Overview

The minimization algorithms begin with a bounded region known to contain a minimum. The region is described by an lower bound @math{a} and an upper bound @math{b}, with an estimate of the minimum @math{x}.

@image{min-interval}

The value of the function at @math{x} must be less than the value of the function at the ends of the interval,

This condition guarantees that a minimum is contained somewhere within the interval. On each iteration a new point @math{x'} is selected using one of the available algorithms. If the new point is a better estimate of the minimum, @math{f(x') < f(x)}, then the current estimate of the minimum @math{x} is updated. The new point also allows the size of the bounded interval to be reduced, by choosing the most compact set of points which satisfies the constraint @math{f(a) > f(x) < f(b)}. The interval is reduced until it encloses the true minimum to a desired tolerance. This provides a best estimate of the location of the minimum and a rigorous error estimate.

Several bracketing algorithms are available within a single framework. The user provides a high-level driver for the algorithm, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,

• initialize minimizer state, s, for algorithm T
• update s using the iteration T
• test s for convergence, and repeat iteration if necessary

The state for the minimizers is held in a gsl_min_fminimizer struct. The updating procedure uses only function evaluations (not derivatives).