Polyhedral Homotopy Continuation Method for solving sparse polynomial system, optimized by only tracing real zeros
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// This file is part of Bertini 2.0.
//
// heun_euler.hpp is free software: you can redistribute it and/or modify
// it under the terms of the GNU General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
//(at your option) any later version.
//
// heun_euler.hpp is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU General Public License
// along with heun_euler.hpp. If not, see <http://www.gnu.org/licenses/>.
//
// predictor.hpp
//
// copyright 2015
// James B. Collins
// West Texas A&M University
// Department of Mathematics
// Spring 2016
/**
\file base_predictor.hpp
\brief Contains a base class for all ODE predictors.
*/
#ifndef BERTINI_BASE_PREDICTORS_HPP
#define BERTINI_BASE_PREDICTORS_HPP
#include <Eigen/LU>
#include <boost/type_index.hpp>
#include "mpfr_extensions.hpp"
#include "system.hpp"
#include "trackers/amp_criteria.hpp"
#include "trackers/config.hpp"
namespace bertini {
namespace tracking {
namespace predict {
using Predictor = Predictor;
/**
\brief Get the Bertini2 default predictor.
Currently set to Euler, though this will change in future versions.
*/
inline Predictor DefaultPredictor() { return Predictor::Euler; }
/**
The lowest order of the predictor. The order of the error estimate is this
plus one.
*/
inline unsigned Order(Predictor predictor_choice) {
switch (predictor_choice) {
case (Predictor::Euler):
return 1;
case (Predictor::HeunEuler):
return 1;
case (Predictor::RK4):
return 4;
case (Predictor::RKF45):
return 4;
case (Predictor::RKCashKarp45):
return 4;
case (Predictor::RKDormandPrince56):
return 5;
case (Predictor::RKVerner67):
return 6;
default: {
throw std::runtime_error("incompatible predictor choice in Order");
}
}
}
inline bool HasErrorEstimate(Predictor predictor_choice) {
switch (predictor_choice) {
case (Predictor::Euler):
return false;
case (Predictor::HeunEuler):
return true;
case (Predictor::RK4):
return false;
case (Predictor::RKF45):
return true;
case (Predictor::RKCashKarp45):
return true;
case (Predictor::RKDormandPrince56):
return true;
case (Predictor::RKVerner67):
return true;
default: {
throw std::runtime_error(
"incompatible predictor choice in HasErrorEstimate");
}
}
}
/**
/class BasePredictor
\brief An interface for all predictors.
## Purpose
Stores all the information needed to implement the predictor method
- Butcher Table
- Number of Stages
- Order of the method.
Also stores information computed during implementation of the method.
## Use
Implement the following pure functions:
- FullStep
- SetErrorEstimate
- SetSizeProportion
*/
class BasePredictor {
public:
BasePredictor(Predictor method) {};
/**
\brief Perform a generic predictor step.
\param next_space The computed prediction.
\param method An enum class selecting the predictor method to use.
\param S The system being solved.
\param current_space The current space variable vector.
\param current_time The current time.
\param delta_t The size of the time step.
\param condition_number_estimate The computed estimate of the condition
number of the Jacobian. \param
num_steps_since_last_condition_number_computation. Updated in this function.
\param frequency_of_CN_estimation How many steps to take between condition
number estimates. \param prec_type The operating precision type. \param
tracking_tolerance How tightly to track the path.
*/
template <typename ComplexType, typename RealType>
SuccessCode Predict(
Vec<ComplexType>& next_space, System const& S,
Vec<ComplexType> const& current_space, ComplexType current_time,
ComplexType const& delta_t, RealType& condition_number_estimate,
unsigned& num_steps_since_last_condition_number_computation,
unsigned frequency_of_CN_estimation, RealType const& tracking_tolerance) {
return FullStep<ComplexType, RealType>(next_space, S, current_space,
current_time, delta_t);
}
/**
\brief Perform a generic predictor step and return size_proportion and
condition number information
\param next_space The computed prediction.
