One dimensional Minimization

# Title: One dimensional Minimization

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

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

## 1 GSL::Min::FMinimizer class

`GSL::Min::FMinimizer.new(t)`
`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 Ruby constant,

  - GSL::Min::FMinimizer::GOLDENSECTION
  - GSL::Min::FMinimizer::BRENT

  ex1)

      include GSL
      s1 = Min::FMinimizer.new(Min::FMinimizer::GOLDENSECTION)

  ex2)

      include GSL::Min
      s2 = FMinimizer.new(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.

## 2 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.

## 3 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.

## 4 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.new { |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.new(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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