[Dick Moores] > Aw, that's just amazing. Well, complex numbers are amazing in many ways. The code is actually obvious, if you understand the motivation. Polar coordinates are more natural for complex * / and **. If you a view a complex number c as a vector in the complex plane (from the origin to c), then c's n'th roots are just n equally spaced spokes on a wheel, with one of the spokes coinciding with c's "natural" angle divided by n. That determines the angles of all the spokes, since there are n spokes altogether and the angular difference between adjacent spokes is constant (2*pi/n). All the seeming obscurity in the code is really just due to converting between rectangular (x, y) and polar (length, angle) representations.
If we worked with polar coordinates all along, the input would be given as a magnitude and angle, say r and a. The n n'th roots then (still in polar coordinates) all have magnitude r**(1./n), and the n angles are a/n, a/n + b, a/n + 2*b, ..., and a/n + (n-1)*b where b=2*pi/n. Piece o' cake! > I put your function in http://www.rcblue.com/Python/croots.py, That's fine by me -- but I'd appreciate it if you stopped claiming there that my name is Bill <wink>. ... > Actually, I'm trying to write a Python script that computes all 3 > roots of a cubic equation. Do you happen to have one tucked > away in your store of wisdom and tricks? (One for real coefficients > will do). I don't, no. You can code one for cubics from Cardano's formula, e.g., http://mathworld.wolfram.com/CubicFormula.html but it's rarely worth the bother -- it's complicated and doesn't generalize. In practice, roots for polynomials beyond quadratics are usually obtained by numerical approximation methods that don't care much about the polynomial's degree. _______________________________________________ Tutor maillist - [EMAIL PROTECTED] http://mail.python.org/mailman/listinfo/tutor
