加法及减法的表达式所对应的数据结构的稳定性与比特复杂度。我们证明比特复杂度仅随表达式中的矩阵操作次数线性增长。此外，我们研究了维护矩阵表达式行列式的比特复杂度，并表明所需比特复杂度取决于矩阵条件数的对数而非其行列式的对数。我们还讨论了秩维护及其与行列式维护的关联。本文结果具有广泛应用，涵盖计算几何（如计算多面体体积）到优化（如使用单纯形算法求解线性规划）。