| // 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: |
| // |
| // * 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) |
| |
| #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_ |