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.

Overview¶ ↑

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,

initialize solver state, s, for algorithm T
update s using the iteration T
test s for convergence, and repeat iteration if necessary

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.

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.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

Providing the function to solve¶ ↑

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

GSL::MultiRoot:Function_fdf.alloc(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.alloc(procf, procdf, n, params)

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

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.

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.

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 examples/multiroot/fsolver3.rb.

Example¶ ↑

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.alloc(proc, 2, params)

fsolver = MultiRoot::FSolver.alloc("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

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.alloc(procf, procdf, n, params)

fdfsolver = MultiRoot::FdfSolver.alloc("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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