Tableau.hh

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/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *\
 * Copyright (c) 2025, Davide Stocco and Enrico Bertolazzi.                                      *
 *                                                                                               *
 * The Sandals project is distributed under the BSD 2-Clause License.                            *
 *                                                                                               *
 * Davide Stocco                                                               Enrico Bertolazzi *
 * University of Trento                                                     University of Trento *
 * e-mail: davide.stocco@unitn.it                             e-mail: enrico.bertolazzi@unitn.it *
\* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
 
#pragma once
 
#ifndef SANDALS_TABLEAU_HH
#define SANDALS_TABLEAU_HH
 
#include <Sandals.hh>
 
namespace Sandals {
 
  /*\
   |   _____     _     _
   |  |_   _|_ _| |__ | | ___  __ _ _   _
   |    | |/ _` | '_ \| |/ _ \/ _` | | | |
   |    | | (_| | |_) | |  __/ (_| | |_| |
   |    |_|\__,_|_.__/|_|\___|\__,_|\__,_|
   |
  \*/

  template <typename Real, Integer S>
  struct Tableau
  {
    SANDALS_BASIC_CONSTANTS(Real) 
    const Real SQRT_EPSILON{std::sqrt(EPSILON)};  \
 
    using Type = enum class type : Integer {ERK = 0, IRK = 1, DIRK = 2}; 
    using Vector = Eigen::Vector<Real, S>;    
    using Matrix = Eigen::Matrix<Real, S, S>; 
 
    std::string name;                
    Type        type;                
    Integer     order;               
    Integer     order_e{-1};         
    Matrix      A;                   
    Vector      b;                   
    Vector      b_e{Vector::Zero()}; 
    Vector      c;                   
    bool        is_embedded{false};  

    bool check(bool verbose = false) const {
 
      #define CMD "Sandals::" << this->name << "::check(...): "
 
      // Check the occupancy of the matrix A
      if (type == Type::ERK && (!this->A.isLowerTriangular() || !this->A.diagonal().isZero())) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "A matrix occupancy not consistent with an ERK method.");
        return false;
      } else if (type == Type::DIRK && (!this->A.isLowerTriangular() || this->A.diagonal().isZero())) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "A matrix occupancy not consistent with a DIRK method.");
        return false;
      } else if (type == Type::IRK && this->A.isZero()) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "A matrix occupancy not consistent with an IRK method.");
        return false;
      }
 
      // Check the order of the method
      Integer computed_order{compute_order(this->A, this->b, this->c, verbose)};
      if (this->order != computed_order) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "method order check failed, " << computed_order << " ≠ "
          << this->order << ".");
        return false;
      }
 
      // Check the embedded method consistency
      if (!this->is_embedded && (this->order_e != -1 || !this->b_e.isZero())) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "embedded method inconsistency.");
        return false;
      } else {
        return true;
      }
 
      // Check the embedded method order
      Integer computed_order_e{compute_order(this->A, this->b_e, this->c, verbose)};
      if (this->order_e != computed_order_e) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "embedded method order check failed, " << computed_order_e
          << " ≠ " << this->order_e << ".");
        return false;
      } else {
        return true;
      }
 
      #undef CMD
    }
 
  private:
    Integer compute_order(Matrix const &A, Vector const &b, Vector const &c, bool verbose = false) const {
 
      #define CMD "Sandals::" << this->name << "::tableau_order(...): "
 
      // Temporary variables initialization
      Integer order{0};
      Real tolerance{std::pow(EPSILON, 2.0/3.0)};
 
      // Precheck consistency
      Real tmp{(A*Vector::Ones() - c).norm()};
      if (tmp > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "precheck failed, ||A*I - c|| = " << tmp << " ≠ 0.");
        return order;
      }
 
      // Check order 1
      tmp = b.sum();
      if (std::abs(tmp - 1.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 1 check failed, a_1 = " << tmp << " ≠ 1.");
        return order;
      }
 
      order = 1; // Order 1 is the highest order that can be checked
 
      // Check order 2
      tmp = b.transpose() * c;
      if (std::abs(tmp - 1.0/2.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 2 check failed, a_1 = " << tmp << " ≠ 1/2.");
        return order;
      }
 
      order = 2; // Order 2 is the highest order that can be checked
 
      // Check order 3
      Vector c_2{c.array().pow(2)};
      tmp = b.transpose() * c_2;
      if (std::abs(tmp - 1.0/3.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 3 check failed, a_1 = " << tmp << " ≠ 1/3.");
        return order;
      }
 
