Halley.hh

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/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *\
 * Copyright (c) 2025, Davide Stocco and Enrico Bertolazzi.                  *
 *                                                                           *
 * The Optimist project is distributed under the BSD 2-Clause License.       *
 *                                                                           *
 * Davide Stocco                                           Enrico Bertolazzi *
 * University of Trento                                 University of Trento *
 * davide.stocco@unitn.it                         enrico.bertolazzi@unitn.it *
\* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
 
#pragma once
 
#ifndef OPTIMIST_ROOTFINDER_HALLEY_HH
#define OPTIMIST_ROOTFINDER_HALLEY_HH
 
#include "Optimist/RootFinder.hh"
 
namespace Optimist {
  namespace RootFinder {
 
    /*\
     |   _   _       _ _
     |  | | | | __ _| | | ___ _   _
     |  | |_| |/ _` | | |/ _ \ | | |
     |  |  _  | (_| | | |  __/ |_| |
     |  |_| |_|\__,_|_|_|\___|\__, |
     |                        |___/
    \*/

    template <typename Scalar>
    class Halley : public RootFinder<Scalar, Halley<Scalar>> {
     public:
      static constexpr bool RequiresFunction{true};
      static constexpr bool RequiresFirstDerivative{true};
      static constexpr bool RequiresSecondDerivative{true};
 
      OPTIMIST_BASIC_CONSTANTS(Scalar)
 
      
      Halley() {}

      constexpr std::string name_impl() const {
        return "Halley";
      }

      template <typename FunctionLambda,
                typename FirstDerivativeLambda,
                typename SecondDerivativeLambda>
      bool solve_impl(FunctionLambda &&function,
                      FirstDerivativeLambda &&first_derivative,
                      SecondDerivativeLambda &&second_derivative,
                      Scalar x_ini,
                      Scalar &x_sol) {
#define CMD "Optimist::RootFinder::Halley::solve(...): "
 
        // Reset internal variables
        this->reset_counters();
 
        // Print header
        if (this->m_verbose) {
          this->header();
        }
 
        // Initialize variables
        bool damped, success;
        Scalar residuals, step_norm;
        Scalar x_old, x_new, function_old, function_new, step_old, step_new;
        Scalar first_derivative_old, second_derivative_old;
 
        // Set initial iteration
        x_old = x_ini;
        success =
            this->evaluate_function(std::forward<FunctionLambda>(function),
                                    x_old,
                                    function_old);
        OPTIMIST_ASSERT(success,
                        CMD "function evaluation failed at the initial point.");
 
        // Algorithm iterations
        Scalar tolerance_residuals{this->m_tolerance};
        Scalar tolerance_step_norm{this->m_tolerance * this->m_tolerance};
        for (this->m_iterations = 1;
             this->m_iterations < this->m_max_iterations;
             ++this->m_iterations) {
          // Evaluate derivatives
          success = this->evaluate_first_derivative(
              std::forward<FirstDerivativeLambda>(first_derivative),
              x_old,
              first_derivative_old);
          OPTIMIST_ASSERT(success,
                          CMD "first derivative evaluation failed at iteration "
                              << this->m_iterations << ".");
          success = this->evaluate_second_derivative(
              std::forward<SecondDerivativeLambda>(second_derivative),
              x_old,
              second_derivative_old);
          OPTIMIST_ASSERT(
              success,
              CMD "second derivative evaluation failed at iteration "
                  << this->m_iterations << ".");
 
          // Calculate step
          if (std::abs(first_derivative_old) < Halley::SQRT_EPSILON) {
            OPTIMIST_WARNING(CMD
                             "close-to-singular first derivative detected.");
            first_derivative_old = (first_derivative_old > 0)
                                       ? Halley::SQRT_EPSILON
                                       : -Halley::SQRT_EPSILON;
          }
          if (std::abs(second_derivative_old) < Halley::SQRT_EPSILON) {
            OPTIMIST_WARNING(CMD
                             "close-to-singular second derivative detected.");
            second_derivative_old = (second_derivative_old > 0)
                                        ? Halley::SQRT_EPSILON
                                        : -Halley::SQRT_EPSILON;
          }
          step_old =
              -(function_old / first_derivative_old) /
              (1.0 - (function_old * second_derivative_old) /
                         (2.0 * first_derivative_old * first_derivative_old));
          OPTIMIST_ASSERT(
              std::isfinite(step_old),
              CMD "step " << this->m_iterations << " is not finite.");
 
          // Check convergence
          residuals = std::abs(function_old);
          step_norm = std::abs(step_old);
          if (this->m_verbose) {
            this->info(residuals);
          }
          if (residuals < tolerance_residuals ||
              step_norm < tolerance_step_norm) {
            this->m_converged = true;
            break;
          }
 
          if (this->m_damped) {
            // Relax the iteration process
            damped = this->damp(std::forward<FunctionLambda>(function),
                                x_old,
                                function_old,
                                step_old,
                                x_new,
                                function_new,
                                step_new);
            OPTIMIST_ASSERT_WARNING(damped, CMD "damping failed.");
          } else {
            // Update point
            x_new = x_old + step_old;
            success =
                this->evaluate_function(std::forward<FunctionLambda>(function),
                                        x_new,
                                        function_new);
            OPTIMIST_ASSERT(success,
                            CMD "function evaluation failed at iteration "
                                << this->m_iterations << ".");
          }
 
          // Update internal variables
          x_old        = x_new;
          function_old = function_new;
          step_old     = step_new;
        }
 
        // Print bottom
        if (this->m_verbose) {
          this->bottom();
        }
 
        // Convergence data
        x_sol = x_old;
        return this->m_converged;
 
#undef CMD
      }
 
    };  // class Halley
 
  }  // namespace RootFinder
 
}  // namespace Optimist
 
#endif  // OPTIMIST_ROOTFINDER_HALLEY_HH