internal/ceres/polynomial.cc - ceres-solver - Git at Google

 // Ceres Solver - A fast non-linear least squares minimizer
 // Copyright 2023 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: moll.markus@arcor.de (Markus Moll)
 //         sameeragarwal@google.com (Sameer Agarwal)

 #include "ceres/polynomial.h"

 #include <cmath>
 #include <cstddef>
 #include <vector>

 #include "Eigen/Dense"
 #include "absl/log/check.h"
 #include "absl/log/log.h"
 #include "ceres/function_sample.h"
 #include "ceres/internal/export.h"

 namespace ceres::internal {

 namespace {

 // Balancing function as described by B. N. Parlett and C. Reinsch,
 // "Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors".
 // In: Numerische Mathematik, Volume 13, Number 4 (1969), 293-304,
 // Springer Berlin / Heidelberg. DOI: 10.1007/BF02165404
 void BalanceCompanionMatrix(Matrix* companion_matrix_ptr) {
   CHECK(companion_matrix_ptr != nullptr);
   Matrix& companion_matrix = *companion_matrix_ptr;
   Matrix companion_matrix_offdiagonal = companion_matrix;
   companion_matrix_offdiagonal.diagonal().setZero();

   const int degree = companion_matrix.rows();

   // gamma <= 1 controls how much a change in the scaling has to
   // lower the 1-norm of the companion matrix to be accepted.
   //
   // gamma = 1 seems to lead to cycles (numerical issues?), so
   // we set it slightly lower.
   const double gamma = 0.9;

   // Greedily scale row/column pairs until there is no change.
   bool scaling_has_changed;
   do {
     scaling_has_changed = false;

     for (int i = 0; i < degree; ++i) {
       const double row_norm = companion_matrix_offdiagonal.row(i).lpNorm<1>();
       const double col_norm = companion_matrix_offdiagonal.col(i).lpNorm<1>();

       // Decompose row_norm/col_norm into mantissa * 2^exponent,
       // where 0.5 <= mantissa < 1. Discard mantissa (return value
       // of frexp), as only the exponent is needed.
       int exponent = 0;
       std::frexp(row_norm / col_norm, &exponent);
       exponent /= 2;

       if (exponent != 0) {
         const double scaled_col_norm = std::ldexp(col_norm, exponent);
         const double scaled_row_norm = std::ldexp(row_norm, -exponent);
         if (scaled_col_norm + scaled_row_norm < gamma * (col_norm + row_norm)) {
           // Accept the new scaling. (Multiplication by powers of 2 should not
           // introduce rounding errors (ignoring non-normalized numbers and
           // over- or underflow))
           scaling_has_changed = true;
           companion_matrix_offdiagonal.row(i) *= std::ldexp(1.0, -exponent);
           companion_matrix_offdiagonal.col(i) *= std::ldexp(1.0, exponent);
         }
       }
     }
   } while (scaling_has_changed);

   companion_matrix_offdiagonal.diagonal() = companion_matrix.diagonal();
   companion_matrix = companion_matrix_offdiagonal;
   VLOG(3) << "Balanced companion matrix is\n" << companion_matrix;
 }

 void BuildCompanionMatrix(const Vector& polynomial,
                           Matrix* companion_matrix_ptr) {
   CHECK(companion_matrix_ptr != nullptr);
   Matrix& companion_matrix = *companion_matrix_ptr;

   const int degree = polynomial.size() - 1;

   companion_matrix.resize(degree, degree);
   companion_matrix.setZero();
   companion_matrix.diagonal(-1).setOnes();
   companion_matrix.col(degree - 1) = -polynomial.reverse().head(degree);
 }

