1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2010 Gael Guennebaud <gael.guennebaud@inria.fr>
5 //
6 // This Source Code Form is subject to the terms of the Mozilla
7 // Public License v. 2.0. If a copy of the MPL was not distributed
8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9
10 // The computeRoots function included in this is based on materials
11 // covered by the following copyright and license:
12 //
13 // Geometric Tools, LLC
14 // Copyright (c) 1998-2010
15 // Distributed under the Boost Software License, Version 1.0.
16 //
17 // Permission is hereby granted, free of charge, to any person or organization
18 // obtaining a copy of the software and accompanying documentation covered by
19 // this license (the "Software") to use, reproduce, display, distribute,
20 // execute, and transmit the Software, and to prepare derivative works of the
21 // Software, and to permit third-parties to whom the Software is furnished to
22 // do so, all subject to the following:
23 //
24 // The copyright notices in the Software and this entire statement, including
25 // the above license grant, this restriction and the following disclaimer,
26 // must be included in all copies of the Software, in whole or in part, and
27 // all derivative works of the Software, unless such copies or derivative
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30 //
31 // THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
32 // IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
33 // FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
34 // SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
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37 // DEALINGS IN THE SOFTWARE.
38
39 #include <iostream>
40 #include <Eigen/Core>
41 #include <Eigen/Eigenvalues>
42 #include <Eigen/Geometry>
43 #include <bench/BenchTimer.h>
44
45 using namespace Eigen;
46 using namespace std;
47
48 template<typename Matrix, typename Roots>
computeRoots(const Matrix & m,Roots & roots)49 inline void computeRoots(const Matrix& m, Roots& roots)
50 {
51 typedef typename Matrix::Scalar Scalar;
52 const Scalar s_inv3 = 1.0/3.0;
53 const Scalar s_sqrt3 = internal::sqrt(Scalar(3.0));
54
55 // The characteristic equation is x^3 - c2*x^2 + c1*x - c0 = 0. The
56 // eigenvalues are the roots to this equation, all guaranteed to be
57 // real-valued, because the matrix is symmetric.
58 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);
59 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);
60 Scalar c2 = m(0,0) + m(1,1) + m(2,2);
61
62 // Construct the parameters used in classifying the roots of the equation
63 // and in solving the equation for the roots in closed form.
64 Scalar c2_over_3 = c2*s_inv3;
65 Scalar a_over_3 = (c1 - c2*c2_over_3)*s_inv3;
66 if (a_over_3 > Scalar(0))
67 a_over_3 = Scalar(0);
68
69 Scalar half_b = Scalar(0.5)*(c0 + c2_over_3*(Scalar(2)*c2_over_3*c2_over_3 - c1));
70
71 Scalar q = half_b*half_b + a_over_3*a_over_3*a_over_3;
72 if (q > Scalar(0))
73 q = Scalar(0);
74
75 // Compute the eigenvalues by solving for the roots of the polynomial.
76 Scalar rho = internal::sqrt(-a_over_3);
77 Scalar theta = std::atan2(internal::sqrt(-q),half_b)*s_inv3;
78 Scalar cos_theta = internal::cos(theta);
79 Scalar sin_theta = internal::sin(theta);
80 roots(0) = c2_over_3 + Scalar(2)*rho*cos_theta;
81 roots(1) = c2_over_3 - rho*(cos_theta + s_sqrt3*sin_theta);
82 roots(2) = c2_over_3 - rho*(cos_theta - s_sqrt3*sin_theta);
83
84 // Sort in increasing order.
85 if (roots(0) >= roots(1))
86 std::swap(roots(0),roots(1));
87 if (roots(1) >= roots(2))
88 {
89 std::swap(roots(1),roots(2));
90 if (roots(0) >= roots(1))
91 std::swap(roots(0),roots(1));
92 }
93 }
94
95 template<typename Matrix, typename Vector>
eigen33(const Matrix & mat,Matrix & evecs,Vector & evals)96 void eigen33(const Matrix& mat, Matrix& evecs, Vector& evals)
97 {
98 typedef typename Matrix::Scalar Scalar;
99 // Scale the matrix so its entries are in [-1,1]. The scaling is applied
100 // only when at least one matrix entry has magnitude larger than 1.
