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# One dimensional root-finding and the solver classes

One-dimensional root finding algorithms can be divided into two classes, `root bracketing` and `root polishing`. The state for bracketing solvers is held in a `GSL::Root::FSolver` object. The updating procedure uses only function evaluations (not derivatives). The state for root polishing solvers is held in a `GSL::Root::FdfSolver` object. The updates require both the function and its derivative (hence the name fdf) to be supplied by the user.

Contents:

1. Solver classes

2. Methods

   1. Methods to solve equations

   2. GSL::Function_fdf class: Providing the function to solve

   3. Search Stopping Parameters

3. Higher-level interface

4. Example

## Solver classes

- GSL::Root::FSolver.alloc(T)

  This creates a equation solver with a root bracketing algorithm of type `T`. The type `T` is given by a String or a constant,

  - `"bisection"` or `GSL::Root::FSolver::BISECION`

  - `"falsepos"` or `GSL::Root::FSolver::FALSEPOS`

  - `"brent"` or `GSL::Root::FSolver::BRENT`.

  - Ex:

        include GSL::Root
        s1 = FSolver.alloc("bisection")
        s2 = FSolver.alloc("brent")
        s3 = FSolver.alloc(FSolver::BISECTION)
        s4 = FSolver.alloc(FSolver::BRENT)

- GSL::Root::FdfSolver.alloc(T)

  This creates a derivative-based solver of type `T`. The type `T` is given by a String or a constant,

  - `"newton"` or `GSL::Root::FdfSolver::NEWTON`

  - `"secant"` or `GSL::Root::FdfSolver::SECANT`

  - `"steffenson"` or `GSL::Root::FdfSolver::STEFFENSON`.

## Methods

- GSL::Root::FSolver#set(f, xl, xu)

  This initialize the solver `self` to use the function `f`, and the initial search interval `xl, xu`. The function to be solved `f` is given an instanse of the GSL::Function class.

- GSL::Root::FdfSolver#set(fdf, r)

  This initializes, or reinitializes, an existing solver `self` to use the function and derivative `fdf` and the initial guess `r`. Here `fdf` is a `GSL::Function_fdf` object (see below).

### Methods to solve equations

- GSL::Root::FSolver#iterate

- GSL::Root::FdfSolver#iterate

  This performs a single iteration of the solver. If the iteration encounters an unexpected problem then an error code will be returned ( `GSL::EBADFUNC` or `GSL::EZERODIV` ).

- GSL::Root::FSolver#root

- GSL::Root::FdfSolver#root

  Returns the current estimate of the root.

- GSL::Root::FSolver#name

- GSL::Root::FdfSolver#name

  This returns the name of the algorithm.

- GSL::Root::FSolver#x_lower

- GSL::Root::FSolver#x_upper

  Return the current bracketing interval for the solver.

### GSL::Function_fdf class: Providing the function to solve

The `FSolver` object require an instance of the GSL::Function class, which is already introduced elsewhere. The `FdfSolver` which uses the root-polishing algorithm requires not only the function to solve, but also procedures to calculate the derivatives. This is given by the `GSL::Function_fdf` class.

- GSL::Function_fdf.alloc()

- GSL::Function_fdf.alloc(f, df)

- GSL::Function_fdf.alloc(f, df, fdf)

  Constructors. Here `f, df` are Ruby `Proc` objects which return a single value. The option `fdf` must return an array which contain the values of the function and its derivative.

- GSL::Function_fdf#set(f, df)

- GSL::Function_fdf#set(f, df, fdf)

  This initializes or reinitializes the `Function_fdf` object `self` by two or three `Proc` objects `f, df` and `fdf`.

  - ex: A quadratic equation a*x*x + b*x + c = 0:

        # Returns a value of the function
        f = Proc.new { |x, params|
          a = params[0]; b = params[1]; c = params[2]
          (a*x + b)*x + c
        }
        # Calculate the derivative
        df = Proc.new { |x, params|
          a = params[0]; b = params[1]
          2*a*x + b
        }

        function_fdf = Function_fdf.alloc(f, df)

- GSL::Function_fdf#set(f, df, params...)

- GSL::Function_fdf#set(f, df, fdf, params...)

  This sets or resets the procedures and the constant parameters in the function.

- GSL::Function_fdf#set_params(...)

  This sets or resets the constant parameters in the function.

