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：完整函数的带参数定积分满足线性偏微分方程的完整系统。当我们将参数限制于一维曲线时，该方程组转化为关于参数空间曲线的线性常微分方程。通过数值求解该线性常微分方程，即可实现积分的求值。这种数值求定积分的方法称为完整梯度方法（HGM），对于统计中多种归一化常数的计算具有实用价值。本文将讨论并比较求解线性常微分方程以计算归一化常数的方法。