ceres-solver/examples/nist.cc

// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2017 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
//   this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
//   this list of conditions and the following disclaimer in the documentation
//   and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
//   used to endorse or promote products derived from this software without
//   specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
// Author: sameeragarwal@google.com (Sameer Agarwal)
//
// The National Institute of Standards and Technology has released a
// set of problems to test non-linear least squares solvers.
//
// More information about the background on these problems and
// suggested evaluation methodology can be found at:
//
//   http://www.itl.nist.gov/div898/strd/nls/nls_info.shtml
//
// The problem data themselves can be found at
//
//   http://www.itl.nist.gov/div898/strd/nls/nls_main.shtml
//
// The problems are divided into three levels of difficulty, Easy,
// Medium and Hard. For each problem there are two starting guesses,
// the first one far away from the global minimum and the second
// closer to it.
//
// A problem is considered successfully solved, if every components of
// the solution matches the globally optimal solution in at least 4
// digits or more.
//
// This dataset was used for an evaluation of Non-linear least squares
// solvers:
//
// P. F. Mondragon & B. Borchers, A Comparison of Nonlinear Regression
// Codes, Journal of Modern Applied Statistical Methods, 4(1):343-351,
// 2005.
//
// The results from Mondragon & Borchers can be summarized as
//               Excel  Gnuplot  GaussFit  HBN  MinPack
// Average LRE     2.3      4.3       4.0  6.8      4.4
//      Winner       1        5        12   29       12
//
// Where the row Winner counts, the number of problems for which the
// solver had the highest LRE.

// In this file, we implement the same evaluation methodology using
// Ceres. Currently using Levenberg-Marquardt with DENSE_QR, we get
//
//               Excel  Gnuplot  GaussFit  HBN  MinPack  Ceres
// Average LRE     2.3      4.3       4.0  6.8      4.4    9.4
//      Winner       0        0         5   11        2     41

#include <fstream>
#include <iostream>
#include <iterator>

#include "Eigen/Core"
#include "ceres/ceres.h"
#include "ceres/tiny_solver.h"
#include "ceres/tiny_solver_cost_function_adapter.h"
#include "gflags/gflags.h"
#include "glog/logging.h"

DEFINE_bool(use_tiny_solver, false, "Use TinySolver instead of Ceres::Solver");
DEFINE_string(nist_data_dir, "", "Directory containing the NIST non-linear"
              "regression examples");
DEFINE_string(minimizer, "trust_region",
              "Minimizer type to use, choices are: line_search & trust_region");
DEFINE_string(trust_region_strategy, "levenberg_marquardt",
              "Options are: levenberg_marquardt, dogleg");
DEFINE_string(dogleg, "traditional_dogleg",
              "Options are: traditional_dogleg, subspace_dogleg");
DEFINE_string(linear_solver, "dense_qr", "Options are: "
              "sparse_cholesky, dense_qr, dense_normal_cholesky and"
              "cgnr");
DEFINE_string(preconditioner, "jacobi", "Options are: "
              "identity, jacobi");
DEFINE_string(line_search, "wolfe",
              "Line search algorithm to use, choices are: armijo and wolfe.");
DEFINE_string(line_search_direction, "lbfgs",
              "Line search direction algorithm to use, choices: lbfgs, bfgs");
DEFINE_int32(max_line_search_iterations, 20,
             "Maximum number of iterations for each line search.");
DEFINE_int32(max_line_search_restarts, 10,
             "Maximum number of restarts of line search direction algorithm.");
DEFINE_string(line_search_interpolation, "cubic",
              "Degree of polynomial aproximation in line search, "
              "choices are: bisection, quadratic & cubic.");
DEFINE_int32(lbfgs_rank, 20,
             "Rank of L-BFGS inverse Hessian approximation in line search.");
DEFINE_bool(approximate_eigenvalue_bfgs_scaling, false,
            "Use approximate eigenvalue scaling in (L)BFGS line search.");
DEFINE_double(sufficient_decrease, 1.0e-4,
              "Line search Armijo sufficient (function) decrease factor.");
DEFINE_double(sufficient_curvature_decrease, 0.9,
              "Line search Wolfe sufficient curvature decrease factor.");
DEFINE_int32(num_iterations, 10000, "Number of iterations");
DEFINE_bool(nonmonotonic_steps, false, "Trust region algorithm can use"
            " nonmonotic steps");
DEFINE_double(initial_trust_region_radius, 1e4, "Initial trust region radius");
DEFINE_bool(use_numeric_diff, false,
            "Use numeric differentiation instead of automatic "
            "differentiation.");
DEFINE_string(numeric_diff_method, "ridders", "When using numeric "
              "differentiation, selects algorithm. Options are: central, "
              "forward, ridders.");
DEFINE_double(ridders_step_size, 1e-9, "Initial step size for Ridders "
              "numeric differentiation.");
DEFINE_int32(ridders_extrapolations, 3, "Maximal number of Ridders "
             "extrapolations.");

