Multidimensional Root-Finding

# Title: Multidimensional Root-Finding

## 1 Initializing the Solver

Two types of solvers are available. The solver itself depends only on the dimension of the problem and the algorithm and can be reused for different problems. The `FdfSolver` requires derivatives of the function to solve.

`GSL::MultiRoot::FSolver.new(T, n)`
`GSL::MultiRoot::FSolver.alloc(T, n)`
  This creates an instance of the `FSolver` class of type T for a system of n dimensions. The type is given by a constant or a string,
  - GSL::MultiRoot:FSolver::HYBRIDS, or "hybrids"
  - GSL::MultiRoot:FSolver::HYBRID, or "hybrid"
  - GSL::MultiRoot:FSolver::DNEWTON, or "dnewton"
  - GSL::MultiRoot:FSolver::BROYDEN, or "broyden"
`GSL::MultiRoot::FdfSolver.new(T, n)`
`GSL::MultiRoot::FdfSolver.alloc(T, n)`
  This creates an instance of the `FdfSolver` class of type T for a system of n dimensions. The type is given by a constant,
  - GSL::MultiRoot:FdfSolver::HYBRIDSJ, or "hybridsj"
  - GSL::MultiRoot:FdfSolver::HYBRIDJ, or "hybridj",
  - GSL::MultiRoot:FdfSolver::NEWTON, or "newton",
  - GSL::MultiRoot:FdfSolver::GNEWTON, or "gnewton
`GSL::MultiRoot::FSolver#set(func, x)`
  This method sets, or resets, an existing solver self to use the function func and the initial guess x. Here x is a `Vector`, and func is a `MultiRoot:Function` object.
`GSL::MultiRoot::FdfSolver#set(func_fdf, x)`
  This method sets, or resets, an existing solver self to use the function func_fdf and the initial guess x. Here x is a `Vector`, and func_fdf is a `MultiRoot:Function_fdf` object.
`GSL::MultiRoot::FSolver#name`
`GSL::MultiRoot::FdfSolver#name`

## 2 Providing the function to solve

`GSL::MultiRoot:Function.new(proc, dim, params)`

  See example below:

      # x: vector, current guess
      # params: a scalar or an array
      # f: vector, function value
      proc = Proc.new { |x, params, f|
        a = params[0]; b = params[1]
        x0 = x[0]; x1 = x[1]
        f[0] = a*(1 - x0)
        f[1] = b*(x1 - x0*x0)
      }

      params = [1.0, 10.0]
      func = MultiRoot::Function.new(proc, 2, params)
      fsolver = MultiRoot::FSolver.new("broyden", 2)
      x = [-10, -5]    # initial guess
      fsolver.set(func, x)

`GSL::MultiRoot:Function_fdf.new(proc, dim, params)`

  See the example below:

      procf = Proc.new { |x, params, f|
        a = params[0]; b = params[1]
        x0 = x[0]; x1 = x[1]
        f[0] = a*(1 - x0)
        f[1] = b*(x1 - x0*x0)
      }

      procdf = Proc.new { |x, params, jac|
        a = params[0]; b = params[1]
        jac.set(0, 0, -a)
        jac.set(0, 1, 0)
        jac.set(1, 0, -2*b*x[0])
        jac.set(1, 1, b)
      }

      params = [1.0, 10.0]
      func_fdf = MultiRoot::Function_fdf.new(procf, procdf, n, params)

      fdfsolver = MultiRoot::FdfSolver.new("gnewton", n)
      x = [-10.0, -5.0]
      fdfsolver.set(func_fdf, x)

## 3 Iteration

`GSL::MultiRoot::FSolver#interate`
`GSL::MultiRoot::FdfSolver#interate`

  These methods perform a single iteration of the solver 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 or its derivative evaluated to Inf or NaN.
  - GSL_ENOPROG: the iteration is not making any progress, preventing the algorithm from continuing.

  The solver maintains a current best estimate of the root at all times. This information can be accessed with the following auxiliary methods.

