add a small bench demoing the possibilities of a direct 3x3 eigen decomposition

bench/eig33.cpp

@@ -0,0 +1,139 @@
+#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& rkA, Roots& adRoot)
+{
+  typedef typename Matrix::Scalar Scalar;
+  const Scalar msInv3 = 1.0/3.0;
+  const Scalar msRoot3 = ei_sqrt(Scalar(3.0));
+
+  Scalar dA00 = rkA(0,0);
+  Scalar dA01 = rkA(0,1);
+  Scalar dA02 = rkA(0,2);
+  Scalar dA11 = rkA(1,1);
+  Scalar dA12 = rkA(1,2);
+  Scalar dA22 = rkA(2,2);
+
+  // 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 dC0 = dA00*dA11*dA22 + Scalar(2)*dA01*dA02*dA12 - dA00*dA12*dA12 - dA11*dA02*dA02 - dA22*dA01*dA01;
+  Scalar dC1 = dA00*dA11 - dA01*dA01 + dA00*dA22 - dA02*dA02 + dA11*dA22 - dA12*dA12;
+  Scalar dC2 = dA00 + dA11 + dA22;
+
+  // Construct the parameters used in classifying the roots of the equation
+  // and in solving the equation for the roots in closed form.
+  Scalar dC2Div3 = dC2*msInv3;
+  Scalar dADiv3 = (dC1 - dC2*dC2Div3)*msInv3;
+  if (dADiv3 > Scalar(0))
+    dADiv3 = Scalar(0);
+
+  Scalar dMBDiv2 = Scalar(0.5)*(dC0 + dC2Div3*(Scalar(2)*dC2Div3*dC2Div3 - dC1));
+
+  Scalar dQ = dMBDiv2*dMBDiv2 + dADiv3*dADiv3*dADiv3;
+  if (dQ > Scalar(0))
+    dQ = Scalar(0);
+
+  // Compute the eigenvalues by solving for the roots of the polynomial.
+  Scalar dMagnitude = ei_sqrt(-dADiv3);
+  Scalar dAngle = std::atan2(ei_sqrt(-dQ),dMBDiv2)*msInv3;
+  Scalar dCos = ei_cos(dAngle);
+  Scalar dSin = ei_sin(dAngle);
+  adRoot(0) = dC2Div3 + 2.f*dMagnitude*dCos;
+  adRoot(1) = dC2Div3 - dMagnitude*(dCos + msRoot3*dSin);
+  adRoot(2) = dC2Div3 - dMagnitude*(dCos - msRoot3*dSin);
+
+  // Sort in increasing order.
+  if (adRoot(0) >= adRoot(1))
+    std::swap(adRoot(0),adRoot(1));
+  if (adRoot(1) >= adRoot(2))
+  {
+    std::swap(adRoot(1),adRoot(2));
+    if (adRoot(0) >= adRoot(1))
+      std::swap(adRoot(0),adRoot(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();
+    
+    // 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";
+}