namespace ceres {
namespace examples {
namespace {

using Eigen::Dynamic;
using Eigen::RowMajor;
typedef Eigen::Matrix<double, Dynamic, 1> Vector;
typedef Eigen::Matrix<double, Dynamic, Dynamic, RowMajor> Matrix;

using std::atof;
using std::atoi;
using std::cout;
using std::ifstream;
using std::string;
using std::vector;

void SplitStringUsingChar(const string& full,
                          const char delim,
                          vector<string>* result) {
  std::back_insert_iterator< vector<string> > it(*result);

  const char* p = full.data();
  const char* end = p + full.size();
  while (p != end) {
    if (*p == delim) {
      ++p;
    } else {
      const char* start = p;
      while (++p != end && *p != delim) {
        // Skip to the next occurence of the delimiter.
      }
      *it++ = string(start, p - start);
    }
  }
}

bool GetAndSplitLine(ifstream& ifs, vector<string>* pieces) {
  pieces->clear();
  char buf[256];
  ifs.getline(buf, 256);
  SplitStringUsingChar(string(buf), ' ', pieces);
  return true;
}

void SkipLines(ifstream& ifs, int num_lines) {
  char buf[256];
  for (int i = 0; i < num_lines; ++i) {
    ifs.getline(buf, 256);
  }
}

class NISTProblem {
 public:
  explicit NISTProblem(const string& filename) {
    ifstream ifs(filename.c_str(), ifstream::in);
    CHECK(ifs) << "Unable to open : " << filename;

    vector<string> pieces;
    SkipLines(ifs, 24);
    GetAndSplitLine(ifs, &pieces);
    const int kNumResponses = atoi(pieces[1].c_str());

    GetAndSplitLine(ifs, &pieces);
    const int kNumPredictors = atoi(pieces[0].c_str());

    GetAndSplitLine(ifs, &pieces);
    const int kNumObservations = atoi(pieces[0].c_str());

    SkipLines(ifs, 4);
    GetAndSplitLine(ifs, &pieces);
    const int kNumParameters = atoi(pieces[0].c_str());
    SkipLines(ifs, 8);

    // Get the first line of initial and final parameter values to
    // determine the number of tries.
    GetAndSplitLine(ifs, &pieces);
    const int kNumTries = pieces.size() - 4;

    predictor_.resize(kNumObservations, kNumPredictors);
    response_.resize(kNumObservations, kNumResponses);
    initial_parameters_.resize(kNumTries, kNumParameters);
    final_parameters_.resize(1, kNumParameters);

    // Parse the line for parameter b1.
    int parameter_id = 0;
    for (int i = 0; i < kNumTries; ++i) {
      initial_parameters_(i, parameter_id) = atof(pieces[i + 2].c_str());
    }
    final_parameters_(0, parameter_id) = atof(pieces[2 + kNumTries].c_str());

