// Ceres Solver - A fast non-linear least squares minimizer

// Copyright 2015 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:

//

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//   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.

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//   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

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//

// Author: moll.markus@arcor.de (Markus Moll)

//         sameeragarwal@google.com (Sameer Agarwal)

#ifndef CERES_INTERNAL_POLYNOMIAL_SOLVER_H_

#define CERES_INTERNAL_POLYNOMIAL_SOLVER_H_

#include <vector>

#include "ceres/internal/eigen.h"

#include "ceres/internal/port.h"

namespace ceres {

namespace internal {

struct FunctionSample;

// All polynomials are assumed to be the form

//

//   sum_{i=0}^N polynomial(i) x^{N-i}.

//

// and are given by a vector of coefficients of size N + 1.

// Evaluate the polynomial at x using the Horner scheme.

inline double EvaluatePolynomial(const Vector& polynomial, double x) {

  double v = 0.0;

  for (int i = 0; i < polynomial.size(); ++i) {

    v = v * x + polynomial(i);

  }

  return v;

}

// Use the companion matrix eigenvalues to determine the roots of the

// polynomial.

//

// This function returns true on success, false otherwise.

// Failure indicates that the polynomial is invalid (of size 0) or

// that the eigenvalues of the companion matrix could not be computed.

// On failure, a more detailed message will be written to LOG(ERROR).

// If real is not NULL, the real parts of the roots will be returned in it.

// Likewise, if imaginary is not NULL, imaginary parts will be returned in it.

bool FindPolynomialRoots(const Vector& polynomial,

                         Vector* real,

                         Vector* imaginary);

// Return the derivative of the given polynomial. It is assumed that

// the input polynomial is at least of degree zero.

Vector DifferentiatePolynomial(const Vector& polynomial);

// Find the minimum value of the polynomial in the interval [x_min,

// x_max]. The minimum is obtained by computing all the roots of the

// derivative of the input polynomial. All real roots within the

// interval [x_min, x_max] are considered as well as the end points

// x_min and x_max. Since polynomials are differentiable functions,

// this ensures that the true minimum is found.

void MinimizePolynomial(const Vector& polynomial,

                        double x_min,

                        double x_max,

                        double* optimal_x,

                        double* optimal_value);

// Given a set of function value and/or gradient samples, find a

// polynomial whose value and gradients are exactly equal to the ones

// in samples.

//

// Generally speaking,

//

// degree = # values + # gradients - 1

//

// Of course its possible to sample a polynomial any number of times,

// in which case, generally speaking the spurious higher order

// coefficients will be zero.

Vector FindInterpolatingPolynomial(const std::vector<FunctionSample>& samples);

// Interpolate the function described by samples with a polynomial,

// and minimize it on the interval [x_min, x_max]. Depending on the

// input samples, it is possible that the interpolation or the root

// finding algorithms may fail due to numerical difficulties. But the

// function is guaranteed to return its best guess of an answer, by

// considering the samples and the end points as possible solutions.

void MinimizeInterpolatingPolynomial(const std::vector<FunctionSample>& samples,

                                     double x_min,

                                     double x_max,

                                     double* optimal_x,

                                     double* optimal_value);

}  // namespace internal

}  // namespace ceres

#endif  // CERES_INTERNAL_POLYNOMIAL_SOLVER_H_