Definite integrals with parameters of holonomic functions satisfy holonomic systems of linear partial differential equations. When we restrict parameters to a one dimensional curve, the system becomes a linear ordinary differential equation (ODE) with respect to a curve in the parameter space. We can evaluate the integral by solving the linear ODE numerically. This approach to evaluate numerically definite integrals is called the holonomic gradient method (HGM) and it is useful to evaluate several normalizing constants in statistics. We will discuss and compare methods to solve linear ODE's to evaluate normalizing constants.

翻译：具有参数的完整函数的定积分满足完整线性偏微分方程组。当我们将参数限制在一维曲线上时，该方程组变为关于参数空间中的曲线的线性常微分方程。我们可以通过数值求解该线性常微分方程来评估积分。这种数值评估定积分的方法称为完整梯度方法（HGM），对于评估统计学中的多个归一化常数非常有用。我们将讨论并比较求解线性常微分方程以评估归一化常数的各种方法。