=begin = Multidimensional Root-Finding == 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 (({samples/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 (()) (()) (()) (()) =end