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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2010 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
// The computeRoots function included in this is based on materials
// covered by the following copyright and license:
//
// Geometric Tools, LLC
// Copyright (c) 1998-2010
// Distributed under the Boost Software License, Version 1.0.
//
// Permission is hereby granted, free of charge, to any person or organization
// obtaining a copy of the software and accompanying documentation covered by
// this license (the "Software") to use, reproduce, display, distribute,
// execute, and transmit the Software, and to prepare derivative works of the
// Software, and to permit third-parties to whom the Software is furnished to
// do so, all subject to the following:
//
// The copyright notices in the Software and this entire statement, including
// the above license grant, this restriction and the following disclaimer,
// must be included in all copies of the Software, in whole or in part, and
// all derivative works of the Software, unless such copies or derivative
// works are solely in the form of machine-executable object code generated by
// a source language processor.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
// SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
// FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE,
// ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
// DEALINGS IN THE SOFTWARE.
#include <iostream>
#include <Eigen/Core>
#include <Eigen/Eigenvalues>
#include <Eigen/Geometry>
#include <bench/BenchTimer.h>
using namespace Eigen;
using namespace std;
template<typename Matrix, typename Roots>
inline void computeRoots(const Matrix& m, Roots& roots)
{
typedef typename Matrix::Scalar Scalar;
const Scalar s_inv3 = 1.0/3.0;
const Scalar s_sqrt3 = internal::sqrt(Scalar(3.0));
// The characteristic equation is x^3 - c2*x^2 + c1*x - c0 = 0. The
// eigenvalues are the roots to this equation, all guaranteed to be
// real-valued, because the matrix is symmetric.
Scalar c0 = m(0,0)*m(1,1)*m(2,2) + Scalar(2)*m(0,1)*m(0,2)*m(1,2) - m(0,0)*m(1,2)*m(1,2) - m(1,1)*m(0,2)*m(0,2) - m(2,2)*m(0,1)*m(0,1);
Scalar c1 = m(0,0)*m(1,1) - m(0,1)*m(0,1) + m(0,0)*m(2,2) - m(0,2)*m(0,2) + m(1,1)*m(2,2) - m(1,2)*m(1,2);
Scalar c2 = m(0,0) + m(1,1) + m(2,2);
// Construct the parameters used in classifying the roots of the equation
// and in solving the equation for the roots in closed form.
Scalar c2_over_3 = c2*s_inv3;
Scalar a_over_3 = (c1 - c2*c2_over_3)*s_inv3;
if (a_over_3 > Scalar(0))
a_over_3 = Scalar(0);
Scalar half_b = Scalar(0.5)*(c0 + c2_over_3*(Scalar(2)*c2_over_3*c2_over_3 - c1));
Scalar q = half_b*half_b + a_over_3*a_over_3*a_over_3;
if (q > Scalar(0))
q = Scalar(0);
// Compute the eigenvalues by solving for the roots of the polynomial.
Scalar rho = internal::sqrt(-a_over_3);
Scalar theta = std::atan2(internal::sqrt(-q),half_b)*s_inv3;
Scalar cos_theta = internal::cos(theta);
Scalar sin_theta = internal::sin(theta);
roots(0) = c2_over_3 + Scalar(2)*rho*cos_theta;
roots(1) = c2_over_3 - rho*(cos_theta + s_sqrt3*sin_theta);
roots(2) = c2_over_3 - rho*(cos_theta - s_sqrt3*sin_theta);
// Sort in increasing order.
if (roots(0) >= roots(1))
std::swap(roots(0),roots(1));
if (roots(1) >= roots(2))
{
std::swap(roots(1),roots(2));
if (roots(0) >= roots(1))
std::swap(roots(0),roots(1));
}
}
template<typename Matrix, typename Vector>
void eigen33(const Matrix& mat, Matrix& evecs, Vector& evals)
{
typedef typename Matrix::Scalar Scalar;
// Scale the matrix so its entries are in [-1,1]. The scaling is applied
// only when at least one matrix entry has magnitude larger than 1.
Scalar scale = mat.cwiseAbs()/*.template triangularView<Lower>()*/.maxCoeff();
scale = std::max(scale,Scalar(1));
Matrix scaledMat = mat / scale;
// Compute the eigenvalues
// scaledMat.setZero();
computeRoots(scaledMat,evals);
// compute the eigen vectors
// **here we assume 3 differents eigenvalues**
// "optimized version" which appears to be slower with gcc!
// Vector base;
// Scalar alpha, beta;
// base << scaledMat(1,0) * scaledMat(2,1),
// scaledMat(1,0) * scaledMat(2,0),
// -scaledMat(1,0) * scaledMat(1,0);
// for(int k=0; k<2; ++k)
// {
// alpha = scaledMat(0,0) - evals(k);
// beta = scaledMat(1,1) - evals(k);
// evecs.col(k) = (base + Vector(-beta*scaledMat(2,0), -alpha*scaledMat(2,1), alpha*beta)).normalized();
// }
// evecs.col(2) = evecs.col(0).cross(evecs.col(1)).normalized();
// // naive version
// Matrix tmp;
// tmp = scaledMat;
// tmp.diagonal().array() -= evals(0);
// evecs.col(0) = tmp.row(0).cross(tmp.row(1)).normalized();
//
// tmp = scaledMat;
// tmp.diagonal().array() -= evals(1);
// evecs.col(1) = tmp.row(0).cross(tmp.row(1)).normalized();
//
// tmp = scaledMat;
// tmp.diagonal().array() -= evals(2);
// evecs.col(2) = tmp.row(0).cross(tmp.row(1)).normalized();
// a more stable version:
if((evals(2)-evals(0))<=Eigen::NumTraits<Scalar>::epsilon())
{
evecs.setIdentity();
}
else
{
Matrix tmp;
tmp = scaledMat;
tmp.diagonal ().array () -= evals (2);
evecs.col (2) = tmp.row (0).cross (tmp.row (1)).normalized ();
tmp = scaledMat;
tmp.diagonal ().array () -= evals (1);
evecs.col(1) = tmp.row (0).cross(tmp.row (1));
Scalar n1 = evecs.col(1).norm();
if(n1<=Eigen::NumTraits<Scalar>::epsilon())
evecs.col(1) = evecs.col(2).unitOrthogonal();
else
evecs.col(1) /= n1;
// make sure that evecs[1] is orthogonal to evecs[2]
evecs.col(1) = evecs.col(2).cross(evecs.col(1).cross(evecs.col(2))).normalized();
evecs.col(0) = evecs.col(2).cross(evecs.col(1));
}
// Rescale back to the original size.
evals *= scale;
}
int main()
{
BenchTimer t;
int tries = 10;
int rep = 400000;
typedef Matrix3f Mat;
typedef Vector3f Vec;
Mat A = Mat::Random(3,3);
A = A.adjoint() * A;
SelfAdjointEigenSolver<Mat> eig(A);
BENCH(t, tries, rep, eig.compute(A));
std::cout << "Eigen: " << t.best() << "s\n";
Mat evecs;
Vec evals;
BENCH(t, tries, rep, eigen33(A,evecs,evals));
std::cout << "Direct: " << t.best() << "s\n\n";
std::cerr << "Eigenvalue/eigenvector diffs:\n";
std::cerr << (evals - eig.eigenvalues()).transpose() << "\n";
for(int k=0;k<3;++k)
if(evecs.col(k).dot(eig.eigenvectors().col(k))<0)
evecs.col(k) = -evecs.col(k);
std::cerr << evecs - eig.eigenvectors() << "\n\n";
}
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