 // Remove leading terms with zero coefficients.
 Vector RemoveLeadingZeros(const Vector& polynomial_in) {
   int i = 0;
   while (i < (polynomial_in.size() - 1) && polynomial_in(i) == 0.0) {
     ++i;
   }
   return polynomial_in.tail(polynomial_in.size() - i);
 }

 void FindLinearPolynomialRoots(const Vector& polynomial,
                                Vector* real,
                                Vector* imaginary) {
   CHECK_EQ(polynomial.size(), 2);
   if (real != nullptr) {
     real->resize(1);
     (*real)(0) = -polynomial(1) / polynomial(0);
   }

   if (imaginary != nullptr) {
     imaginary->setZero(1);
   }
 }

 void FindQuadraticPolynomialRoots(const Vector& polynomial,
                                   Vector* real,
                                   Vector* imaginary) {
   CHECK_EQ(polynomial.size(), 3);
   const double a = polynomial(0);
   const double b = polynomial(1);
   const double c = polynomial(2);
   const double D = b * b - 4 * a * c;
   const double sqrt_D = sqrt(fabs(D));
   if (real != nullptr) {
     real->setZero(2);
   }
   if (imaginary != nullptr) {
     imaginary->setZero(2);
   }

   // Real roots.
   if (D >= 0) {
     if (real != nullptr) {
       // Stable quadratic roots according to BKP Horn.
       // http://people.csail.mit.edu/bkph/articles/Quadratics.pdf
       if (b >= 0) {
         (*real)(0) = (-b - sqrt_D) / (2.0 * a);
         (*real)(1) = (2.0 * c) / (-b - sqrt_D);
       } else {
         (*real)(0) = (2.0 * c) / (-b + sqrt_D);
         (*real)(1) = (-b + sqrt_D) / (2.0 * a);
       }
     }
     return;
   }

   // Use the normal quadratic formula for the complex case.
   if (real != nullptr) {
     (*real)(0) = -b / (2.0 * a);
     (*real)(1) = -b / (2.0 * a);
   }
   if (imaginary != nullptr) {
     (*imaginary)(0) = sqrt_D / (2.0 * a);
     (*imaginary)(1) = -sqrt_D / (2.0 * a);
   }
 }
 }  // namespace

 bool FindPolynomialRoots(const Vector& polynomial_in,
                          Vector* real,
                          Vector* imaginary) {
   if (polynomial_in.size() == 0) {
     LOG(ERROR) << "Invalid polynomial of size 0 passed to FindPolynomialRoots";
     return false;
   }

   Vector polynomial = RemoveLeadingZeros(polynomial_in);
   const int degree = polynomial.size() - 1;

   VLOG(3) << "Input polynomial: " << polynomial_in.transpose();
   if (polynomial.size() != polynomial_in.size()) {
     VLOG(3) << "Trimmed polynomial: " << polynomial.transpose();
   }

   // Is the polynomial constant?
   if (degree == 0) {
     LOG(WARNING) << "Trying to extract roots from a constant "
                  << "polynomial in FindPolynomialRoots";
     // We return true with no roots, not false, as if the polynomial is constant
     // it is correct that there are no roots. It is not the case that they were
     // there, but that we have failed to extract them.
     return true;
   }

   // Linear
   if (degree == 1) {
     FindLinearPolynomialRoots(polynomial, real, imaginary);
     return true;
   }

   // Quadratic
   if (degree == 2) {
     FindQuadraticPolynomialRoots(polynomial, real, imaginary);
     return true;
   }

   // The degree is now known to be at least 3. For cubic or higher
   // roots we use the method of companion matrices.

   // Divide by leading term
   const double leading_term = polynomial(0);
   polynomial /= leading_term;

   // Build and balance the companion matrix to the polynomial.
   Matrix companion_matrix(degree, degree);
   BuildCompanionMatrix(polynomial, &companion_matrix);
   BalanceCompanionMatrix(&companion_matrix);

   // Find its (complex) eigenvalues.
   Eigen::EigenSolver<Matrix> solver(companion_matrix, false);
   if (solver.info() != Eigen::Success) {
     LOG(ERROR) << "Failed to extract eigenvalues from companion matrix.";
     return false;
   }

   // Output roots
   if (real != nullptr) {
     *real = solver.eigenvalues().real();
   } else {
     LOG(WARNING) << "nullptr pointer passed as real argument to "
                  << "FindPolynomialRoots. Real parts of the roots will not "
                  << "be returned.";
   }
   if (imaginary != nullptr) {
     *imaginary = solver.eigenvalues().imag();
   }
   return true;
 }

 Vector DifferentiatePolynomial(const Vector& polynomial) {
   const int degree = polynomial.rows() - 1;
   CHECK_GE(degree, 0);

