You may enjoy https://people.maths.ox.ac.uk/trefethen/mythspaper.pdf and http://www.chebfun.org/ATAP/ and in general http://www.chebfun.org/

The paper was great, thanks for sharing! I made this notebook while learning about quadrature to figure out how to compute elliptic integrals for arclength and to integrate joint distributions. After reading that paper, I think what I've written is accurate, but please let me know if you saw something I should correct. Maybe in the future, I can add more methods, like the Clenshaw–Curtis quadrature, which seems comparable to gaussian quadrature for some kinds of problems. Also in this notebook, I'm calculating the graphed polynomial with a Vandermonde matrix, which as the paper points out, has an exponentially increasing rounding error. I had not heard of the barycentric method for calculating polynomial interpolation. I'd love to figure out how to implement that so I can graph polynomials order 100.

Your notebook is fine. Just thought you might enjoy. I found the ATAP book quite insightful and all of the other publications from the Chebfun team are also great. Barycentric Lagrange interpolation is super easy to implement; I should try to make a notebook sometime. In the mean time here is a Desmos plot: https://www.desmos.com/calculator/qijzjcpi4a

Once you have computed nodes, values, and weights there is an implementation of (complex-valued) barycentric evaluation of a rational function in the 'reval' function of https://observablehq.com/@jrus/scpie

I finally got around to implementing the barycentric Lagrange polynomials, and the work great! In the first graph, before, I had to cap the number of points for a single Gaussian interval at 20 because of the exponential error of the Vandermonde matrices, but now I can include all 100 points. Thank you so much! If you ever make a notebook about it, let me know so I can link to it.