      Vector Ac{A * c};
      tmp = b.transpose() * Ac;
      if (std::abs(tmp - 1.0/6.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 3 check failed, a_2 = " << tmp << " ≠ 1/6.");
        return order;
      }
 
      order = 3; // Order 3 is the highest order that can be checked
 
      // Check order 4
      Vector c_3{c.array().pow(3)};
      tmp = b.transpose() * c_3;
      if (std::abs(tmp - 1.0/4.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 4 check failed, a_1 = " << tmp << " ≠ 1/4.");
        return order;
      }
 
      Vector cAc{c.array() * Ac.array()};
      tmp = b.transpose() * cAc;
      if (std::abs(tmp - 1.0/8.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 4 check failed, a_2 = " << tmp << " ≠ 1/8.");
        return order;
      }
 
      Vector bA{(b.transpose() * A).transpose()};
      tmp = bA.transpose() * c_2;
      if (std::abs(tmp - 1.0/12.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 4 check failed, a_3 = " << tmp << " ≠ 1/12.");
        return order;
      }
 
      tmp = b.transpose() * A * A * c;
      if (std::abs(tmp - 1.0/24.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 4 check failed, a_4 = " << tmp << " ≠ 1/24.");
        return order;
      }
 
      order = 4; // Order 4 is the highest order that can be checked
 
      // Check order 5
      Vector c_4{c.array().pow(4)};
      tmp = b.transpose() * c_4;
      if (std::abs(tmp - 1.0/5.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 5 check failed, a_1 = " << tmp << " ≠ 1/5.");
        return order;
      }
 
      Vector b_c2{b.array() * c_2.array()};
      tmp = b_c2.transpose() * Ac;
      if (std::abs(tmp - 1.0/10.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 5 check failed, a_2 = " << tmp << " ≠ 1/10.");
        return order;
      }
 
      Vector b_Ac{b.array() * Ac.array()};
      tmp = b_Ac.transpose() * Ac;
      if (std::abs(tmp - 1.0/20.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 5 check failed, a_3 = " << tmp << " ≠ 1/20.");
        return order;
      }
 
      Vector b_c{b.array() * c.array()};
      Vector Ac2{A * c_2};
      tmp = b_c.transpose() * Ac2;
      if (std::abs(tmp - 1.0/15.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 5 check failed, a_4 = " << tmp << " ≠ 1/15.");
        return order;
      }
 
      Vector Ac3{A * c_3};
      tmp = b.transpose() * Ac3;
      if (std::abs(tmp - 1.0/20.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 5 check failed, a_5 = " << tmp << " ≠ 1/20.");
        return order;
      }
 
      Vector AAc{A * Ac};
      tmp = b_c.transpose() * AAc;
      if (std::abs(tmp - 1.0/30.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 5 check failed, a_6 = " << tmp << " ≠ 1/30.");
        return order;
      }
 
      tmp = bA.transpose() * (c.array() * Ac.array()).matrix();
      if (std::abs(tmp - 1.0/40.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 5 check failed, a_7 = " << tmp << " ≠ 1/40.");
        return order;
      }
 
      tmp = bA.transpose() * Ac2;
      if (std::abs(tmp - 1.0/60.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 5 check failed, a_8 = " << tmp << " ≠ 1/60.");
        return order;
      }
 
      tmp = bA.transpose() * AAc;
      if (std::abs(tmp - 1.0/120.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 5 check failed, a_9 = " << tmp << " ≠ 1/120.");
        return order;
      }
 
      order = 5; // Order 5 is the highest order that can be checked
 
      // Check order 6
      Vector c_5{c.array().pow(5)};
      tmp = b.transpose() * c_5;
      if (std::abs(tmp - 1.0/6.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_1 = " << tmp << " ≠ 1/6.");
        return order;
      }
 
      tmp = (b.array() * c_3.array()).matrix().transpose() * Ac;
      if (std::abs(tmp - 1.0/12.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_2 = " << tmp << " ≠ 1/12.");
        return order;
      }
 
