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196 lines
7.1 KiB
C++
196 lines
7.1 KiB
C++
// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2010 Gael Guennebaud <gael.guennebaud@inria.fr>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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// The computeRoots function included in this is based on materials
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// covered by the following copyright and license:
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//
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// Geometric Tools, LLC
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// Copyright (c) 1998-2010
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// Distributed under the Boost Software License, Version 1.0.
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//
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// Permission is hereby granted, free of charge, to any person or organization
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// obtaining a copy of the software and accompanying documentation covered by
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// this license (the "Software") to use, reproduce, display, distribute,
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// execute, and transmit the Software, and to prepare derivative works of the
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// Software, and to permit third-parties to whom the Software is furnished to
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// do so, all subject to the following:
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//
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// The copyright notices in the Software and this entire statement, including
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// the above license grant, this restriction and the following disclaimer,
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// must be included in all copies of the Software, in whole or in part, and
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// all derivative works of the Software, unless such copies or derivative
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// works are solely in the form of machine-executable object code generated by
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// a source language processor.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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// FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
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// SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
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// FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE,
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// ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
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// DEALINGS IN THE SOFTWARE.
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#include <iostream>
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#include <Eigen/Core>
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#include <Eigen/Eigenvalues>
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#include <Eigen/Geometry>
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#include <bench/BenchTimer.h>
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using namespace Eigen;
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using namespace std;
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template<typename Matrix, typename Roots>
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inline void computeRoots(const Matrix& m, Roots& roots)
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{
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typedef typename Matrix::Scalar Scalar;
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const Scalar s_inv3 = 1.0/3.0;
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const Scalar s_sqrt3 = std::sqrt(Scalar(3.0));
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// The characteristic equation is x^3 - c2*x^2 + c1*x - c0 = 0. The
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// eigenvalues are the roots to this equation, all guaranteed to be
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// real-valued, because the matrix is symmetric.
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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);
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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);
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Scalar c2 = m(0,0) + m(1,1) + m(2,2);
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// Construct the parameters used in classifying the roots of the equation
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// and in solving the equation for the roots in closed form.
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Scalar c2_over_3 = c2*s_inv3;
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Scalar a_over_3 = (c1 - c2*c2_over_3)*s_inv3;
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if (a_over_3 > Scalar(0))
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a_over_3 = Scalar(0);
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Scalar half_b = Scalar(0.5)*(c0 + c2_over_3*(Scalar(2)*c2_over_3*c2_over_3 - c1));
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Scalar q = half_b*half_b + a_over_3*a_over_3*a_over_3;
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if (q > Scalar(0))
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q = Scalar(0);
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// Compute the eigenvalues by solving for the roots of the polynomial.
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Scalar rho = std::sqrt(-a_over_3);
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Scalar theta = std::atan2(std::sqrt(-q),half_b)*s_inv3;
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Scalar cos_theta = std::cos(theta);
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Scalar sin_theta = std::sin(theta);
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roots(2) = c2_over_3 + Scalar(2)*rho*cos_theta;
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roots(0) = c2_over_3 - rho*(cos_theta + s_sqrt3*sin_theta);
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roots(1) = c2_over_3 - rho*(cos_theta - s_sqrt3*sin_theta);
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}
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template<typename Matrix, typename Vector>
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void eigen33(const Matrix& mat, Matrix& evecs, Vector& evals)
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{
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typedef typename Matrix::Scalar Scalar;
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// Scale the matrix so its entries are in [-1,1]. The scaling is applied
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// only when at least one matrix entry has magnitude larger than 1.
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Scalar shift = mat.trace()/3;
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Matrix scaledMat = mat;
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scaledMat.diagonal().array() -= shift;
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Scalar scale = scaledMat.cwiseAbs()/*.template triangularView<Lower>()*/.maxCoeff();
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scale = std::max(scale,Scalar(1));
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scaledMat/=scale;
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// Compute the eigenvalues
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// scaledMat.setZero();
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computeRoots(scaledMat,evals);
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// compute the eigen vectors
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// **here we assume 3 differents eigenvalues**
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// "optimized version" which appears to be slower with gcc!
