11d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Ceres Solver - A fast non-linear least squares minimizer 21d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Copyright 2012 Google Inc. All rights reserved. 31d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// http://code.google.com/p/ceres-solver/ 41d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// 51d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Redistribution and use in source and binary forms, with or without 61d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// modification, are permitted provided that the following conditions are met: 71d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// 81d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// * Redistributions of source code must retain the above copyright notice, 91d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// this list of conditions and the following disclaimer. 101d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// * Redistributions in binary form must reproduce the above copyright notice, 111d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// this list of conditions and the following disclaimer in the documentation 121d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// and/or other materials provided with the distribution. 131d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// * Neither the name of Google Inc. nor the names of its contributors may be 141d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// used to endorse or promote products derived from this software without 151d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// specific prior written permission. 161d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// 171d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 181d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 191d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 201d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE 211d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR 221d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF 231d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS 241d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN 251d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 261d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE 271d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// POSSIBILITY OF SUCH DAMAGE. 281d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// 291d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Author: moll.markus@arcor.de (Markus Moll) 301d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// sameeragarwal@google.com (Sameer Agarwal) 311d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 321d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling#ifndef CERES_INTERNAL_POLYNOMIAL_SOLVER_H_ 331d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling#define CERES_INTERNAL_POLYNOMIAL_SOLVER_H_ 341d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 351d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling#include <vector> 361d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling#include "ceres/internal/eigen.h" 371d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling#include "ceres/internal/port.h" 381d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 391d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberlingnamespace ceres { 401d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberlingnamespace internal { 411d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 421d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// All polynomials are assumed to be the form 431d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// 441d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// sum_{i=0}^N polynomial(i) x^{N-i}. 451d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// 461d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// and are given by a vector of coefficients of size N + 1. 471d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 481d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Evaluate the polynomial at x using the Horner scheme. 491d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberlinginline double EvaluatePolynomial(const Vector& polynomial, double x) { 501d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double v = 0.0; 511d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling for (int i = 0; i < polynomial.size(); ++i) { 521d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling v = v * x + polynomial(i); 531d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling } 541d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling return v; 551d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling} 561d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 571d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Use the companion matrix eigenvalues to determine the roots of the 581d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// polynomial. 591d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// 601d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// This function returns true on success, false otherwise. 611d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Failure indicates that the polynomial is invalid (of size 0) or 621d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// that the eigenvalues of the companion matrix could not be computed. 631d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// On failure, a more detailed message will be written to LOG(ERROR). 641d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// If real is not NULL, the real parts of the roots will be returned in it. 651d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Likewise, if imaginary is not NULL, imaginary parts will be returned in it. 661d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberlingbool FindPolynomialRoots(const Vector& polynomial, 671d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling Vector* real, 681d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling Vector* imaginary); 691d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 701d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Return the derivative of the given polynomial. It is assumed that 711d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// the input polynomial is at least of degree zero. 721d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha HaeberlingVector DifferentiatePolynomial(const Vector& polynomial); 731d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 741d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Find the minimum value of the polynomial in the interval [x_min, 751d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// x_max]. The minimum is obtained by computing all the roots of the 761d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// derivative of the input polynomial. All real roots within the 771d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// interval [x_min, x_max] are considered as well as the end points 781d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// x_min and x_max. Since polynomials are differentiable functions, 791d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// this ensures that the true minimum is found. 801d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberlingvoid MinimizePolynomial(const Vector& polynomial, 811d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double x_min, 821d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double x_max, 831d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double* optimal_x, 841d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double* optimal_value); 851d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 861d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Structure for storing sample values of a function. 871d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// 881d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Clients can use this struct to communicate the value of the 891d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// function and or its gradient at a given point x. 901d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberlingstruct FunctionSample { 911d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling FunctionSample() 921d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling : x(0.0), 931d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling value(0.0), 941d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling value_is_valid(false), 951d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling gradient(0.0), 961d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling gradient_is_valid(false) { 971d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling } 981d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 991d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double x; 1001d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double value; // value = f(x) 1011d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling bool value_is_valid; 1021d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double gradient; // gradient = f'(x) 1031d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling bool gradient_is_valid; 1041d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling}; 1051d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 1061d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Given a set of function value and/or gradient samples, find a 1071d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// polynomial whose value and gradients are exactly equal to the ones 1081d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// in samples. 1091d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// 1101d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Generally speaking, 1111d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// 1121d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// degree = # values + # gradients - 1 1131d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// 1141d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Of course its possible to sample a polynomial any number of times, 1151d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// in which case, generally speaking the spurious higher order 1161d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// coefficients will be zero. 1171d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha HaeberlingVector FindInterpolatingPolynomial(const vector<FunctionSample>& samples); 1181d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 1191d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// Interpolate the function described by samples with a polynomial, 1201d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// and minimize it on the interval [x_min, x_max]. Depending on the 1211d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// input samples, it is possible that the interpolation or the root 1221d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// finding algorithms may fail due to numerical difficulties. But the 1231d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// function is guaranteed to return its best guess of an answer, by 1241d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling// considering the samples and the end points as possible solutions. 1251d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberlingvoid MinimizeInterpolatingPolynomial(const vector<FunctionSample>& samples, 1261d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double x_min, 1271d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double x_max, 1281d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double* optimal_x, 1291d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling double* optimal_value); 1301d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 1311d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling} // namespace internal 1321d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling} // namespace ceres 1331d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling 1341d2624a10e2c559f8ba9ef89eaa30832c0a83a96Sascha Haeberling#endif // CERES_INTERNAL_POLYNOMIAL_SOLVER_H_ 135