On Convergence Rates of Root-Seeking Methods

[Claudio L.]

(03-11-2018 11:36 PM)Paul Dale Wrote:  Rather than Newton's which requires the derivative, did you mean secant?

I did mean Newton, with a formula to approximate the derivative, or if the expression has a derivative the system can determine it algebraically and use it.

(03-11-2018 11:36 PM)Paul Dale Wrote:  Dieter is correct, I'd recommend Brent's method for a bracket solver. It is guaranteed to get a solution and is quadratically convergent almost always. There was a modification in the 34S to also include the Ridder's method in addition to Brent's secant, inverse quadratic and bisection methods. Testing indicated that it was beneficial, although I don't remember the conditions under which it is used.

There is quite a bit of code before the searching can begin. If the initial estimates don't bracket the solution, a bisection step is done before switching to more advanced methods. There might be a forced secant step after the bisection, I'm not sure anymore. If the first two function evaluations are equal, it does a bisection step and if that produces a third equal value a right then left step exponentially increasing interval scan is done. When there is only one initial estimate, it stars with the interval scan.


Pauli

Brent seems like a powerhouse combo of other methods. If I implement Brent, does it make any sense to offer the user a choice to use bisection, etc.? It will probably end up being the default method, unless complex mode is enabled, then something like Muller seems to be the only choice?