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// Vector base;
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// Scalar alpha, beta;
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// base << scaledMat(1,0) * scaledMat(2,1),
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// scaledMat(1,0) * scaledMat(2,0),
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// -scaledMat(1,0) * scaledMat(1,0);
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// for(int k=0; k<2; ++k)
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// {
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// alpha = scaledMat(0,0) - evals(k);
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// beta = scaledMat(1,1) - evals(k);
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// evecs.col(k) = (base + Vector(-beta*scaledMat(2,0), -alpha*scaledMat(2,1), alpha*beta)).normalized();
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// }
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// evecs.col(2) = evecs.col(0).cross(evecs.col(1)).normalized();
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// // naive version
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// Matrix tmp;
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// tmp = scaledMat;
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// tmp.diagonal().array() -= evals(0);
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// evecs.col(0) = tmp.row(0).cross(tmp.row(1)).normalized();
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//
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// tmp = scaledMat;
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// tmp.diagonal().array() -= evals(1);
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// evecs.col(1) = tmp.row(0).cross(tmp.row(1)).normalized();
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//
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// tmp = scaledMat;
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// tmp.diagonal().array() -= evals(2);
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// evecs.col(2) = tmp.row(0).cross(tmp.row(1)).normalized();
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// a more stable version:
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if((evals(2)-evals(0))<=Eigen::NumTraits<Scalar>::epsilon())
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{
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evecs.setIdentity();
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}
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else
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{
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Matrix tmp;
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tmp = scaledMat;
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tmp.diagonal ().array () -= evals (2);
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evecs.col (2) = tmp.row (0).cross (tmp.row (1)).normalized ();
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tmp = scaledMat;
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tmp.diagonal ().array () -= evals (1);
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evecs.col(1) = tmp.row (0).cross(tmp.row (1));
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Scalar n1 = evecs.col(1).norm();
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if(n1<=Eigen::NumTraits<Scalar>::epsilon())
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evecs.col(1) = evecs.col(2).unitOrthogonal();
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else
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evecs.col(1) /= n1;
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// make sure that evecs[1] is orthogonal to evecs[2]
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evecs.col(1) = evecs.col(2).cross(evecs.col(1).cross(evecs.col(2))).normalized();
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evecs.col(0) = evecs.col(2).cross(evecs.col(1));
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}
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// Rescale back to the original size.
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evals *= scale;
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evals.array()+=shift;
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}
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int main()
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{
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BenchTimer t;
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int tries = 10;
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int rep = 400000;
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typedef Matrix3d Mat;
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typedef Vector3d Vec;
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Mat A = Mat::Random(3,3);
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A = A.adjoint() * A;
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// Mat Q = A.householderQr().householderQ();
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// A = Q * Vec(2.2424567,2.2424566,7.454353).asDiagonal() * Q.transpose();
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SelfAdjointEigenSolver<Mat> eig(A);
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BENCH(t, tries, rep, eig.compute(A));
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std::cout << "Eigen iterative: " << t.best() << "s\n";
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BENCH(t, tries, rep, eig.computeDirect(A));
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std::cout << "Eigen direct : " << t.best() << "s\n";
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Mat evecs;
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Vec evals;
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BENCH(t, tries, rep, eigen33(A,evecs,evals));
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std::cout << "Direct: " << t.best() << "s\n\n";
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// std::cerr << "Eigenvalue/eigenvector diffs:\n";
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// std::cerr << (evals - eig.eigenvalues()).transpose() << "\n";
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// for(int k=0;k<3;++k)
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// if(evecs.col(k).dot(eig.eigenvectors().col(k))<0)
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// evecs.col(k) = -evecs.col(k);
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// std::cerr << evecs - eig.eigenvectors() << "\n\n";
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}
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