\param method An enum class selecting the predictor method to use.
\param size_proportion $a$ in AMP2 paper.
\param norm_J The computed estimate of the norm of the Jacobian matrix.
\param norm_J_inverse The computed estimate of the norm of the inverse of the
Jacobian matrix. \param S The system being solved. \param current_space The
current space variable vector. \param current_time The current time. \param
delta_t The size of the time step. \param condition_number_estimate The
computed estimate of the condition number of the Jacobian. \param
num_steps_since_last_condition_number_computation. Updated in this function.
\param frequency_of_CN_estimation How many steps to take between condition
number estimates. \param prec_type The operating precision type. \param
tracking_tolerance How tightly to track the path. \param AMP_config The
settings for adaptive multiple precision.
*/
template <typename ComplexType, typename RealType>
SuccessCode Predict(
Vec<ComplexType>& next_space, RealType& size_proportion, RealType& norm_J,
RealType& norm_J_inverse, System const& S,
Vec<ComplexType> const& current_space, ComplexType current_time,
ComplexType const& delta_t, RealType& condition_number_estimate,
unsigned& num_steps_since_last_condition_number_computation,
unsigned frequency_of_CN_estimation, RealType const& tracking_tolerance,
AdaptiveMultiplePrecisionConfig const& AMP_config) {
auto success_code = Predict<ComplexType, RealType>(
next_space, S, current_space, current_time, delta_t,
condition_number_estimate,
num_steps_since_last_condition_number_computation,
frequency_of_CN_estimation, tracking_tolerance);
if (success_code != SuccessCode::Success) return success_code;
// Calculate condition number and updated if needed
Eigen::PartialPivLU<Mat<ComplexType>>& dhdxref =
std::get<Eigen::PartialPivLU<Mat<ComplexType>>>(dh_dx_);
Mat<ComplexType>& LUref = std::get<Mat<ComplexType>>(LU_);
Vec<ComplexType> randy = RandomOfUnits<ComplexType>(S.NumVariables());
Vec<ComplexType> temp_soln = LUref.solve(randy);
norm_J = dhdxref.norm();
norm_J_inverse = temp_soln.norm();
if (num_steps_since_last_condition_number_computation >=
frequency_of_CN_estimation) {
condition_number_estimate = norm_J * norm_J_inverse;
num_steps_since_last_condition_number_computation =
1; // reset the counter to 1
} else // no need to compute the condition number
num_steps_since_last_condition_number_computation++;
// Set size_proportion
SetSizeProportion<ComplexType, RealType>(size_proportion, delta_t);
// AMP Criteria
if (!amp::CriterionA(norm_J, norm_J_inverse,
AMP_config)) // AMP_criterion_A != ok
return SuccessCode::HigherPrecisionNecessary;
else if (!amp::CriterionC(norm_J_inverse, current_space, tracking_tolerance,
AMP_config)) // AMP_criterion_C != ok
return SuccessCode::HigherPrecisionNecessary;
return success_code;
}
/**
\brief Perform a generic predictor step and return error estimate,
size_proportion and condition number information
\param next_space The computed prediction.
\param method An enum class selecting the predictor method to use.
\param error_estimate Estimate of the error from an embedded method.
\param size_proportion $a$ in AMP2 paper.
\param norm_J The computed estimate of the norm of the Jacobian matrix.
\param norm_J_inverse The computed estimate of the norm of the inverse of the
Jacobian matrix. \param S The system being solved. \param current_space The
current space variable vector. \param current_time The current time. \param
delta_t The size of the time step. \param condition_number_estimate The
computed estimate of the condition number of the Jacobian. \param
num_steps_since_last_condition_number_computation. Updated in this function.
\param frequency_of_CN_estimation How many steps to take between condition
number estimates. \param prec_type The operating precision type. \param
tracking_tolerance How tightly to track the path. \param AMP_config The
settings for adaptive multiple precision.
