min.rdoc

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Last Update: Sun Nov 14 14:53:48 -0800 2010

One dimensional Minimization

This chapter describes routines for finding minima of arbitrary one-dimensional functions.

Contents:

  1. Introduction
  2. GSL::Min::FMinimizer class
  3. Iteration
  4. Stopping Parameters
  5. Examples

Introduction

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, 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 (or solver) state, s, for algorithm T
  • update s using the iteration T
  • test s for convergence, and repeat iteration if necessary

The state of the minimizers is held in a GSL::Min::FMinimizer object. The updating procedure use only function evaluations (not derivatives). The function to minimize is given as an instance of the GSL::Function class to the minimizer.

GSL::Min::FMinimizer class


  • GSL::Min::FMinimizer.alloc(t)

    These method create an instance of the GSL::Min::FMinimizer class of type t. The type t is given by a String,

    • "goldensection"
    • "brent"
    • "quad_golden"

    or by a Ruby constant,

    • GSL::Min::FMinimizer::GOLDENSECTION
    • GSL::Min::FMinimizer::BRENT
    • GSL::Min::FMinimizer::QUAD_GOLDEN (GSL-1.13)

    ex)

        include GSL::Min
        s = FMinimizer.alloc(FMinimizer::BRENT)
    

  • GSL::Min::FMinimizer#set(f, xmin, xlow, xup)

    This method sets, or resets, an existing minimizer self to use the function f (given by a GSL::Function object) and the initial search interval [xlow, xup], with a guess for the location of the minimum xmin.

    If the interval given does not contain a minimum, then the method returns an error code of GSL::FAILURE.


  • GSL::Min::FMinimizer#set_with_values(f, xmin, fmin, xlow, flow, xup, fup)

    This method is equivalent to Fminimizer#set but uses the values fmin, flowe and fup instead of computing f(xmin), f(xlow) and f(xup).


  • GSL::Min::FMinimizer#name

    This returns the name of the minimizer.

Iteration


  • GSL::Min::FMinimizer#iterate

    This method performs a single iteration of the minimizer self. If the iteration encounters an unexpected problem then an error code will be returned,

    • GSL::EBADFUNC: the iteration encountered a singular point where the function evaluated to Inf or NaN.
    • GSL::FAILURE: the algorithm could not improve the current best approximation or bounding interval.

    The minimizer maintains a current best estimate of the position of the minimum at all times, and the current interval bounding the minimum. This information can be accessed with the following auxiliary methods


  • GSL::Min::FMinimizer#x_minimum

    Returns the current estimate of the position of the minimum for the minimizer self.


  • GSL::Min::FMinimizer#x_upper
  • GSL::Min::FMinimizer#x_lower

    Return the current upper and lower bound of the interval for the minimizer self.


  • GSL::Min::FMinimizer#f_minimum
  • GSL::Min::FMinimizer#f_upper
  • GSL::Min::FMinimizer#f_lower

    Return the value of the function at the current estimate of the minimum and at the upper and lower bounds of interval for the minimizer self.

Stopping Parameters


  • GSL::Min::FMinimizer#test_interval(epsabs, epsrel)
  • GSL::Min.test_interval(xlow, xup, epsabs, epsrel)

    These methoeds test for the convergence of the interval

    xlow, xup
    with absolute error epsabs and relative

    error epsrel. The test returns GSL::SUCCESS if the following condition is achieved,

      |a - b| < epsabs + epsrel min(|a|,|b|)
    

    when the interval x = [a,b] does not include the origin. If the interval includes the origin then min(|a|,|b|) is replaced by zero (which is the minimum value of |x| over the interval). This ensures that the relative error is accurately estimated for minima close to the origin.

    This condition on the interval also implies that any estimate of the minimum x_m in the interval satisfies the same condition with respect to the true minimum x_m^*,

      |x_m - x_m^*| < epsabs + epsrel x_m^*
    

    assuming that the true minimum x_m^* is contained within the interval.

Example

To find the minimum of the function f(x) = cos(x) + 1.0:

     #!/usr/bin/env ruby
     require("gsl")
     include GSL::Min

     fn1 = Function.alloc { |x| Math::cos(x) + 1.0 }

     iter = 0;  max_iter = 500
     m = 2.0             # initial guess
     m_expected = Math::PI
     a = 0.0
     b = 6.0

     gmf = FMinimizer.alloc(FMinimizer::BRENT)
     gmf.set(fn1, m, a, b)

     printf("using %s method\n", gmf.name)
     printf("%5s [%9s, %9s] %9s %10s %9s\n", "iter", "lower", "upper", "min",
            "err", "err(est)")

     printf("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n", iter, a, b, m, m - m_expected, b - a)

     begin
       iter += 1
       status = gmf.iterate
       status = gmf.test_interval(0.001, 0.0)
       puts("Converged:") if status == GSL::SUCCESS
       a = gmf.x_lower
       b = gmf.x_upper
       m = gmf.x_minimum
       printf("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
              iter, a, b, m, m - m_expected, b - a);
     end while status == GSL::CONTINUE and iter < max_iter

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