   // Degree zero polynomials are constants, and their derivative does
   // not result in a smaller degree polynomial, just a degree zero
   // polynomial with value zero.
   if (degree == 0) {
     return Eigen::VectorXd::Zero(1);
   }

   Vector derivative(degree);
   for (int i = 0; i < degree; ++i) {
     derivative(i) = (degree - i) * polynomial(i);
   }

   return derivative;
 }

 void MinimizePolynomial(const Vector& polynomial,
                         const double x_min,
                         const double x_max,
                         double* optimal_x,
                         double* optimal_value) {
   // Find the minimum of the polynomial at the two ends.
   //
   // We start by inspecting the middle of the interval. Technically
   // this is not needed, but we do this to make this code as close to
   // the minFunc package as possible.
   *optimal_x = (x_min + x_max) / 2.0;
   *optimal_value = EvaluatePolynomial(polynomial, *optimal_x);

   const double x_min_value = EvaluatePolynomial(polynomial, x_min);
   if (x_min_value < *optimal_value) {
     *optimal_value = x_min_value;
     *optimal_x = x_min;
   }

   const double x_max_value = EvaluatePolynomial(polynomial, x_max);
   if (x_max_value < *optimal_value) {
     *optimal_value = x_max_value;
     *optimal_x = x_max;
   }

   // If the polynomial is linear or constant, we are done.
   if (polynomial.rows() <= 2) {
     return;
   }

   const Vector derivative = DifferentiatePolynomial(polynomial);
   Vector roots_real;
   if (!FindPolynomialRoots(derivative, &roots_real, nullptr)) {
     LOG(WARNING) << "Unable to find the critical points of "
                  << "the interpolating polynomial.";
     return;
   }

   // This is a bit of an overkill, as some of the roots may actually
   // have a complex part, but its simpler to just check these values.
   for (int i = 0; i < roots_real.rows(); ++i) {
     const double root = roots_real(i);
     if ((root < x_min) || (root > x_max)) {
       continue;
     }

     const double value = EvaluatePolynomial(polynomial, root);
     if (value < *optimal_value) {
       *optimal_value = value;
       *optimal_x = root;
     }
   }
 }

 Vector FindInterpolatingPolynomial(const std::vector<FunctionSample>& samples) {
   const int num_samples = samples.size();
   int num_constraints = 0;
   for (int i = 0; i < num_samples; ++i) {
     if (samples[i].value_is_valid) {
       ++num_constraints;
     }
     if (samples[i].gradient_is_valid) {
       ++num_constraints;
     }
   }

   const int degree = num_constraints - 1;

   Matrix lhs = Matrix::Zero(num_constraints, num_constraints);
   Vector rhs = Vector::Zero(num_constraints);

   int row = 0;
   for (int i = 0; i < num_samples; ++i) {
     const FunctionSample& sample = samples[i];
     if (sample.value_is_valid) {
       for (int j = 0; j <= degree; ++j) {
         lhs(row, j) = pow(sample.x, degree - j);
       }
       rhs(row) = sample.value;
       ++row;
     }

     if (sample.gradient_is_valid) {
       for (int j = 0; j < degree; ++j) {
         lhs(row, j) = (degree - j) * pow(sample.x, degree - j - 1);
       }
       rhs(row) = sample.gradient;
       ++row;
     }
   }

   // TODO(sameeragarwal): This is a hack.
   // https://github.com/ceres-solver/ceres-solver/issues/248
   Eigen::FullPivLU<Matrix> lu(lhs);
   return lu.setThreshold(0.0).solve(rhs);
 }

 void MinimizeInterpolatingPolynomial(const std::vector<FunctionSample>& samples,
                                      double x_min,
                                      double x_max,
                                      double* optimal_x,
                                      double* optimal_value) {
   const Vector polynomial = FindInterpolatingPolynomial(samples);
   MinimizePolynomial(polynomial, x_min, x_max, optimal_x, optimal_value);
   for (const auto& sample : samples) {
     if ((sample.x < x_min) || (sample.x > x_max)) {
       continue;
     }

     const double value = EvaluatePolynomial(polynomial, sample.x);
     if (value < *optimal_value) {
       *optimal_x = sample.x;
       *optimal_value = value;
     }
   }
 }

