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Rigorous numerical integration of algebraic functions

Nils Bruin, Linden Disney-Hogg, Wuqian Effie Gao

TL;DR

This work tackles the certified numerical evaluation of line integrals of algebraic functions along straight segments by modeling the integrand as a root of a polynomial $f(z,g)$ and bounding the complex variation of $g$ via Fujiwara's bound. The authors develop a rigorous Gauss-Legendre quadrature framework with adaptive path splitting, using ellipse-based error control and local bounds $M_j$ to compute the required number of nodes per segment, and provide asymptotic analyses showing substantial gains when critical points approach the path. They demonstrate practical efficacy by computing period matrices of algebraic Riemann surfaces with certified accuracy, implementing the method in SageMath 9.6, and comparing to heuristic approaches on random plane quartics. The contributions bridge numerical analysis and algebraic geometry, offering a scalable, certified tool for high-precision period computations and Abel–Jacobi map evaluations on complex curves. The approach provides clear guidance on when path splitting yields asymptotic and practical benefits, and it situates the method within broader alternatives for node computation and quadratures.

Abstract

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.

Rigorous numerical integration of algebraic functions

TL;DR

This work tackles the certified numerical evaluation of line integrals of algebraic functions along straight segments by modeling the integrand as a root of a polynomial and bounding the complex variation of via Fujiwara's bound. The authors develop a rigorous Gauss-Legendre quadrature framework with adaptive path splitting, using ellipse-based error control and local bounds to compute the required number of nodes per segment, and provide asymptotic analyses showing substantial gains when critical points approach the path. They demonstrate practical efficacy by computing period matrices of algebraic Riemann surfaces with certified accuracy, implementing the method in SageMath 9.6, and comparing to heuristic approaches on random plane quartics. The contributions bridge numerical analysis and algebraic geometry, offering a scalable, certified tool for high-precision period computations and Abel–Jacobi map evaluations on complex curves. The approach provides clear guidance on when path splitting yields asymptotic and practical benefits, and it situates the method within broader alternatives for node computation and quadratures.

Abstract

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.
Paper Structure (15 sections, 11 theorems, 45 equations, 4 figures)

This paper contains 15 sections, 11 theorems, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Error bounds on numerical integration
  3. Gauss-Legendre integration
  4. Bounding the variation of an algebraic function
  5. Path splitting strategy
  6. Sensitivity to nearby critical points
  7. Asymptotics for a critical point approaching the integration path
  8. Integration paths passing between close-by critical points
  9. Comparison of performance on periods of smooth plane quartic curves
  10. Further Computational Considerations
  11. Caching Nodes
  12. Alternative Methods for Calculating Nodes
  13. Alternative Integration Quadratures
  14. Rounding Error
  15. Slow Convergence

Key Result

Proposition 1.1

Take $g(z)$ with a critical point at $z=0$ such that $\lim_{z\to 0} g(z)z^v$ exists and suppose that the function is bounded on the annuli $\delta \leq |z|<2$ for $\delta>1$. Choose an absolute error tolerance $E_\mathrm{tol}>0$. Then using Strategy strat:main, we can compute an approximation $I_q$

Figures (4)

  • Figure 1: Example configuration of circles and ellipses for both strategies when there is a single critical point at $0.3+0.4i$.
  • Figure 2: $\log(N^{(i)}(x+iy))$ for $x,y\in(0,1)$ for Strategies \ref{['strat:main']} and \ref{['strat:ref']}, with $E_\mathrm{tol}=2^{-100}$ and $v=\frac{1}{2}$.
  • Figure 3: Proxy and actual bounds on number of nodes.
  • Figure 4: Ratio of runtimes of the rigorous methods ($t_i$) to the heuristic method ($t_h$). A reference time of $t_{\min} \approx 1.2$ seconds is taken.

Theorems & Definitions (23)

  • Proposition 1.1
  • Theorem 2.1: Rabinowitz1969NeurohrThesis
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4: Ahlfors1979, pp.124-126
  • Corollary 2.5
  • Lemma 2.6: Fujiwara1916
  • Corollary 2.7
  • Remark 2.8
  • ...and 13 more