    // Parse the remaining parameter lines.
    for (int parameter_id = 1; parameter_id < kNumParameters; ++parameter_id) {
     GetAndSplitLine(ifs, &pieces);
     // b2, b3, ....
     for (int i = 0; i < kNumTries; ++i) {
       initial_parameters_(i, parameter_id) = atof(pieces[i + 2].c_str());
     }
     final_parameters_(0, parameter_id) = atof(pieces[2 + kNumTries].c_str());
    }

    // Certfied cost
    SkipLines(ifs, 1);
    GetAndSplitLine(ifs, &pieces);
    certified_cost_ = atof(pieces[4].c_str()) / 2.0;

    // Read the observations.
    SkipLines(ifs, 18 - kNumParameters);
    for (int i = 0; i < kNumObservations; ++i) {
      GetAndSplitLine(ifs, &pieces);
      // Response.
      for (int j = 0; j < kNumResponses; ++j) {
        response_(i, j) =  atof(pieces[j].c_str());
      }

      // Predictor variables.
      for (int j = 0; j < kNumPredictors; ++j) {
        predictor_(i, j) =  atof(pieces[j + kNumResponses].c_str());
      }
    }
  }

  Matrix initial_parameters(int start) const { return initial_parameters_.row(start); }  // NOLINT
  Matrix final_parameters() const  { return final_parameters_; }
  Matrix predictor()        const { return predictor_;         }
  Matrix response()         const { return response_;          }
  int predictor_size()      const { return predictor_.cols();  }
  int num_observations()    const { return predictor_.rows();  }
  int response_size()       const { return response_.cols();   }
  int num_parameters()      const { return initial_parameters_.cols(); }
  int num_starts()          const { return initial_parameters_.rows(); }
  double certified_cost()   const { return certified_cost_; }

 private:
  Matrix predictor_;
  Matrix response_;
  Matrix initial_parameters_;
  Matrix final_parameters_;
  double certified_cost_;
};

#define NIST_BEGIN(CostFunctionName)                          \
  struct CostFunctionName {                                   \
  CostFunctionName(const double* const x,                     \
                   const double* const y,                     \
                   const int n)                               \
      : x_(x), y_(y), n_(n) {}                                \
    const double* x_;                                         \
    const double* y_;                                         \
    const int n_;                                             \
    template <typename T>                                     \
    bool operator()(const T* const b, T* residual) const {    \
      for (int i = 0; i < n_; ++i) {                          \
        const T x(x_[i]);                                     \
        residual[i] = y_[i] - (

#define NIST_END ); } return true; }};

// y = b1 * (b2+x)**(-1/b3)  +  e
NIST_BEGIN(Bennet5)
  b[0] * pow(b[1] + x, -1.0 / b[2])
NIST_END

// y = b1*(1-exp[-b2*x])  +  e
NIST_BEGIN(BoxBOD)
  b[0] * (1.0 - exp(-b[1] * x))
NIST_END

// y = exp[-b1*x]/(b2+b3*x)  +  e
NIST_BEGIN(Chwirut)
  exp(-b[0] * x) / (b[1] + b[2] * x)
NIST_END

// y  = b1*x**b2  +  e
NIST_BEGIN(DanWood)
  b[0] * pow(x, b[1])
NIST_END

// y = b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 )
//     + b6*exp( -(x-b7)**2 / b8**2 ) + e
NIST_BEGIN(Gauss)
  b[0] * exp(-b[1] * x) +
  b[2] * exp(-pow((x - b[3])/b[4], 2)) +
  b[5] * exp(-pow((x - b[6])/b[7], 2))
NIST_END

// y = b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x)  +  e
NIST_BEGIN(Lanczos)
  b[0] * exp(-b[1] * x) + b[2] * exp(-b[3] * x) + b[4] * exp(-b[5] * x)
NIST_END

// y = (b1+b2*x+b3*x**2+b4*x**3) /
//     (1+b5*x+b6*x**2+b7*x**3)  +  e
NIST_BEGIN(Hahn1)
  (b[0] + b[1] * x + b[2] * x * x + b[3] * x * x * x) /
  (1.0 + b[4] * x + b[5] * x * x + b[6] * x * x * x)
NIST_END

