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# 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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