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Diffstat (limited to 'internal/ceres/polynomial.h')
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diff --git a/internal/ceres/polynomial.h b/internal/ceres/polynomial.h new file mode 100644 index 0000000..42ffdcb --- /dev/null +++ b/internal/ceres/polynomial.h @@ -0,0 +1,134 @@ +// Ceres Solver - A fast non-linear least squares minimizer +// Copyright 2012 Google Inc. All rights reserved. +// http://code.google.com/p/ceres-solver/ +// +// 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 { + +// 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); + +// Structure for storing sample values of a function. +// +// Clients can use this struct to communicate the value of the +// function and or its gradient at a given point x. +struct FunctionSample { + FunctionSample() + : x(0.0), + value(0.0), + value_is_valid(false), + gradient(0.0), + gradient_is_valid(false) { + } + + double x; + double value; // value = f(x) + bool value_is_valid; + double gradient; // gradient = f'(x) + bool gradient_is_valid; +}; + +// 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 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 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_ |