// y = (b1 + b2*x + b3*x**2) /
//    (1 + b4*x + b5*x**2)  +  e
NIST_BEGIN(Kirby2)
  (b[0] + b[1] * x + b[2] * x * x) /
  (1.0 + b[3] * x + b[4] * x * x)
NIST_END

// y = b1*(x**2+x*b2) / (x**2+x*b3+b4)  +  e
NIST_BEGIN(MGH09)
  b[0] * (x * x + x * b[1]) / (x * x + x * b[2] + b[3])
NIST_END

// y = b1 * exp[b2/(x+b3)]  +  e
NIST_BEGIN(MGH10)
  b[0] * exp(b[1] / (x + b[2]))
NIST_END

// y = b1 + b2*exp[-x*b4] + b3*exp[-x*b5]
NIST_BEGIN(MGH17)
  b[0] + b[1] * exp(-x * b[3]) + b[2] * exp(-x * b[4])
NIST_END

// y = b1*(1-exp[-b2*x])  +  e
NIST_BEGIN(Misra1a)
  b[0] * (1.0 - exp(-b[1] * x))
NIST_END

// y = b1 * (1-(1+b2*x/2)**(-2))  +  e
NIST_BEGIN(Misra1b)
  b[0] * (1.0 - 1.0/ ((1.0 + b[1] * x / 2.0) * (1.0 + b[1] * x / 2.0)))  // NOLINT
NIST_END

// y = b1 * (1-(1+2*b2*x)**(-.5))  +  e
NIST_BEGIN(Misra1c)
  b[0] * (1.0 - pow(1.0 + 2.0 * b[1] * x, -0.5))
NIST_END

// y = b1*b2*x*((1+b2*x)**(-1))  +  e
NIST_BEGIN(Misra1d)
  b[0] * b[1] * x / (1.0 + b[1] * x)
NIST_END

const double kPi = 3.141592653589793238462643383279;
// pi = 3.141592653589793238462643383279E0
// y =  b1 - b2*x - arctan[b3/(x-b4)]/pi  +  e
NIST_BEGIN(Roszman1)
  b[0] - b[1] * x - atan2(b[2], (x - b[3])) / kPi
NIST_END

// y = b1 / (1+exp[b2-b3*x])  +  e
NIST_BEGIN(Rat42)
  b[0] / (1.0 + exp(b[1] - b[2] * x))
NIST_END

// y = b1 / ((1+exp[b2-b3*x])**(1/b4))  +  e
NIST_BEGIN(Rat43)
  b[0] / pow(1.0 + exp(b[1] - b[2] * x), 1.0 / b[3])
NIST_END

// y = (b1 + b2*x + b3*x**2 + b4*x**3) /
//    (1 + b5*x + b6*x**2 + b7*x**3)  +  e
NIST_BEGIN(Thurber)
  (b[0] + b[1] * x + b[2] * x * x  + b[3] * x * x * x) /
  (1.0 + b[4] * x + b[5] * x * x + b[6] * x * x * x)
NIST_END

// y = b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 )
//        + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 )
//        + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 )  + e
NIST_BEGIN(ENSO)
  b[0] + b[1] * cos(2.0 * kPi * x / 12.0) +
         b[2] * sin(2.0 * kPi * x / 12.0) +
         b[4] * cos(2.0 * kPi * x / b[3]) +
         b[5] * sin(2.0 * kPi * x / b[3]) +
         b[7] * cos(2.0 * kPi * x / b[6]) +
         b[8] * sin(2.0 * kPi * x / b[6])
NIST_END