 }  // namespace ceres::internal
	// Ceres Solver - A fast non-linear least squares minimizer
	// Copyright 2023 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: moll.markus@arcor.de (Markus Moll)
	// sameeragarwal@google.com (Sameer Agarwal)

	#include "ceres/polynomial.h"

	#include <cmath>
	#include <cstddef>
	#include <vector>

	#include "Eigen/Dense"
	#include "absl/log/check.h"
	#include "absl/log/log.h"
	#include "ceres/function_sample.h"
	#include "ceres/internal/export.h"

	namespace ceres::internal {

	namespace {

	// Balancing function as described by B. N. Parlett and C. Reinsch,
	// "Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors".
	// In: Numerische Mathematik, Volume 13, Number 4 (1969), 293-304,
	// Springer Berlin / Heidelberg. DOI: 10.1007/BF02165404
	void BalanceCompanionMatrix(Matrix* companion_matrix_ptr) {
	CHECK(companion_matrix_ptr != nullptr);
	Matrix& companion_matrix = *companion_matrix_ptr;
	Matrix companion_matrix_offdiagonal = companion_matrix;
	companion_matrix_offdiagonal.diagonal().setZero();

	const int degree = companion_matrix.rows();

	// gamma <= 1 controls how much a change in the scaling has to
	// lower the 1-norm of the companion matrix to be accepted.
	//
	// gamma = 1 seems to lead to cycles (numerical issues?), so
	// we set it slightly lower.
	const double gamma = 0.9;

	// Greedily scale row/column pairs until there is no change.
	bool scaling_has_changed;
	do {
	scaling_has_changed = false;

	for (int i = 0; i < degree; ++i) {
	const double row_norm = companion_matrix_offdiagonal.row(i).lpNorm<1>();
	const double col_norm = companion_matrix_offdiagonal.col(i).lpNorm<1>();

	// Decompose row_norm/col_norm into mantissa * 2^exponent,
	// where 0.5 <= mantissa < 1. Discard mantissa (return value
	// of frexp), as only the exponent is needed.
	int exponent = 0;
	std::frexp(row_norm / col_norm, &exponent);
	exponent /= 2;

	if (exponent != 0) {
	const double scaled_col_norm = std::ldexp(col_norm, exponent);
	const double scaled_row_norm = std::ldexp(row_norm, -exponent);
	if (scaled_col_norm + scaled_row_norm < gamma * (col_norm + row_norm)) {
	// Accept the new scaling. (Multiplication by powers of 2 should not
	// introduce rounding errors (ignoring non-normalized numbers and
	// over- or underflow))
	scaling_has_changed = true;
	companion_matrix_offdiagonal.row(i) *= std::ldexp(1.0, -exponent);
	companion_matrix_offdiagonal.col(i) *= std::ldexp(1.0, exponent);
	}
	}
	}
	} while (scaling_has_changed);

	companion_matrix_offdiagonal.diagonal() = companion_matrix.diagonal();
	companion_matrix = companion_matrix_offdiagonal;
	VLOG(3) << "Balanced companion matrix is\n" << companion_matrix;
	}

	void BuildCompanionMatrix(const Vector& polynomial,
	Matrix* companion_matrix_ptr) {
	CHECK(companion_matrix_ptr != nullptr);
	Matrix& companion_matrix = *companion_matrix_ptr;

	const int degree = polynomial.size() - 1;

	companion_matrix.resize(degree, degree);
	companion_matrix.setZero();
	companion_matrix.diagonal(-1).setOnes();
	companion_matrix.col(degree - 1) = -polynomial.reverse().head(degree);
	}

