TITLE: Computing Gauss Legendre quadrature for large $N$
QUESTION [14 upvotes]: I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it, and why should it work?
To those who need some background:
The goal is to approximate an integral by a discrete interpolating sum:
$$\int\limits_{(-1,1)} f(x) dx = \sum\limits_{j=-N}^N f(x_j)w^j $$
The question is, how to choose  $\{ x_j, w^j \} _j$ appropriately.The Gauss Legendre quadrature tells you (for good reasons) to choose $x_j $ to be the roots of the $n$-th Legendre polynomial.
Problem : A straightforward computation of the Legendre polynomial for high $N$ is highly unstable, as it involves "big" coefficients of alternating signs.
EDIT: 
I've found a very simple code that computes weight and abscissas using eigenvalues of a symmetric matrix, but doesn't seem use Golub Welsch. The matrix is
$$\forall 1\leq j\leq N-1 \,  A_{j,j+1} = A_{j+1,j}  = \frac{j}{\sqrt{4j^2 -1}}$$ with all other entries are zero. The discussion about it was split to another post.

REPLY [7 votes]: To add to Fredrik Johansson's answer: A nice history of algorithms for computing Gauss quadrature rules can be found in this SIAM News article by Alex Townsend.  Therein, it is stated that the "final chapter" was written by Ignace Bogaert in this SISC paper, which gives an algorithm that is even faster and more accurate than the algorithm of Hale & Townsend.
There is a free open-source implementation at https://sourceforge.net/projects/fastgausslegendrequadrature/