      Vector Ac_2{Ac.array().pow(2)};
      tmp = b_c.transpose() * Ac_2;
      if (std::abs(tmp - 1.0/24.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_3 = " << tmp << " ≠ 1/24.");
        return order;
      }
 
      tmp = b_c2.transpose() * Ac2;
      if (std::abs(tmp - 1.0/18.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_4 = " << tmp << " ≠ 1/18.");
        return order;
      }
 
      tmp = b.transpose() * (Ac2.array() * Ac.array()).matrix();
      if (std::abs(tmp - 1.0/36.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_5 = " << tmp << " ≠ 1/36.");
        return order;
      }
 
      tmp = b_c.transpose() * Ac3;
      if (std::abs(tmp - 1.0/24.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_6 = " << tmp << " ≠ 1/24.");
        return order;
      }
 
      Vector Ac4{A * c_4};
      tmp = b.transpose() * Ac4;
      if (std::abs(tmp - 1.0/30.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_7 = " << tmp << " ≠ 1/30.");
        return order;
      }
 
      Vector bc2A{A.transpose() * b_c2};
      tmp = bc2A.transpose() * Ac;
      if (std::abs(tmp - 1.0/36.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_8 = " << tmp << " ≠ 1/36.");
        return order;
      }
 
      tmp = b_Ac.transpose() * AAc;
      if (std::abs(tmp - 1.0/72.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_9 = " << tmp << " ≠ 1/72.");
        return order;
      }
 
      Vector bcA{b_c.transpose() * A};
      tmp = bcA.transpose() * cAc;
      if (std::abs(tmp - 1.0/48.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_10 = " << tmp << " ≠ 1/48.");
        return order;
      }
 
      tmp = (bA.array() * c_2.array()).matrix().transpose() * Ac;
      if (std::abs(tmp - 1.0/60.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_11 = " << tmp << " ≠ 1/60.");
        return order;
      }
 
      tmp = bA.transpose() * Ac_2;
      if (std::abs(tmp - 1.0/120.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_12 = " << tmp << " ≠ 1/120.");
        return order;
      }
 
      tmp = bcA.transpose() * Ac2;
      if (std::abs(tmp - 1.0/72.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_13 = " << tmp << " ≠ 1/72.");
        return order;
      }
 
      Vector bA_c{bA.array() * c.array()};
      tmp = bA_c.transpose() * Ac2;
      if (std::abs(tmp - 1.0/90.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_14 = " << tmp << " ≠ 1/90.");
        return order;
      }
 
      tmp = bA.transpose() * Ac3;
      if (std::abs(tmp - 1.0/120.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_15 = " << tmp << " ≠ 1/120.");
        return order;
      }
 
      tmp = bcA.transpose() * AAc;
      if (std::abs(tmp - 1.0/144.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_16 = " << tmp << " ≠ 1/144.");
        return order;
      }
 
      tmp = bA_c.transpose() * AAc;
      if (std::abs(tmp - 1.0/180.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_17 = " << tmp << " ≠ 1/180.");
        return order;
      }
 
      Vector AcAc{A * cAc};
      tmp = bA.transpose() * AcAc;
      if (std::abs(tmp - 1.0/240.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_18 = " << tmp << " ≠ 1/240.");
        return order;
      }
 
      tmp = bA.transpose() * A * A * c_2;
      if (std::abs(tmp - 1.0/360.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_19 = " << tmp << " ≠ 1/360.");
        return order;
      }
 
      tmp = bA.transpose() * A * A * Ac;
      if (std::abs(tmp - 1.0/720.0) > tolerance) {
        SANDALS_ASSERT_WARNING(!verbose, CMD "order 6 check failed, a_20 = " << tmp << " ≠ 1/720.");
        return order;
      }
 
      order = 6; // Order 6 is the highest order that can be checked with this method
 
      return order;
 
      #undef CMD
    }
  }; // struct Tableau
 
} // namespace Sandals
 
#endif // SANDALS_TABLEAU_HH