diff --git a/bench/eig33.cpp b/bench/eig33.cpp
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+++ b/bench/eig33.cpp
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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";
+}