	// Remove leading terms with zero coefficients.
	Vector RemoveLeadingZeros(const Vector& polynomial_in) {
	int i = 0;
	while (i < (polynomial_in.size() - 1) && polynomial_in(i) == 0.0) {
	++i;
	}
	return polynomial_in.tail(polynomial_in.size() - i);
	}

	void FindLinearPolynomialRoots(const Vector& polynomial,
	Vector* real,
	Vector* imaginary) {
	CHECK_EQ(polynomial.size(), 2);
	if (real != nullptr) {
	real->resize(1);
	(*real)(0) = -polynomial(1) / polynomial(0);
	}

	if (imaginary != nullptr) {
	imaginary->setZero(1);
	}
	}

	void FindQuadraticPolynomialRoots(const Vector& polynomial,
	Vector* real,
	Vector* imaginary) {
	CHECK_EQ(polynomial.size(), 3);
	const double a = polynomial(0);
	const double b = polynomial(1);
	const double c = polynomial(2);
	const double D = b * b - 4 * a * c;
	const double sqrt_D = sqrt(fabs(D));
	if (real != nullptr) {
	real->setZero(2);
	}
	if (imaginary != nullptr) {
	imaginary->setZero(2);
	}

	// Real roots.
	if (D >= 0) {
	if (real != nullptr) {
	// Stable quadratic roots according to BKP Horn.
	// http://people.csail.mit.edu/bkph/articles/Quadratics.pdf
	if (b >= 0) {
	(real)(0) = (-b - sqrt_D) / (2.0 a);
	(real)(1) = (2.0 c) / (-b - sqrt_D);
	} else {
	(real)(0) = (2.0 c) / (-b + sqrt_D);
	(real)(1) = (-b + sqrt_D) / (2.0 a);
	}
	}
	return;
	}

	// Use the normal quadratic formula for the complex case.
	if (real != nullptr) {
	(real)(0) = -b / (2.0 a);
	(real)(1) = -b / (2.0 a);
	}
	if (imaginary != nullptr) {
	(imaginary)(0) = sqrt_D / (2.0 a);
	(imaginary)(1) = -sqrt_D / (2.0 a);
	}
	}
	} // namespace

	bool FindPolynomialRoots(const Vector& polynomial_in,
	Vector* real,
	Vector* imaginary) {
	if (polynomial_in.size() == 0) {
	LOG(ERROR) << "Invalid polynomial of size 0 passed to FindPolynomialRoots";
	return false;
	}

	Vector polynomial = RemoveLeadingZeros(polynomial_in);
	const int degree = polynomial.size() - 1;

	VLOG(3) << "Input polynomial: " << polynomial_in.transpose();
	if (polynomial.size() != polynomial_in.size()) {
	VLOG(3) << "Trimmed polynomial: " << polynomial.transpose();
	}

	// Is the polynomial constant?
	if (degree == 0) {
	LOG(WARNING) << "Trying to extract roots from a constant "
	<< "polynomial in FindPolynomialRoots";
	// We return true with no roots, not false, as if the polynomial is constant
	// it is correct that there are no roots. It is not the case that they were
	// there, but that we have failed to extract them.
	return true;
	}

	// Linear
	if (degree == 1) {
	FindLinearPolynomialRoots(polynomial, real, imaginary);
	return true;
	}

	// Quadratic
	if (degree == 2) {
	FindQuadraticPolynomialRoots(polynomial, real, imaginary);
	return true;
	}

	// The degree is now known to be at least 3. For cubic or higher
	// roots we use the method of companion matrices.

	// Divide by leading term
	const double leading_term = polynomial(0);
	polynomial /= leading_term;

	// Build and balance the companion matrix to the polynomial.
	Matrix companion_matrix(degree, degree);
	BuildCompanionMatrix(polynomial, &companion_matrix);
	BalanceCompanionMatrix(&companion_matrix);

	// Find its (complex) eigenvalues.
	Eigen::EigenSolver<Matrix> solver(companion_matrix, false);
	if (solver.info() != Eigen::Success) {
	LOG(ERROR) << "Failed to extract eigenvalues from companion matrix.";
	return false;
	}

	// Output roots
	if (real != nullptr) {
	*real = solver.eigenvalues().real();
	} else {
	LOG(WARNING) << "nullptr pointer passed as real argument to "
	<< "FindPolynomialRoots. Real parts of the roots will not "
	<< "be returned.";
	}
	if (imaginary != nullptr) {
	*imaginary = solver.eigenvalues().imag();
	}
	return true;
	}

	Vector DifferentiatePolynomial(const Vector& polynomial) {
	const int degree = polynomial.rows() - 1;
	CHECK_GE(degree, 0);

