This paper analyzes the computational complexity of validated interval methods for uncertain nonlinear systems and steady-state enclosure. Interval analysis produces guaranteed enclosures that account for uncertainty and round-off, but its adoption is often limited by computational cost in high dimensions. We develop an algorithm-level worst-case framework that makes explicit the dependence on the problem dimension $n$, the initial search region size $\mathrm{Vol}(X_0)$, the target tolerance $\varepsilon$, and the costs of validated primitives (inclusion-function evaluation, Jacobian evaluation, and interval linear algebra). Within this framework, we derive worst-case time and space bounds for interval bisection, subdivision$+$filter, interval constraint propagation, interval Newton, and interval Krawczyk, and identify dominant cost drivers. We also show that the computation of the determinant and inverse of interval matrices via naive Laplace expansion exhibits factorial growth with increasing matrix dimension, motivating specialized interval linear algebra. We complement the worst-case bounds with computational results on two application-motivated biochemical steady-state models (a Hill-type regulatory network and an enzyme-saturation-based winner-take-all circuit) in dimensions $n\in\{2,5,10\}$, including instances that process millions of boxes. The resulting analysis and experiments support the practical design of validated solvers for uncertainty-aware steady-state screening tasks such as robust operating-point certification and multistability assessment.

翻译：本文分析了针对不确定非线性系统及稳态包络的验证性区间方法的计算复杂度。区间分析能够产生考虑不确定性和舍入误差的保证性包络，但其在高维问题中的计算成本常常限制其应用。我们提出了一种算法层面的最坏情况分析框架，显式揭示了该复杂度与问题维度$n$、初始搜索区域大小$\mathrm{Vol}(X_0)$、目标容差$\varepsilon$以及验证性原语（包含函数求值、雅可比矩阵求值和区间线性代数）计算成本之间的依赖关系。在此框架内，我们推导了区间二分法、细分+滤波法、区间约束传播法、区间牛顿法及区间克劳奇克法的最坏情况时间和空间复杂度界限，并识别了主导成本因素。研究同时表明，通过朴素拉普拉斯展开计算区间矩阵的行列式和逆矩阵会因矩阵维度的增加而呈现阶乘增长，这促使我们需要采用专门的区间线性代数方法。我们以两个受应用启发的生化稳态模型（希尔型调控网络和基于酶饱和的赢家通吃电路）在维度$n\in\{2,5,10\}$下的计算结果（包括处理数百万个盒子的实例）对上述最坏情况界限进行了补充。所得到的分析与实验结果为设计用于不确定感知稳态筛选任务（如鲁棒工作点认证和多稳态评估）的验证性求解器提供了实用指导。