acb_calc_integrate_gl_auto_deg

With $\Delta = (b - a)/2$ and $m = (a + b)/2$, the integral is $\int_{-1}^{1} g$ with $g(t) = \Delta f(\Delta t + m)$. If $g$ is holomorphic and $|g| \le M$ inside the ellipse with foci $\pm 1$ and semi-axes $X$ and $Y = \sqrt{X^2 - 1}$, $\rho = X + Y$, the error of the $n$-point Gauss–Legendre rule is at most

a slightly weakened form of a bound of Trefethen [Tre2008]. $M$ is obtained by evaluating $f$ with order 1 on the rectangle $[\pm X] + [\pm Y] i$ mapped to the segment. The function tries $X = 2, 4, 8, 32, 512, \ldots$ (doubling the exponent) as long as $M$ is finite, and for each $X$ the smallest degree $n$ from the sequence $1, 2, 4, 6, 8, 12, 16, 22, 32, \ldots$ (steps of about $\sqrt{2}$) that meets the tolerance; the smallest $n$ found is used. The nodes and weights are computed by arb_hypgeom_legendre_p_ui_root [JM2018] (Roots of Legendre polynomials and Gauss–Legendre nodes) and cached (per thread), and the evaluations are distributed over threads when flint_get_num_threads() is greater than one.

@misc{flint-doc-acb-calc-acb-calc-integrate-gl-auto-deg,
  author       = {{The FLINT team}},
  title        = {{acb\_calc\_integrate\_gl\_auto\_deg (acb\_calc.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/acb_calc/acb_calc_integrate_gl_auto_deg.html},
  note         = {Generated 2026-11-02}
}