Newton.hh

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

    template <typename Real, Integer N>
    class Newton : public RootFinder<Real, N, Newton<Real, N>, true>
    {
    public:
      static constexpr bool requires_function{true};
      static constexpr bool requires_first_derivative{true};
      static constexpr bool requires_second_derivative{false};
 
      OPTIMIST_BASIC_CONSTANTS(Real)
 
      using typename RootFinder<Real, N, Newton<Real, N>, true>::Vector;
      using typename RootFinder<Real, N, Newton<Real, N>, true>::Matrix;
 
    private:
      Eigen::FullPivLU<Matrix> m_lu; 
 
    public:
      Newton() {}

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

      template <typename FunctionLambda, typename JacobianLambda>
      bool solve_impl(FunctionLambda && function, JacobianLambda && jacobian, Vector const & x_ini,
        Vector & x_sol)
      {
        #define CMD "Optimist::RootFinder::Newton::solve(...): "
 
        // Setup internal variables
        this->reset();
 
        // Print header
        if (this->m_verbose) {this->header();}
 
        // Initialize variables
        bool damped, success;
        Real residuals, step_norm;
        Vector x_old, x_new, function_old, function_new, step_old, step_new;
        Matrix jacobian_old;
 
        // Set initial iteration
        x_old = x_ini;
        success = this->evaluate_function(std::forward<FunctionLambda>(function), x_old, function_old);
        OPTIMIST_ASSERT_WARNING(success,
          CMD "function evaluation failed at the initial point.");
 
        // Algorithm iterations
        Real tolerance_residuals{this->m_tolerance};
        Real tolerance_step_norm{this->m_tolerance * this->m_tolerance};
        for (this->m_iterations = 1; this->m_iterations < this->m_max_iterations; ++this->m_iterations)
        {
          // Store trace
          this->store_trace(x_old);
 
          // Calculate step
          success = this->evaluate_jacobian(jacobian, x_old, jacobian_old);
          OPTIMIST_ASSERT_WARNING(success,
            CMD "jacobian evaluation failed at iteration " << this->m_iterations << ".");
          this->m_lu.compute(jacobian_old);
          OPTIMIST_ASSERT(this->m_lu.rank() == N,
            "Optimist::RootFinder::Newton::solve(...): singular Jacobian detected.");
          step_old = -this->m_lu.solve(function_old);
 
          // Check convergence
          residuals = function_old.norm();
          step_norm = step_old.norm();
          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,
              "Optimist::RootFinder::Newton::solve(...): 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_WARNING(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 Newton
 
  } // namespace RootFinder
 
} // namespace Optimist
 
#endif // OPTIMIST_ROOTFINDER_NEWTON_HH