Multidimensional Minimization

# Title: Multidimensional Minimization

## 1 Initializing the Multidimensional Minimizer

`GSL::MultiMin::FdfMinimizer.new(type, n)`
`GSL::MultiMin::FMinimizer.new(type, n)`
  These method create a minimizer of type type for an n-dimension function. The type is given by a name of the algorithm, or by a Ruby constant.
  - `GSL::MultiMin::FdfMinimizer::CONJUGATE_FR` or `"conjugate_fr"`
  - `GSL::MultiMin::FdfMinimizer::CONJUGATE_PR` or `"conjugate_pr"`
  - `GSL::MultiMin::FdfMinimizer::VECTOR_BFGS` or `"vector_bfgs"`
  - `GSL::MultiMin::FdfMinimizer::STEEPEST_DESCENT` or `"steepest_descent"`
  - `GSL::MultiMin::FMinimizer::NMSIMPLEX` or `"nmsimplex"`

  - ex:

        include GSL::MultiMin
        m1 = FdfMinimizer.new(FdfMinimizer::CONJUGATE_FR, 2)
        m2 = FdfMinimizer.new("steepest_descent", 4)
        m3 = FMinimizer.new(FMinimizer::NMSIMPLEX, 3)
        m4 = FMinimizer.new("nmsimplex", 2)

`GSL::MultiMin::FdfMinimizer#set(func, x, step_size, tol)`
  This method initializes the minimizer self to minimize the function fdf (the `GSL::MultiMin::Function_fdf` class, see below) starting from the initial point x (`GSL::Vector`). The size of the first trial step is given by step_size (`Vector`). The accuracy of the line minimization is specified by tol.
`GSL::MultiMin::FMinimizer#set(func, x, step_size)`
  This method initializes the minimizer self to minimize the function func, starting from the initial point x (Vector). The size of the initial trial steps is given in vector step_size.
`GSL::MultiMin::FdfMinimizer#name`
`GSL::MultiMin::FMinimizer#name`
  These return the name of the minimizer self.

## 2 Providing a function to minimize

You must provide a parametric function of `n` variables for the minimizers to operate on. You may also need to provide a routine which calculates the gradient of the function. In order to allow for general parameters the functions are defined by the classes, `GSL::MultiMin::Function_fdf` and `GSL::MultiMin::Function`.

`GSL::MultiMin:Function_fdf.new(proc_f, proc_df, n)`
`GSL::MultiMin:Function_fdf.new(proc_f, proc_df, proc_fdf, n)`
`GSL::MultiMin:Function_fdf#set_procs(proc_f, proc_df)`
`GSL::MultiMin:Function_fdf#set_procs(proc_f, proc_df, n)`
`GSL::MultiMin:Function_fdf#set_procs(proc_f, proc_df, proc_fdf, n)`
`GSL::MultiMin:Function_fdf#set_params(params)`

  See example below.

      include GSL::MultiMin

      my_f = Proc.new { |v, params|
        x = v[0]; y = v[1]
        p0 = params[0]; p1 = params[1]
        10.0*(x - p0)*(x - p0) + 20.0*(y - p1)*(y - p1) + 30.0
      }

      my_df = Proc.new { |v, params, df|
        x = v[0]; y = v[1]
        p0 = params[0]; p1 = params[1]
        df[0] = 20.0*(x-p0)
        df[1] = 40.0*(y-p1)
      }

      my_func = Function_fdf.new(my_f, my_df, 2)
      my_func.set_params([1.0, 2.0])      # parameters

`GSL::MultiMin:Function.new(proc_f, n)`
`GSL::MultiMin:Function#set_proc(proc_f)`
`GSL::MultiMin:Function#set_proc(proc_f, n)`
`GSL::MultiMin:Function#set_params(params)`

  See example below.

      include GSL::MultiMin

      np = 2
      my_f = Proc.new { |v, params|
        x = v[0]; y = v[1]
        p0 = params[0]; p1 = params[1]
        10.0*(x - p0)*(x - p0) + 20.0*(y - p1)*(y - p1) + 30.0
      }

      my_func = Function.new(my_f, np)
      my_func.set_params([1.0, 2.0])      # parameters

