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arXiv2026

Greedy Thiele continued-fraction approximation on continuum domains in the complex plane

We describe an adaptive greedy algorithm for Thiele continued-fraction (TCF) approximation of a function defined on a continuum domain in the complex plane. The algorithm iteratively selects interpolation nodes from an adaptively refined set of sample points on the domain boundary. We also present new algorithms for evaluating Thiele continued fractions and their accessory weights using only a single floating-point division. Numerical experiments comparing the greedy TCF method with the AAA algorithm on several challenging functions defined on the interval $[-1,1]$ and on the unit circle show that continuum TCF is consistently faster than AAA, by factors ranging from 8 to 40.

math.NA↗

arXiv2026

The Discrete Harmonic Center of a Quadrilateral

Triangulate a simple quadrilateral by connecting all vertices to an additional point. If the vertices carry values, the piecewise linear function can be assigned a Dirichlet energy. We show that the minimal Dirichlet energy as a function of the location of the inserted point is convex, and the location of the minimum is independent of the values at the corners - a quadrilateral has a discrete harmonic center, characterized by an equilibrium of currents across the inserted edges. It turns out that the fixed points of the Möbius involution swapping opposite corners of the quadrilateral are critical points of this energy, so the discrete harmonic center is Möbius-covariant. For tangential and cyclic quadrilaterals the center admits simple closed forms related to the circle centers. The center and its data-independence generalize to polytopes with d + 2 vertices in dimension d, but the conformal characterizations are special to four points in the plane.

math.DG↗