[ldtasceh] Practical compass straightedge for impossible constructions?

Are there a compass and straightedge techniques for trisecting an angle, duplicating a cube, and constructing a regular heptagon (or n-gon), which theoretically take an infinite number of steps, but in practice converge very rapidly?  I think the goal is Newton's method for polynomial roots done with compass and straightedge, converging quadratically.

What about squaring the circle?  The goal here is a compass and straightedge translation of the Gauss Legendre algorithm.