Module rgsl::types::minimizer [−] [src]

One dimensional Minimization

This chapter describes routines for finding minima of arbitrary one-dimensional functions. The library provides low level components for a variety of iterative minimizers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the algorithms. Each class of methods uses the same framework, so that you can switch between minimizers at runtime without needing to recompile your program. Each instance of a minimizer keeps track of its own state, allowing the minimizers to be used in multi-threaded programs.

Overview

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

The value of the function at x must be less than the value of the function at the ends of the interval,

f(a) > f(x) < f(b) This condition guarantees that a minimum is contained somewhere within the interval. On each iteration a new point x' is selected using one of the available algorithms. If the new point is a better estimate of the minimum, i.e. where f(x') < f(x), then the current estimate of the minimum 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 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).

Caveats

Note that minimization functions can only search for one minimum at a time. When there are several minima in the search area, the first minimum to be found will be returned; however it is difficult to predict which of the minima this will be. In most cases, no error will be reported if you try to find a minimum in an area where there is more than one.

With all minimization algorithms it can be difficult to determine the location of the minimum to full numerical precision. The behavior of the function in the region of the minimum x* can be approximated by a Taylor expansion,

y = f(x*) + (1/2) f''(x*) (x - x*)2

and the second term of this expansion can be lost when added to the first term at finite precision. This magnifies the error in locating x*, making it proportional to \sqrt \epsilon (where \epsilon is the relative accuracy of the floating point numbers). For functions with higher order minima, such as x4, the magnification of the error is correspondingly worse. The best that can be achieved is to converge to the limit of numerical accuracy in the function values, rather than the location of the minimum itself.

Providing the function to minimize

You must provide a continuous function of one variable for the minimizers to operate on. In order to allow for general parameters the functions are defined by a gsl_function data type (see Providing the function to solve).

Iteration

The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any minimizer of the corresponding type. The same functions work for all minimizers so that different methods can be substituted at runtime without modifications to the code.

Stopping Parameters

A minimization procedure should stop when one of the following conditions is true:

  • A minimum has been found to within the user-specified precision.
  • A user-specified maximum number of iterations has been reached.
  • An error has occurred.

The handling of these conditions is under user control. The function below allows the user to test the precision of the current result.

Minimization Algorithms

The minimization algorithms described in this section require an initial interval which is guaranteed to contain a minimum—if a and b are the endpoints of the interval and x is an estimate of the minimum then f(a) > f(x) < f(b). This ensures that the function has at least one minimum somewhere in the interval. If a valid initial interval is used then these algorithm cannot fail, provided the function is well-behaved.

Structs

Minimizer
MinimizerType