// y = (b1/b2) * exp[-0.5*((x-b3)/b2)**2]  +  e
NIST_BEGIN(Eckerle4)
  b[0] / b[1] * exp(-0.5 * pow((x - b[2])/b[1], 2))
NIST_END

struct Nelson {
 public:
  Nelson(const double* const x, const double* const y, const int n)
      : x_(x), y_(y), n_(n) {}

  template <typename T>
  bool operator()(const T* const b, T* residual) const {
    // log[y] = b1 - b2*x1 * exp[-b3*x2]  +  e
    for (int i = 0; i < n_; ++i) {
      residual[i] = log(y_[i]) - (b[0] - b[1] * x_[2 * i] * exp(-b[2] * x_[2 * i + 1]));
    }
    return true;
  }

 private:
  const double* x_;
  const double* y_;
  const int n_;
};

static void SetNumericDiffOptions(ceres::NumericDiffOptions* options) {
  options->max_num_ridders_extrapolations = FLAGS_ridders_extrapolations;
  options->ridders_relative_initial_step_size = FLAGS_ridders_step_size;
}

void SetMinimizerOptions(ceres::Solver::Options* options) {
  CHECK(
      ceres::StringToMinimizerType(FLAGS_minimizer, &options->minimizer_type));
  CHECK(ceres::StringToLinearSolverType(FLAGS_linear_solver,
                                        &options->linear_solver_type));
  CHECK(ceres::StringToPreconditionerType(FLAGS_preconditioner,
                                          &options->preconditioner_type));
  CHECK(ceres::StringToTrustRegionStrategyType(
      FLAGS_trust_region_strategy, &options->trust_region_strategy_type));
  CHECK(ceres::StringToDoglegType(FLAGS_dogleg, &options->dogleg_type));
  CHECK(ceres::StringToLineSearchDirectionType(
      FLAGS_line_search_direction, &options->line_search_direction_type));
  CHECK(ceres::StringToLineSearchType(FLAGS_line_search,
                                      &options->line_search_type));
  CHECK(ceres::StringToLineSearchInterpolationType(
      FLAGS_line_search_interpolation,
      &options->line_search_interpolation_type));

  options->max_num_iterations = FLAGS_num_iterations;
  options->use_nonmonotonic_steps = FLAGS_nonmonotonic_steps;
  options->initial_trust_region_radius = FLAGS_initial_trust_region_radius;
  options->max_lbfgs_rank = FLAGS_lbfgs_rank;
  options->line_search_sufficient_function_decrease = FLAGS_sufficient_decrease;
  options->line_search_sufficient_curvature_decrease =
      FLAGS_sufficient_curvature_decrease;
  options->max_num_line_search_step_size_iterations =
      FLAGS_max_line_search_iterations;
  options->max_num_line_search_direction_restarts =
      FLAGS_max_line_search_restarts;
  options->use_approximate_eigenvalue_bfgs_scaling =
      FLAGS_approximate_eigenvalue_bfgs_scaling;
  options->function_tolerance = std::numeric_limits<double>::epsilon();
  options->gradient_tolerance = std::numeric_limits<double>::epsilon();
  options->parameter_tolerance = std::numeric_limits<double>::epsilon();
}

string JoinPath(const string& dirname, const string& basename) {
#ifdef _WIN32
    static const char separator = '\\';
#else
    static const char separator = '/';
#endif  // _WIN32

  if ((!basename.empty() && basename[0] == separator) || dirname.empty()) {
    return basename;
  } else if (dirname[dirname.size() - 1] == separator) {
    return dirname + basename;
  } else {
    return dirname + string(&separator, 1) + basename;
  }
}

