1// Ceres Solver - A fast non-linear least squares minimizer 2// Copyright 2012 Google Inc. All rights reserved. 3// http://code.google.com/p/ceres-solver/ 4// 5// Redistribution and use in source and binary forms, with or without 6// modification, are permitted provided that the following conditions are met: 7// 8// * Redistributions of source code must retain the above copyright notice, 9// this list of conditions and the following disclaimer. 10// * Redistributions in binary form must reproduce the above copyright notice, 11// this list of conditions and the following disclaimer in the documentation 12// and/or other materials provided with the distribution. 13// * Neither the name of Google Inc. nor the names of its contributors may be 14// used to endorse or promote products derived from this software without 15// specific prior written permission. 16// 17// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 18// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 19// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 20// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE 21// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR 22// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF 23// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS 24// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN 25// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 26// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE 27// POSSIBILITY OF SUCH DAMAGE. 28// 29// Author: moll.markus@arcor.de (Markus Moll) 30// sameeragarwal@google.com (Sameer Agarwal) 31 32#ifndef CERES_INTERNAL_POLYNOMIAL_SOLVER_H_ 33#define CERES_INTERNAL_POLYNOMIAL_SOLVER_H_ 34 35#include <vector> 36#include "ceres/internal/eigen.h" 37#include "ceres/internal/port.h" 38 39namespace ceres { 40namespace internal { 41 42// All polynomials are assumed to be the form 43// 44// sum_{i=0}^N polynomial(i) x^{N-i}. 45// 46// and are given by a vector of coefficients of size N + 1. 47 48// Evaluate the polynomial at x using the Horner scheme. 49inline double EvaluatePolynomial(const Vector& polynomial, double x) { 50 double v = 0.0; 51 for (int i = 0; i < polynomial.size(); ++i) { 52 v = v * x + polynomial(i); 53 } 54 return v; 55} 56 57// Use the companion matrix eigenvalues to determine the roots of the 58// polynomial. 59// 60// This function returns true on success, false otherwise. 61// Failure indicates that the polynomial is invalid (of size 0) or 62// that the eigenvalues of the companion matrix could not be computed. 63// On failure, a more detailed message will be written to LOG(ERROR). 64// If real is not NULL, the real parts of the roots will be returned in it. 65// Likewise, if imaginary is not NULL, imaginary parts will be returned in it. 66bool FindPolynomialRoots(const Vector& polynomial, 67 Vector* real, 68 Vector* imaginary); 69 70// Return the derivative of the given polynomial. It is assumed that 71// the input polynomial is at least of degree zero. 72Vector DifferentiatePolynomial(const Vector& polynomial); 73 74// Find the minimum value of the polynomial in the interval [x_min, 75// x_max]. The minimum is obtained by computing all the roots of the 76// derivative of the input polynomial. All real roots within the 77// interval [x_min, x_max] are considered as well as the end points 78// x_min and x_max. Since polynomials are differentiable functions, 79// this ensures that the true minimum is found. 80void MinimizePolynomial(const Vector& polynomial, 81 double x_min, 82 double x_max, 83 double* optimal_x, 84 double* optimal_value); 85 86// Structure for storing sample values of a function. 87// 88// Clients can use this struct to communicate the value of the 89// function and or its gradient at a given point x. 90struct FunctionSample { 91 FunctionSample() 92 : x(0.0), 93 value(0.0), 94 value_is_valid(false), 95 gradient(0.0), 96 gradient_is_valid(false) { 97 } 98 string ToDebugString() const; 99 100 double x; 101 double value; // value = f(x) 102 bool value_is_valid; 103 double gradient; // gradient = f'(x) 104 bool gradient_is_valid; 105}; 106 107// Given a set of function value and/or gradient samples, find a 108// polynomial whose value and gradients are exactly equal to the ones 109// in samples. 110// 111// Generally speaking, 112// 113// degree = # values + # gradients - 1 114// 115// Of course its possible to sample a polynomial any number of times, 116// in which case, generally speaking the spurious higher order 117// coefficients will be zero. 118Vector FindInterpolatingPolynomial(const vector<FunctionSample>& samples); 119 120// Interpolate the function described by samples with a polynomial, 121// and minimize it on the interval [x_min, x_max]. Depending on the 122// input samples, it is possible that the interpolation or the root 123// finding algorithms may fail due to numerical difficulties. But the 124// function is guaranteed to return its best guess of an answer, by 125// considering the samples and the end points as possible solutions. 126void MinimizeInterpolatingPolynomial(const vector<FunctionSample>& samples, 127 double x_min, 128 double x_max, 129 double* optimal_x, 130 double* optimal_value); 131 132} // namespace internal 133} // namespace ceres 134 135#endif // CERES_INTERNAL_POLYNOMIAL_SOLVER_H_ 136