Newton or Secant?

[ttw]

There are several texts on the subject and there should be some stuff on the internet. Basically, Newton's method is faster per step (log of the error gets cut in half if the approximation is near enough to a root; the secant method cuts the log of the error by the Golden Section.) On the other hand, Newton's method nominally takes 2 function evaluations per step while the secant method only uses 1 function evaluation. It's possible that the derivative used in Newton's method can be had for little cost which would be an advantage for Newton's method. Normally the convergence region of the secant method is wider that that of Newton's.

There is another (poorer) method that sometimes is called the secant method but the proper name is Reguli Falsi. In this case, one does not use the last two iterations like the secant method but uses the last two iterations which have different signs.