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// All rights reserved.
//
// 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 ETH Zurich and UNC Chapel Hill 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 HOLDERS 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: Johannes L. Schoenberger (jsch-at-demuc-dot-de)
#ifndef COLMAP_SRC_BASE_POLYNOMIAL_H_
#define COLMAP_SRC_BASE_POLYNOMIAL_H_
#include <Eigen/Core>
namespace colmap {
// 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.
//
// The implementation is based on COLMAP's old polynomial functionality and is
// inspired by Ceres-Solver's/Theia's implementation to support complex
// polynomials. The companion matrix implementation is based on NumPy.
// Evaluate the polynomial for the given coefficients at x using the Horner
// scheme. This function is templated such that the polynomial may be evaluated
// at real and/or imaginary points.
template <typename T>
T EvaluatePolynomial(const Eigen::VectorXd& coeffs, const T& x);
// Find the root of polynomials of the form: a * x + b = 0.
// The real and/or imaginary variable may be NULL if the output is not needed.
bool FindLinearPolynomialRoots(const Eigen::VectorXd& coeffs,
Eigen::VectorXd* real, Eigen::VectorXd* imag);
// Find the roots of polynomials of the form: a * x^2 + b * x + c = 0.
// The real and/or imaginary variable may be NULL if the output is not needed.
bool FindQuadraticPolynomialRoots(const Eigen::VectorXd& coeffs,
Eigen::VectorXd* real, Eigen::VectorXd* imag);
// Find the roots of a polynomial using the Durand-Kerner method, based on:
//
// https://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method
//
// The Durand-Kerner is comparatively fast but often unstable/inaccurate.
// The real and/or imaginary variable may be NULL if the output is not needed.
bool FindPolynomialRootsDurandKerner(const Eigen::VectorXd& coeffs,
Eigen::VectorXd* real,
Eigen::VectorXd* imag);
// Find the roots of a polynomial using the companion matrix method, based on:
//
// R. A. Horn & C. R. Johnson, Matrix Analysis. Cambridge,
// UK: Cambridge University Press, 1999, pp. 146-7.
//
// Compared to Durand-Kerner, this method is slower but more stable/accurate.
// The real and/or imaginary variable may be NULL if the output is not needed.
bool FindPolynomialRootsCompanionMatrix(const Eigen::VectorXd& coeffs,
Eigen::VectorXd* real,
Eigen::VectorXd* imag);
////////////////////////////////////////////////////////////////////////////////
// Implementation
////////////////////////////////////////////////////////////////////////////////
template <typename T>
T EvaluatePolynomial(const Eigen::VectorXd& coeffs, const T& x) {
T value = 0.0;
for (Eigen::VectorXd::Index i = 0; i < coeffs.size(); ++i) {
value = value * x + coeffs(i);
}
return value;
}
} // namespace colmap
#endif // COLMAP_SRC_BASE_POLYNOMIAL_H_
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