This paper analyses the computational complexity of validated interval methods for uncertain nonlinear systems. 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 the dependence on the initial search volume $\mathrm{Vol}(X_0)$, the target tolerance $\varepsilon$, and the costs of validated primitives explicit (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. The bounds quantify the scaling with $\mathrm{Vol}(X_0)$ and $\varepsilon$ for validated steady-state enclosure and highlight dominant cost drivers. We also show that determinant and inverse computation for interval matrices via naive Laplace expansion is factorial in the matrix dimension, motivating specialised interval linear algebra. Finally, interval Newton and interval Krawczyk have comparable leading-order costs; Krawczyk is typically cheaper in practice because it inverts a real midpoint matrix rather than an interval matrix. These results support the practical design of solvers for validated steady-state analysis in applications such as biochemical reaction network modelling, robust parameter estimation, and other uncertainty-aware computations in systems and synthetic biology.

翻译：本文分析了已验证区间方法求解不确定非线性系统的计算复杂度。区间分析能够产生考虑不确定性和舍入误差的有保证包络，但其在高维问题中的应用常受限于计算成本。我们建立了一种算法层次的最坏情况框架，显式地揭示了计算复杂度对初始搜索体积 $\mathrm{Vol}(X_0)$、目标容差 $\varepsilon$ 以及已验证原语（包含函数评估、雅可比矩阵评估和区间线性代数）成本的依赖关系。在该框架下，我们推导了区间二分法、细分+过滤法、区间约束传播法、区间牛顿法和区间Krawczyk法的最坏情况时间与空间界。这些界量化了已验证稳态包络计算中随 $\mathrm{Vol}(X_0)$ 和 $\varepsilon$ 的缩放规律，并突出了主要成本驱动因素。我们还表明，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导致关于矩阵维度的阶乘级复杂度，这促使需要专门的区间线性代数方法。最后，区间牛顿法和区间Krawczyk法的主导阶成本相当；但在实践中，Krawczyk法通常更廉价，因其仅需对实数中点矩阵求逆而非区间矩阵。这些结果支持在生化反应网络建模、鲁棒参数估计以及系统与合成生物学中的其他不确定性感知计算等应用中，设计用于已验证稳态分析求解器的实践方案。