Numerical errors are insidious, difficult to predict and inherent in different levels of critical systems design. Indeed, numerical algorithms generally constitute approximations of an ideal mathematical model, which itself constitutes an approximation of a physical reality which has undergone multiple measurement errors. To this are added rounding errors due to computer arithmetic implementations, often neglected even if they can significantly distort the results obtained. This applies to Runge–Kutta methods used for the numerical integration of ordinary differential equations, that are ubiquitous to model fundamental laws of physics, chemistry, biology or economy. We provide a Coq formalization of the rounding error analysis of Runge–Kutta methods applied to linear systems and implemented in floating-point arithmetic. We propose a generic methodology to build a bound on the error accumulated over the iterations, taking gradual underflow into account. We then instantiate this methodology for two classic Runge–Kutta methods, namely Euler and RK2. The formalization of the results include the definition of matrix norms, the proof of rounding error bounds of matrix operations and the formalization of the generic results and their applications on examples. In order to support the proposed approach, we provide numerical experiments on examples coming from nuclear physics applications.

数字错误是隐蔽的、难以预测的，并且是不同级别的关键系统设计中固有的。事实上，数值算法通常构成理想数学模型的近似，该理想数学模型本身构成经历了多个测量误差的物理现实的近似。除此之外，由于计算机算术实现而增加了舍入误差，即使它们会显着扭曲所获得的结果，但通常会被忽略。这适用于用于常微分方程数值积分的龙格-库塔方法，该方法普遍用于模拟物理、化学、生物或经济的基本定律。我们提供了应用于线性系统并以浮点运算实现的龙格-库塔方法的舍入误差分析的 Coq 形式化。我们提出了一种通用方法来建立迭代过程中累积的误差的界限，并考虑到逐渐下溢。然后，我们将这种方法实例化为两种经典的龙格-库塔方法，即 Euler 和 RK2。结果的形式化包括矩阵范数的定义、矩阵运算的舍入误差界限的证明以及通用结果的形式化及其在示例中的应用。为了支持所提出的方法，我们提供了来自核物理应用示例的数值实验。