We present an algorithm which uses Fujiwara's inequality to bound algebraic functions over ellipses of a certain type, allowing us to concretely implement a rigorous Gauss-Legendre integration method for algebraic functions over a line segment. We consider path splitting strategies to improve convergence of the method and show that these yield significant practical and asymptotic benefits. We implemented these methods to compute period matrices of algebraic Riemann surfaces and these are available in SageMath.

翻译：我们提出一种算法，利用藤原不等式来约束特定椭圆上的代数函数，从而能够具体实现对线段上代数函数的严格高斯-勒让德积分方法。我们考虑路径分割策略以改进该方法的收敛性，并证明这些策略能带来显著的实用性和渐近性优势。我们将这些方法实现用于计算代数黎曼曲面的周期矩阵，这些实现已集成于SageMath中。