  - Ex: x*x - 5 == 0

        function_fdf.set_params([1, 0, -5])

### Search Stopping Parameters

- GSL::Root::test_interval(xl, xu, epsrel, epsabs)

  This function tests for the convergence of the interval `[xl, xu]` 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 roots close to the origin.

  This condition on the interval also implies that any estimate of the root r in the interval satisfies the same condition with respect to the true root r0,

      |r - r0| < epsabs + epsrel r0

  assuming that the true root r0 is contained within the interval.

- GSL::Root::test_delta(x1, x0, epsrel, epsabs)

  This function tests for the convergence of the sequence ..., `x0, x1` with absolute error `epsabs` and relative error `epsrel`. The test returns `GSL::SUCCESS` if the following condition is achieved,

      |x_1 - x_0| < epsabs + epsrel |x_1|

  and returns `GSL::CONTINUE` otherwise.

- GSL::Root::test_residual(f, epsabs)

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

      |f| < 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, |f(x)|, is small enough.

## High-level interface

- GSL::Root:FSolver.solve(func, [xl, xu], [epsabs = 0, epsrel = 1e-6])

  This method try to find a root of the function `func` between the interval `[xl, xu]`, with the accuracy `[epsabs, epsrel]` (optional). An array of 3 elements is returned, as [`root, iterations, status`].

- GSL::Root:FdfSolver.solve(func, x0, [epsabs = 0, epsrel = 1e-6])

  This method try to find a root of the function `func` around `x0`, with the accuracy `[epsabs, epsrel]` (optional). An array of 3 elements is returned, as [`root, iterations, status`].

- GSL::Function#fsolve([xl, xu])

- GSL::Function#fsolve(xl, xu)

  These methods try to find a root of `f(x) = 0` between the interval `[xl, xh]` using Brent's algorithm. An array of 3 elements is returned, as [`root, iterations, status`].

  - ex:

        f = Function.alloc { |x| x*x - 5 }
        f.fsolve([0, 5])             <----- 2.23606797749979

## Example

This example is equivalent to the one found in the GSL manual, using the Brent's algorithm to solve the equation x^2 - 5 = 0.

    #!/usr/bin/env ruby
    require "gsl"

    #solver = Root::FSolver.alloc("bisection")
    #solver = Root::FSolver.alloc("falsepos")
    solver = Root::FSolver.alloc(Root::FSolver::BRENT)
    puts "using #{solver.name} method"

    func = GSL::Function.alloc { |x, params|      # Define a function to solve
      a = params[0]; b = params[1]; c = params[2]
      (a*x + b)*x + c
    }
    expected = Math::sqrt(5.0)

    func.set_params([1, 0, -5])

    printf("%5s [%9s, %9s] %9s %10s %9s\n",
            "iter", "lower", "upper", "root",
            "err", "err(est)")

    solver.set(func, 0.0, 5.0)              # initialize the solver
    i = 1
    begin
      status = solver.iterate
      r = solver.root
      xl = solver.x_lower
      xu = solver.x_upper
      status = Root.test_interval(xl, xu, 0, 0.001)   # Check convergence
      if status == GSL::SUCCESS
        printf("Converged:\n")
      end
      printf("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
             i, xl, xu, r, r - expected, xu - xl)

      i += 1
    end while status != GSL::SUCCESS

The following is an another version, using the `FdfSolver` with the Newton-Raphson algorithm.

    #!/usr/bin/env ruby
    require "gsl"

    f = Proc.new { |x, params|
      a = params[0]; b = params[1]; c = params[2]
      (a*x + b)*x + c
    }

    df = Proc.new { |x, params|
      a = params[0]; b = params[1]
      2.0*a*x + b
    }

    function_fdf = Function_fdf.alloc(f, df)
    params = [1, 0, -5]
    function_fdf.set_params(params)

    solver = Root::FdfSolver.alloc(Root::FdfSolver::NEWTON)
    puts "using #{solver.name} method"

    expected = Math::sqrt(5.0)
    x = 5.0
    solver.set(function_fdf, x)

    printf("%-5s %10s %10s %10s\n", "iter", "root", "err", "err(est)")
    iter = 0
    begin
      iter += 1
      status = solver.iterate
      x0 = x
      x = solver.root

      status = Root::test_delta(x, x0, 0, 1e-3)

      if status == GSL::SUCCESS
        printf("Converged:\n")
      end

      printf("%5d %10.7f %+10.7f %10.7f\n", iter, x, x - expected, x - x0)
    end while status != GSL::SUCCESS

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