roots.rdoc

Path: rdoc/roots.rdoc
Last Update: Sun Nov 14 14:53:48 -0800 2010

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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