template <typename Model, int num_parameters>
CostFunction* CreateCostFunction(const Matrix& predictor,
                                 const Matrix& response,
                                 const int num_observations) {
  Model* model =
      new Model(predictor.data(), response.data(), num_observations);
  ceres::CostFunction* cost_function = NULL;
  if (FLAGS_use_numeric_diff) {
    ceres::NumericDiffOptions options;
    SetNumericDiffOptions(&options);
    if (FLAGS_numeric_diff_method == "central") {
      cost_function = new NumericDiffCostFunction<Model,
                                                  ceres::CENTRAL,
                                                  ceres::DYNAMIC,
                                                  num_parameters>(
          model,
          ceres::TAKE_OWNERSHIP,
          num_observations,
          options);
    } else if (FLAGS_numeric_diff_method == "forward") {
      cost_function = new NumericDiffCostFunction<Model,
                                                  ceres::FORWARD,
                                                  ceres::DYNAMIC,
                                                  num_parameters>(
          model,
          ceres::TAKE_OWNERSHIP,
          num_observations,
          options);
    } else if (FLAGS_numeric_diff_method == "ridders") {
      cost_function = new NumericDiffCostFunction<Model,
                                                  ceres::RIDDERS,
                                                  ceres::DYNAMIC,
                                                  num_parameters>(
          model,
          ceres::TAKE_OWNERSHIP,
          num_observations,
          options);
    } else {
      LOG(ERROR) << "Invalid numeric diff method specified";
      return 0;
    }
  } else {
    cost_function =
        new ceres::AutoDiffCostFunction<Model, ceres::DYNAMIC, num_parameters>(
            model, num_observations);
  }
  return cost_function;
}

double ComputeLRE(const Matrix& expected, const Matrix& actual) {
  // Compute the LRE by comparing each component of the solution
  // with the ground truth, and taking the minimum.
  const double kMaxNumSignificantDigits = 11;
  double log_relative_error = kMaxNumSignificantDigits + 1;
  for (int i = 0; i < expected.cols(); ++i) {
    const double tmp_lre = -std::log10(std::fabs(expected(i) - actual(i)) /
                                       std::fabs(expected(i)));
    // The maximum LRE is capped at 11 - the precision at which the
    // ground truth is known.
    //
    // The minimum LRE is capped at 0 - no digits match between the
    // computed solution and the ground truth.
    log_relative_error =
        std::min(log_relative_error,
                 std::max(0.0, std::min(kMaxNumSignificantDigits, tmp_lre)));
  }
  return log_relative_error;
}

template <typename Model, int num_parameters>
int RegressionDriver(const string& filename) {
  NISTProblem nist_problem(JoinPath(FLAGS_nist_data_dir, filename));
  CHECK_EQ(num_parameters, nist_problem.num_parameters());

  Matrix predictor = nist_problem.predictor();
  Matrix response = nist_problem.response();
  Matrix final_parameters = nist_problem.final_parameters();

  printf("%s\n", filename.c_str());

  // Each NIST problem comes with multiple starting points, so we
  // construct the problem from scratch for each case and solve it.
  int num_success = 0;
  for (int start = 0; start < nist_problem.num_starts(); ++start) {
    Matrix initial_parameters = nist_problem.initial_parameters(start);
    ceres::CostFunction* cost_function = CreateCostFunction<Model, num_parameters>(
        predictor, response,  nist_problem.num_observations());

    double initial_cost;
    double final_cost;

    if (!FLAGS_use_tiny_solver) {
      ceres::Problem problem;
      problem.AddResidualBlock(cost_function, NULL, initial_parameters.data());
      ceres::Solver::Summary summary;
      ceres::Solver::Options options;
      SetMinimizerOptions(&options);
      Solve(options, &problem, &summary);
      initial_cost = summary.initial_cost;
      final_cost = summary.final_cost;
    } else {
      ceres::TinySolverCostFunctionAdapter<Eigen::Dynamic, num_parameters> cfa(
          *cost_function);
      typedef ceres::TinySolver<
          ceres::TinySolverCostFunctionAdapter<Eigen::Dynamic, num_parameters> >
          Solver;
      Solver solver;
      solver.options.max_num_iterations = FLAGS_num_iterations;
      solver.options.gradient_tolerance =
          std::numeric_limits<double>::epsilon();
      solver.options.parameter_tolerance =
          std::numeric_limits<double>::epsilon();

