Multidimensional Root-Finding

This chapter describes functions for multidimensional root-finding (solving nonlinear systems with n equations in n unknowns). The library provides low level components for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs.

  1. Overview

  2. Initializing the Solver

  3. Providing the function to solve

  4. Iteration

  5. Search Stopping Parameters

  6. Higher Level Interface

  7. Examples

    1. FSolver

    2. FdfSolver


The problem of multidimensional root finding requires the simultaneous solution of n equations, f_i, in n variables, x_i, In general there are no bracketing methods available for n dimensional systems, and no way of knowing whether any solutions exist. All algorithms proceed from an initial guess using a variant of the Newton iteration, where x, f are vector quantities and J is the Jacobian matrix J_{ij} = d f_i / d x_j. Additional strategies can be used to enlarge the region of convergence. These include requiring a decrease in the norm |f| on each step proposed by Newton's method, or taking steepest-descent steps in the direction of the negative gradient of |f|.

Several root-finding algorithms are available within a single framework. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,

The evaluation of the Jacobian matrix can be problematic, either because programming the derivatives is intractable or because computation of the n^2 terms of the matrix becomes too expensive. For these reasons the algorithms provided by the library are divided into two classes according to whether the derivatives are available or not.

The state for solvers with an analytic Jacobian matrix is held in a GSL::MultiRoot::FdfSolver object. The updating procedure requires both the function and its derivatives to be supplied by the user.

The state for solvers which do not use an analytic Jacobian matrix is held in a GSL::MultiRoot::FSolver object. The updating procedure uses only function evaluations (not derivatives). The algorithms estimate the matrix J or J^{-1} by approximate methods.

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.

Providing the function to solve


Search Stopping Parameters

Higher Level Interface

See sample script examples/multiroot/fsolver3.rb.



proc = { |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.alloc(proc, 2, params)

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

iter = 0
  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


n = 2

procf = { |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 = { |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.alloc(procf, procdf, n, params)

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

x = [-10.0, -5.0]

fdfsolver.set(f, x)

iter = 0
  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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