`GSL::MultiRoot::FSolver#root`
`GSL::MultiRoot::FdfSolver#root`
  These methods return the current estimate of the root (Vector) for the solver self.
`GSL::MultiRoot::FSolver#f`
`GSL::MultiRoot::FdfSolver#f`
  These methds return the function value `f(x)` (Vector) at the current estimate of the root for the solver self.
`GSL::MultiRoot::FSolver#dx`
`GSL::MultiRoot::FdfSolver#dx`
  These method return the last step dx (Vector) taken by the solver self.

## 4 Search Stopping Parameters

`GSL::MultiRoot::FSolver#test_delta(epsabs, epsrel)`
`GSL::MultiRoot::FdfSolver#test_delta(epsabs, epsrel)`

  This method tests for the convergence of the sequence by comparing the last step `dx` with the absolute error epsabs and relative error epsrel to the current position `x`. The test returns `GSL::SUCCESS` if the following condition is achieved,

      |dx_i| < epsabs + epsrel |x_i|

  for each component of `x` and returns `GSL::CONTINUE` otherwise.

`GSL::MultiRoot::FSolver#test_residual(epsabs)`
`GSL::MultiRoot::FdfSolver#test_residual(epsabs)`

  This method tests the residual value `f` against the absolute error bound epsabs. The test returns `GSL::SUCCESS` if the following condition is achieved,

      sum_i |f_i| < epsabs

  and returns `GSL::CONTINUE` otherwise. This criterion is suitable for situations where the precise location of the root, `x`, is unimportant provided a value can be found where the residual is small enough.

## 5 Higher Level Interface

`GSL::MultiRoot::Function#solve(x0, max_iter = 1000, eps = 1e-7, type = "hybrids")`
`GSL::MultiRoot::FSolver#solve(max_iter = 1000, eps = 1e-7)`
`GSL::MultiRoot::FSolver.solve(fsolver, max_iter = 1000, eps = 1e-7)`

See sample script `samples/multiroot/fsolver3.rb`.

## 6 Example

### 6.1 FSolver

    proc = Proc.new { |x, params, f|
      a = params[0];  b = params[1]
      x0 = x[0];  x1 = x[1]
      f[0] = a*(1 - x0)
      f[1] = b*(x1 - x0*x0)
    }

    params = [1.0, 10.0]
    func = MultiRoot::Function.new(proc, 2, params)

    fsolver = MultiRoot::FSolver.new("hybrid", 2)
    x = [-10, -5]
    fsolver.set(func, x)

    iter = 0
    begin
      iter += 1
      status = fsolver.iterate
      root = fsolver.root
      f = fsolver.f
      printf("iter = %3u x = % .3f % .3f f(x) = % .3e % .3e\n",
              iter, root[0], root[1], f[0], f[1])
      status = fsolver.test_residual(1e-7)
    end while status == GSL::CONTINUE and iter < 1000

### 6.2 FdfSolver

    n = 2

    procf = Proc.new { |x, params, f|
      a = params[0]; b = params[1]
      x0 = x[0]; x1 = x[1]
      f[0] = a*(1 - x0)
      f[1] = b*(x1 - x0*x0)
    }

    procdf = Proc.new { |x, params, jac|
      a = params[0]; b = params[1]
      jac.set(0, 0, -a)
      jac.set(0, 1, 0)
      jac.set(1, 0, -2*b*x[0])
      jac.set(1, 1, b)
    }

    params = [1.0, 10.0]
    f = MultiRoot::Function_fdf.new(procf, procdf, n, params)

    fdfsolver = MultiRoot::FdfSolver.new("gnewton", n)

    x = [-10.0, -5.0]

    fdfsolver.set(f, x)

    iter = 0
    begin
      iter += 1
      status = fdfsolver.iterate
      root = fdfsolver.root
      f = fdfsolver.f
      printf("iter = %3u x = % .3f % .3f f(x) = % .3e % .3e\n",
              iter, root[0], root[1], f[0], f[1])
      status = fdfsolver.test_residual(1e-7)
    end while status == GSL::CONTINUE and iter < 1000

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