	// Degree zero polynomials are constants, and their derivative does
	// not result in a smaller degree polynomial, just a degree zero
	// polynomial with value zero.
	if (degree == 0) {
	return Eigen::VectorXd::Zero(1);
	}

	Vector derivative(degree);
	for (int i = 0; i < degree; ++i) {
	derivative(i) = (degree - i) * polynomial(i);
	}

	return derivative;
	}

	void MinimizePolynomial(const Vector& polynomial,
	const double x_min,
	const double x_max,
	double* optimal_x,
	double* optimal_value) {
	// Find the minimum of the polynomial at the two ends.
	//
	// We start by inspecting the middle of the interval. Technically
	// this is not needed, but we do this to make this code as close to
	// the minFunc package as possible.
	*optimal_x = (x_min + x_max) / 2.0;
	optimal_value = EvaluatePolynomial(polynomial, optimal_x);

	const double x_min_value = EvaluatePolynomial(polynomial, x_min);
	if (x_min_value < *optimal_value) {
	*optimal_value = x_min_value;
	*optimal_x = x_min;
	}

	const double x_max_value = EvaluatePolynomial(polynomial, x_max);
	if (x_max_value < *optimal_value) {
	*optimal_value = x_max_value;
	*optimal_x = x_max;
	}

	// If the polynomial is linear or constant, we are done.
	if (polynomial.rows() <= 2) {
	return;
	}

	const Vector derivative = DifferentiatePolynomial(polynomial);
	Vector roots_real;
	if (!FindPolynomialRoots(derivative, &roots_real, nullptr)) {
	LOG(WARNING) << "Unable to find the critical points of "
	<< "the interpolating polynomial.";
	return;
	}

	// This is a bit of an overkill, as some of the roots may actually
	// have a complex part, but its simpler to just check these values.
	for (int i = 0; i < roots_real.rows(); ++i) {
	const double root = roots_real(i);
	if ((root < x_min) \|\| (root > x_max)) {
	continue;
	}

	const double value = EvaluatePolynomial(polynomial, root);
	if (value < *optimal_value) {
	*optimal_value = value;
	*optimal_x = root;
	}
	}
	}

	Vector FindInterpolatingPolynomial(const std::vector<FunctionSample>& samples) {
	const int num_samples = samples.size();
	int num_constraints = 0;
	for (int i = 0; i < num_samples; ++i) {
	if (samples[i].value_is_valid) {
	++num_constraints;
	}
	if (samples[i].gradient_is_valid) {
	++num_constraints;
	}
	}

	const int degree = num_constraints - 1;

	Matrix lhs = Matrix::Zero(num_constraints, num_constraints);
	Vector rhs = Vector::Zero(num_constraints);

	int row = 0;
	for (int i = 0; i < num_samples; ++i) {
	const FunctionSample& sample = samples[i];
	if (sample.value_is_valid) {
	for (int j = 0; j <= degree; ++j) {
	lhs(row, j) = pow(sample.x, degree - j);
	}
	rhs(row) = sample.value;
	++row;
	}

	if (sample.gradient_is_valid) {
	for (int j = 0; j < degree; ++j) {
	lhs(row, j) = (degree - j) * pow(sample.x, degree - j - 1);
	}
	rhs(row) = sample.gradient;
	++row;
	}
	}

	// TODO(sameeragarwal): This is a hack.
	// https://github.com/ceres-solver/ceres-solver/issues/248
	Eigen::FullPivLU<Matrix> lu(lhs);
	return lu.setThreshold(0.0).solve(rhs);
	}

	void MinimizeInterpolatingPolynomial(const std::vector<FunctionSample>& samples,
	double x_min,
	double x_max,
	double* optimal_x,
	double* optimal_value) {
	const Vector polynomial = FindInterpolatingPolynomial(samples);
	MinimizePolynomial(polynomial, x_min, x_max, optimal_x, optimal_value);
	for (const auto& sample : samples) {
	if ((sample.x < x_min) \|\| (sample.x > x_max)) {
	continue;
	}

	const double value = EvaluatePolynomial(polynomial, sample.x);
	if (value < *optimal_value) {
	*optimal_x = sample.x;
	*optimal_value = value;
	}
	}
	}

	} // namespace ceres::internal