Newton or Secant?

[Namir]

I have always been partial to Newton's root-seeking method. For many decades, it has been my to-go method. I use Bisection in combination of, or instead of, Newton's method. I have always regarded the Secant method, which is based on a rough approximation of Newton's method, as mathematically inferior to Newton's method. I have to admit that I was surprised to see st least one HP calculator stat pac (and maybe Solver to?????) use the Secant Method.

Recently, I decided to sit down and compare Newton and Secant up close an personal. The Secant method requires one function call per iteration. Newton's method requires at least two function calls (one for the function and one for the approximation of the slope) per iteration. Comparing teh two methods depends on the function whose root you seek, the initial root guess (and how close they are to the true root). The test I conducted showed that teh Secant method does very well in requiring fewer function calls than Newton's method to attain a refined guess for the root. In addition, I saw several cases where the Secant method provided the final refined guess for the root in fewer iterations--a DOUBLE WIN!

So, as a new convert, I say don't discard the Secant method. Give it a try and see if it does well for the problem you are solving. Only if it does not pass the mustard test, then use Newton, Halley, or one of my many new algorithms (<-- cheap plug :-)).

Namir