Jon, That's a good question. Short answer: The exact algorithm in optimize() is a bit more complicated than the bi-section, but conceptually you can think of this as a bi-section on the first derivative of the function (the same properties hold). Long answer: Now the reason why optimize does not in fact require you to enter f'(x) is because it relies on a method that is often referred to as quadratic fit. The way this works is that it fits a quadratic function through the three points of the objective function given by the lower interval bound a, the midpoint m, and the upper interval bound b. The quadratic function that fits these three points is q(lambda) = c2 lambda^2 + c2 lambda + c3 where lambda is the vector lambda = (a,m,b). It's like a parabola fitted through these points. Then you can solve this for lambda_{k+1} = argmin q(lambda) and for this minimization a closed form solution exists (like in the NR). The solution is (subject to key stroke errors): lambda_{k+1} = 0.5 * {(m^2-b^2)Q(a)+ {(b^2-a^2)Q(m)+ {(a^2-m^2)Q(b)}/{(m-b)Q(a)+ {(b-a)Q(m)+(a-m)Q(b)} where Q() is the quadratic fit q() evaluated at either a, m, or b respectively. Then you update the "outside" point (a, m, or b) and continue with the next interval. The properties of this are the same as in the bi-section (unimodality, the min/max needs to be contained in the search interval) but you don't need the analytical derivative. Convergence is about as fast as with the golden section on f'(x). Hope this helps. The c code for the exact implementation is here: http://www.netlib.org/fmm/fmin.f if you want to take a look. Jens