      Eigen::Matrix<double, num_parameters, 1> x;
      x = initial_parameters.transpose();
      typename Solver::Summary summary = solver.Solve(cfa, &x);
      initial_parameters = x;
      initial_cost = summary.initial_cost;
      final_cost = summary.final_cost;
      delete cost_function;
    }

    const double log_relative_error = ComputeLRE(nist_problem.final_parameters(),
                                                 initial_parameters);
    const int kMinNumMatchingDigits = 4;
    if (log_relative_error > kMinNumMatchingDigits) {
      ++num_success;
    }

    printf(
        "start: %d status: %s lre: %4.1f initial cost: %e final cost:%e "
        "certified cost: %e\n",
        start + 1,
        log_relative_error < kMinNumMatchingDigits ? "FAILURE" : "SUCCESS",
        log_relative_error,
        initial_cost,
        final_cost,
        nist_problem.certified_cost());
  }
  return num_success;
}


void SolveNISTProblems() {
  if (FLAGS_nist_data_dir.empty()) {
    LOG(FATAL) << "Must specify the directory containing the NIST problems";
  }

  cout << "Lower Difficulty\n";
  int easy_success = 0;
  easy_success += RegressionDriver<Misra1a, 2>("Misra1a.dat");
  easy_success += RegressionDriver<Chwirut, 3>("Chwirut1.dat");
  easy_success += RegressionDriver<Chwirut, 3>("Chwirut2.dat");
  easy_success += RegressionDriver<Lanczos, 6>("Lanczos3.dat");
  easy_success += RegressionDriver<Gauss, 8>("Gauss1.dat");
  easy_success += RegressionDriver<Gauss, 8>("Gauss2.dat");
  easy_success += RegressionDriver<DanWood, 2>("DanWood.dat");
  easy_success += RegressionDriver<Misra1b, 2>("Misra1b.dat");

  cout << "\nMedium Difficulty\n";
  int medium_success = 0;
  medium_success += RegressionDriver<Kirby2, 5>("Kirby2.dat");
  medium_success += RegressionDriver<Hahn1, 7>("Hahn1.dat");
  medium_success += RegressionDriver<Nelson, 3>("Nelson.dat");
  medium_success += RegressionDriver<MGH17, 5>("MGH17.dat");
  medium_success += RegressionDriver<Lanczos, 6>("Lanczos1.dat");
  medium_success += RegressionDriver<Lanczos, 6>("Lanczos2.dat");
  medium_success += RegressionDriver<Gauss, 8>("Gauss3.dat");
  medium_success += RegressionDriver<Misra1c, 2>("Misra1c.dat");
  medium_success += RegressionDriver<Misra1d, 2>("Misra1d.dat");
  medium_success += RegressionDriver<Roszman1, 4>("Roszman1.dat");
  medium_success += RegressionDriver<ENSO, 9>("ENSO.dat");

  cout << "\nHigher Difficulty\n";
  int hard_success = 0;
  hard_success += RegressionDriver<MGH09, 4>("MGH09.dat");
  hard_success += RegressionDriver<Thurber, 7>("Thurber.dat");
  hard_success += RegressionDriver<BoxBOD, 2>("BoxBOD.dat");
  hard_success += RegressionDriver<Rat42, 3>("Rat42.dat");
  hard_success += RegressionDriver<MGH10, 3>("MGH10.dat");
  hard_success += RegressionDriver<Eckerle4, 3>("Eckerle4.dat");
  hard_success += RegressionDriver<Rat43, 4>("Rat43.dat");
  hard_success += RegressionDriver<Bennet5, 3>("Bennett5.dat");

  cout << "\n";
  cout << "Easy    : " << easy_success << "/16\n";
  cout << "Medium  : " << medium_success << "/22\n";
  cout << "Hard    : " << hard_success << "/16\n";
  cout << "Total   : " << easy_success + medium_success + hard_success
       << "/54\n";
}

}  // namespace
}  // namespace examples
}  // namespace ceres

int main(int argc, char** argv) {
  GFLAGS_NAMESPACE::ParseCommandLineFlags(&argc, &argv, true);
  google::InitGoogleLogging(argv[0]);
  ceres::examples::SolveNISTProblems();
  return 0;
}