*/
template <typename ComplexType, typename RealType>
SuccessCode Predict(
Vec<ComplexType>& next_space, RealType& error_estimate,
RealType& size_proportion, RealType& norm_J, RealType& norm_J_inverse,
System const& S, Vec<ComplexType> const& current_space,
ComplexType current_time, ComplexType const& delta_t,
RealType& condition_number_estimate,
unsigned& num_steps_since_last_condition_number_computation,
unsigned frequency_of_CN_estimation, RealType const& tracking_tolerance,
AdaptiveMultiplePrecisionConfig const& AMP_config) {
// If this is a method without an error estimator, then can't calculate size
// proportion and should throw an error
if (!predict::HasErrorEstimate(predictor_)) {
throw std::runtime_error(
"incompatible predictor choice in ExplicitPredict, no error "
"estimator");
}
auto success_code = Predict<ComplexType, RealType>(
next_space, size_proportion, norm_J, norm_J_inverse, S, current_space,
current_time, delta_t, condition_number_estimate,
num_steps_since_last_condition_number_computation,
frequency_of_CN_estimation, tracking_tolerance, AMP_config);
if (success_code != SuccessCode::Success) return success_code;
SetErrorEstimate<ComplexType, RealType>(error_estimate, delta_t);
return success_code;
}
/**
/brief Sets the local variables to correspond to a particular predictor
method
\param method Enum class that determines the predictor method
*/
virtual void PredictorMethod(Predictor method) = 0;
Predictor PredictorMethod() { return predictor_; }
/**
The lowest order of the predictor. The order of the error estimate is this
plus one.
*/
inline unsigned Order() { return p_; }
inline bool HasErrorEstimate() {
return predict::HasErrorEstimate(predictor_);
}
protected:
///////////////////////////
//
// Protected Members Methods
//
////////////////////
/**
\brief Performs a full prediction step from current_time to current_time +
delta_t
\param next_space The computed prediction space
\param S The homotopy system
\param current_space The current space values
\param current_time The current time values
\param delta_t The time step
\return SuccessCode determining result of the computation
*/
template <typename ComplexType, typename RealType>
virtual SuccessCode FullStep(Vec<ComplexType>& next_space, System const& S,
Vec<ComplexType> const& current_space,
ComplexType current_time,
ComplexType const& delta_t) = 0;
/**
\brief Computes the error estimate of this prediction step.
\param error_estimate Computed error estimate
\param delta_t The time step
\return Success code or the computation
*/
template <typename ComplexType, typename RealType>
virtual SuccessCode SetErrorEstimate(RealType& error_estimate,
ComplexType const& delta_t) = 0;
/**
\brief Compute the size proportion variable for AMP computation
\param size_proportion Computed size proportion
\param delta_t The time step
\return Success code of the computation
*/
template <typename ComplexType, typename RealType>
virtual SuccessCode SetSizeProportion(RealType& size_proportion,
ComplexType const& delta_t) = 0;
/**
\brief Evaluates the RHS of the Davidenko differential equation at a
particular time and space
\param S The homotopy system
\param space The space variable used to evaluate RHS
\param time The time variable used to evaluate RHS
\param K Matrix of stage variables
\param stage Which stage variable(column of K) should be filled by this
computation
\return Success code of this computation
*/
template <typename ComplexType>
virtual SuccessCode EvalRHS(System const& S, Vec<ComplexType> const& space,
ComplexType time, Mat<ComplexType>& K,
int stage) = 0;
///////////////////////////
//
// Protected Data Members
//
////////////////////
Predictor predictor_;
unsigned p_;
std::tuple<Mat<dbl>, Mat<mpfr>> dh_dx_;
std::tuple<Eigen::PartialPivLU<Mat<dbl>>, Eigen::PartialPivLU<Mat<mpfr>>> LU_;
}; // re: class BasePredictor
} // namespace predict
} // namespace tracking
} // namespace bertini
#endif