101
102 Scalar scale = mat.cwiseAbs()/*.template triangularView<Lower>()*/.maxCoeff();
103 scale = std::max(scale,Scalar(1));
104 Matrix scaledMat = mat / scale;
105
106 // Compute the eigenvalues
107 // scaledMat.setZero();
108 computeRoots(scaledMat,evals);
109
110 // compute the eigen vectors
111 // **here we assume 3 differents eigenvalues**
112
113 // "optimized version" which appears to be slower with gcc!
114 // Vector base;
115 // Scalar alpha, beta;
116 // base << scaledMat(1,0) * scaledMat(2,1),
117 // scaledMat(1,0) * scaledMat(2,0),
118 // -scaledMat(1,0) * scaledMat(1,0);
119 // for(int k=0; k<2; ++k)
120 // {
121 // alpha = scaledMat(0,0) - evals(k);
122 // beta = scaledMat(1,1) - evals(k);
123 // evecs.col(k) = (base + Vector(-beta*scaledMat(2,0), -alpha*scaledMat(2,1), alpha*beta)).normalized();
124 // }
125 // evecs.col(2) = evecs.col(0).cross(evecs.col(1)).normalized();
126
127 // // naive version
128 // Matrix tmp;
129 // tmp = scaledMat;
130 // tmp.diagonal().array() -= evals(0);
131 // evecs.col(0) = tmp.row(0).cross(tmp.row(1)).normalized();
132 //
133 // tmp = scaledMat;
134 // tmp.diagonal().array() -= evals(1);
135 // evecs.col(1) = tmp.row(0).cross(tmp.row(1)).normalized();
136 //
137 // tmp = scaledMat;
138 // tmp.diagonal().array() -= evals(2);
139 // evecs.col(2) = tmp.row(0).cross(tmp.row(1)).normalized();
140
141 // a more stable version:
142 if((evals(2)-evals(0))<=Eigen::NumTraits<Scalar>::epsilon())
143 {
144 evecs.setIdentity();
145 }
146 else
147 {
148 Matrix tmp;
149 tmp = scaledMat;
150 tmp.diagonal ().array () -= evals (2);
151 evecs.col (2) = tmp.row (0).cross (tmp.row (1)).normalized ();
152
153 tmp = scaledMat;
154 tmp.diagonal ().array () -= evals (1);
155 evecs.col(1) = tmp.row (0).cross(tmp.row (1));
156 Scalar n1 = evecs.col(1).norm();
157 if(n1<=Eigen::NumTraits<Scalar>::epsilon())
158 evecs.col(1) = evecs.col(2).unitOrthogonal();
159 else
160 evecs.col(1) /= n1;
161
162 // make sure that evecs[1] is orthogonal to evecs[2]
163 evecs.col(1) = evecs.col(2).cross(evecs.col(1).cross(evecs.col(2))).normalized();
164 evecs.col(0) = evecs.col(2).cross(evecs.col(1));
165 }
166
167 // Rescale back to the original size.
168 evals *= scale;
169 }
170
main()171 int main()
172 {
173 BenchTimer t;
174 int tries = 10;
175 int rep = 400000;
176 typedef Matrix3f Mat;
177 typedef Vector3f Vec;
178 Mat A = Mat::Random(3,3);
179 A = A.adjoint() * A;
180
181 SelfAdjointEigenSolver<Mat> eig(A);
182 BENCH(t, tries, rep, eig.compute(A));
183 std::cout << "Eigen: " << t.best() << "s\n";
184
185 Mat evecs;
186 Vec evals;
187 BENCH(t, tries, rep, eigen33(A,evecs,evals));
188 std::cout << "Direct: " << t.best() << "s\n\n";
189
190 std::cerr << "Eigenvalue/eigenvector diffs:\n";
191 std::cerr << (evals - eig.eigenvalues()).transpose() << "\n";
192 for(int k=0;k<3;++k)
193 if(evecs.col(k).dot(eig.eigenvectors().col(k))<0)
194 evecs.col(k) = -evecs.col(k);
195 std::cerr << evecs - eig.eigenvectors() << "\n\n";
196 }
197