## 3 Iteration

`GSL::MultiMin::FdfMinimizer#iterate`
`GSL::MultiMin::FMinimizer#iterate`
  These methods perform a single iteration of the minimizer self. If the iteration encounters an unexpected problem then an error code will be returned. The minimizer maintains a current best estimate of the minimum at all times. This information can be accessed with the following methods,
`GSL::MultiMin::FdfMinimizer#x`
`GSL::MultiMin::FdfMinimizer#minimum`
`GSL::MultiMin::FdfMinimizer#gradient`
`GSL::MultiMin::FMinimizer#x`
`GSL::MultiMin::FMinimizer#minimum`
`GSL::MultiMin::FMinimizer#size`
  These method return the current best estimate of the location of the minimum, the value of the function at that point, its gradient, and minimizer specific characteristic size for the minimizer self.
`GSL::MultiMin::FdfMinimizer#restart`
  This method resets the minimizer self to use the current point as a new starting point.

## 4 Stopping Criteria

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 methods below allow the user to test the precision of the current result.

`GSL::MultiMin::FdfMinimizer#test_gradient(epsabs)`
`GSL::MultiMin::FdfMinimizer.test_gradient(g, epsabs)`

  These method test the norm of the gradient g against the absolute tolerance epsabs. The gradient of a multidimensional function goes to zero at a minimum. The tests return `GSL::SUCCESS` if the following condition is achieved,

      |g| < epsabs

  and returns `GSL::CONTINUE` otherwise. A suitable choice of epsabs can be made from the desired accuracy in the function for small variations in `x`. The relationship between these quantities is given by `\delta f = g \delta x`.

`GSL::MultiMin::FdfMinimizer#test_size(epsabs)`
`GSL::MultiMin::FdfMinimizer.test_size(size, epsabs)`
  These method test the minimizer specific characteristic size (if applicable to the used minimizer) against absolute tolerance epsabs. The tests return (`GSL::SUCCESS` if the size is smaller than tolerance, otherwise `GSL::CONTINUE` is returned.

## 5 Examples

### 5.1 FdfMinimizer

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

    my_f = Proc.new { |v, params|
      x = v[0]; y = v[1]
      p0 = params[0]; p1 = params[1]
      10.0*(x - p0)*(x - p0) + 20.0*(y - p1)*(y - p1) + 30.0
    }

    my_df = Proc.new { |v, params, df|
      x = v[0]; y = v[1]
      p0 = params[0]; p1 = params[1]
      df[0] = 20.0*(x-p0)
      df[1] = 40.0*(y-p1)
    }

    my_func = Function_fdf.new(my_f, my_df, 2)
    my_func.set_params([1.0, 2.0])      # parameters

    x = Vector.new(5.0, 7.0)          # starting point

    minimizer = FdfMinimizer.new("conjugate_fr", 2)
    minimizer.set(my_func, x, 0.01, 1e-4)

    iter = 0
    begin
      iter += 1
      status = minimizer.iterate()
      status = minimizer.test_gradient(1e-3)
      if status == GSL::SUCCESS
        puts("Minimum found at")
      end
      x = minimizer.x
      f = minimizer.f
      printf("%5d %.5f %.5f %10.5f\n", iter, x[0], x[1], f)
    end while status == GSL::CONTINUE and iter < 100

### 5.2 FMinimizer

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

    np = 2

    my_f = Proc.new { |v, params|
      x = v[0]; y = v[1]
      p0 = params[0]; p1 = params[1]
      10.0*(x - p0)*(x - p0) + 20.0*(y - p1)*(y - p1) + 30.0
    }

    my_func = Function.new(my_f, np)
    my_func.set_params([1.0, 2.0])      # parameters

    x = Vector.new([5, 7])
    ss = Vector.new(np)
    ss.set_all(1.0)

    minimizer = FMinimizer.new("nmsimplex", np)
    minimizer.set(my_func, x, ss)

    iter = 0
    begin
      iter += 1
      status = minimizer.iterate()
      status = minimizer.test_size(1e-2)
      if status == GSL::SUCCESS
        puts("converged to minimum at")
      end
      x = minimizer.x
      printf("%5d ", iter);
      for i in 0...np do
        printf("%10.3e ", x[i])
      end
      printf("f() = %7.3f size = %.3f\n", minimizer.fval, minimizer.size);
    end while status == GSL::CONTINUE and iter < 100

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