# minimization using the Brent algorithm (brent.c)

Z F mail4me9999@yahoo.com
Mon Mar 17 22:44:00 GMT 2003

```Dear Fabrice Rossi,

I have never worked with multidimentional minimization problems so you
might be correct that some other algorithms use golden/Brent or even
reuse some data from the previous steps explicitly. I will not comment
on that.

What it means to me that a separate function should be used of 1-D case
where there is no sence of using a first guess for the extremum. You
are not getting the golden algorithm if the initial point is not the
golden section.

Moreover, in 1-D when no knowledge about derivatives of F(x) exist and
Brent fails, there is a method which is faster than the golden section
algorithm. This method does not need the first guess and given the
search interval (a,b) and the required accuracy eps, the number of
function evaluations is fixed. Thus, the user not only gets the fastest
convergence, but he knows how long it will take.

The statement that "choosing the golden section as the bisection ratio
can be shown to provide the fastest convergence for this type of
algorithm." from the GSL manual might not be correct.

Anyone heard of the use fo Fibonacci (spelling?) numbers for the
minimization algorithm?

Lazar

--- Fabrice Rossi <rossi@ufrmd.dauphine.fr> wrote:
> Stated like this, I mostly agree with you, except maybe on
> vocabulary.
> What is called the Brent algorithm or the Golden section search is
> exactly what is (currently) implemented in GSL. But as Michael and
> you
> pointed out, in practical situation, we need a way to start the
> algorithm. The rationnal why this is not included into GSL (well, it
> once was in my first implementation of multidimensional minimization)
> is
> (to my mind) that it is highly problem dependant. Nevertheless, this
> discussion seems to show that a general solution (maybe suboptimal)
> is
> needed.
>
> By the way a simple example why this is problem dependant: take
> multidimensional minimization. In general, you have to perform a line

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