From ebea530782a0c628aea4789e0e16eed353b62030 Mon Sep 17 00:00:00 2001 From: Christoph Hertzberg Date: Sun, 17 May 2015 21:54:32 +0200 Subject: [PATCH] bug #1014: More stable direct computation of eigenvalues and -vectors for 3x3 matrices --- .../src/Eigenvalues/SelfAdjointEigenSolver.h | 181 ++++++++---------- 1 file changed, 75 insertions(+), 106 deletions(-) diff --git a/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h b/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h index fff331d33..1830b99c1 100644 --- a/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h +++ b/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h @@ -540,6 +540,11 @@ template struct direct_selfadjoint_eigenvalues struct direct_selfadjoint_eigenvalues Scalar(0)) - a_over_3 = Scalar(0); + Scalar a_over_3 = (c2*c2_over_3 - c1)*s_inv3; + a_over_3 = numext::maxi(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); + Scalar q = a_over_3*a_over_3*a_over_3 - half_b*half_b; + q = numext::maxi(q, Scalar(0)); // Compute the eigenvalues by solving for the roots of the polynomial. - Scalar rho = sqrt(-a_over_3); - Scalar theta = atan2(sqrt(-q),half_b)*s_inv3; + Scalar rho = sqrt(a_over_3); + Scalar theta = atan2(sqrt(q),half_b)*s_inv3; // since sqrt(q) > 0, atan2 is in [0, pi] and theta is in [0, pi/3] Scalar cos_theta = cos(theta); Scalar sin_theta = 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)) - numext::swap(roots(0),roots(1)); - if (roots(1) >= roots(2)) - { - numext::swap(roots(1),roots(2)); - if (roots(0) >= roots(1)) - numext::swap(roots(0),roots(1)); - } + // roots are already sorted, since cos is monotonically decreasing on [0, pi] + roots(0) = c2_over_3 - rho*(cos_theta + s_sqrt3*sin_theta); // == 2*rho*cos(theta+2pi/3) + roots(1) = c2_over_3 - rho*(cos_theta - s_sqrt3*sin_theta); // == 2*rho*cos(theta+ pi/3) + roots(2) = c2_over_3 + Scalar(2)*rho*cos_theta; } - + + EIGEN_DEVICE_FUNC + static inline bool extract_kernel(MatrixType& mat, Ref res, Ref representative) + { + using std::abs; + Index i0; + // Find non-zero column i0 (by construction, there must exist a non zero coefficient on the diagonal): + mat.diagonal().cwiseAbs().maxCoeff(&i0); + // mat.col(i0) is a good candidate for an orthogonal vector to the current eigenvector, + // so let's save it: + representative = mat.col(i0); + Scalar n0, n1; + VectorType c0, c1; + n0 = (c0 = representative.cross(mat.col((i0+1)%3))).squaredNorm(); + n1 = (c1 = representative.cross(mat.col((i0+2)%3))).squaredNorm(); + if(n0>n1) res = c0/std::sqrt(n0); + else res = c1/std::sqrt(n1); + + return true; + } + EIGEN_DEVICE_FUNC static inline void run(SolverType& solver, const MatrixType& mat, int options) { - using std::sqrt; eigen_assert(mat.cols() == 3 && mat.cols() == mat.rows()); eigen_assert((options&~(EigVecMask|GenEigMask))==0 && (options&EigVecMask)!=EigVecMask @@ -603,116 +616,72 @@ template struct direct_selfadjoint_eigenvalues(); + scaledMat.diagonal().array() -= shift; + Scalar scale = scaledMat.cwiseAbs().maxCoeff(); + if(scale > 0) scaledMat /= scale; // TODO for scale==0 we could save the remaining operations // compute the eigenvalues computeRoots(scaledMat,eivals); - // compute the eigen vectors + // compute the eigenvectors if(computeEigenvectors) { - Scalar safeNorm2 = Eigen::NumTraits::epsilon(); if((eivals(2)-eivals(0))<=Eigen::NumTraits::epsilon()) { + // All three eigenvalues are numerically the same eivecs.setIdentity(); } else { - scaledMat = scaledMat.template selfadjointView(); MatrixType tmp; tmp = scaledMat; + // Compute the eigenvector of the most distinct eigenvalue Scalar d0 = eivals(2) - eivals(1); Scalar d1 = eivals(1) - eivals(0); - int k = d0 > d1 ? 2 : 0; - d0 = d0 > d1 ? d0 : d1; - - tmp.diagonal().array () -= eivals(k); - VectorType cross; - Scalar n; - n = (cross = tmp.row(0).cross(tmp.row(1))).squaredNorm(); - - if(n>safeNorm2) + Index k(0), l(2); + if(d0 > d1) { - eivecs.col(k) = cross / sqrt(n); + std::swap(k,l); + d0 = d1; + } + + // Compute the eigenvector of index k + { + tmp.diagonal().array () -= eivals(k); + // By construction, 'tmp' is of rank 2, and its kernel corresponds to the respective eigenvector. + extract_kernel(tmp, eivecs.col(k), eivecs.col(l)); + } + + // Compute eigenvector of index l + if(d0<=2*Eigen::NumTraits::epsilon()*d1) + { + // If d0 is too small, then the two other eigenvalues are numerically the same, + // and thus we only have to ortho-normalize the near orthogonal vector we saved above. + eivecs.col(l) -= eivecs.col(k).dot(eivecs.col(l))*eivecs.col(l); + eivecs.col(l).normalize(); } else { - n = (cross = tmp.row(0).cross(tmp.row(2))).squaredNorm(); + tmp = scaledMat; + tmp.diagonal().array () -= eivals(l); - if(n>safeNorm2) - { - eivecs.col(k) = cross / sqrt(n); - } - else - { - n = (cross = tmp.row(1).cross(tmp.row(2))).squaredNorm(); - - if(n>safeNorm2) - { - eivecs.col(k) = cross / sqrt(n); - } - else - { - // the input matrix and/or the eigenvaues probably contains some inf/NaN, - // => exit - // scale back to the original size. - eivals *= scale; - - solver.m_info = NumericalIssue; - solver.m_isInitialized = true; - solver.m_eigenvectorsOk = computeEigenvectors; - return; - } - } + VectorType dummy; + extract_kernel(tmp, eivecs.col(l), dummy); } - tmp = scaledMat; - tmp.diagonal().array() -= eivals(1); - - if(d0<=Eigen::NumTraits::epsilon()) - { - eivecs.col(1) = eivecs.col(k).unitOrthogonal(); - } - else - { - n = (cross = eivecs.col(k).cross(tmp.row(0))).squaredNorm(); - if(n>safeNorm2) - { - eivecs.col(1) = cross / sqrt(n); - } - else - { - n = (cross = eivecs.col(k).cross(tmp.row(1))).squaredNorm(); - if(n>safeNorm2) - eivecs.col(1) = cross / sqrt(n); - else - { - n = (cross = eivecs.col(k).cross(tmp.row(2))).squaredNorm(); - if(n>safeNorm2) - eivecs.col(1) = cross / sqrt(n); - else - { - // we should never reach this point, - // if so the last two eigenvalues are likely to be very close to each other - eivecs.col(1) = eivecs.col(k).unitOrthogonal(); - } - } - } - - // make sure that eivecs[1] is orthogonal to eivecs[2] - // FIXME: this step should not be needed - Scalar d = eivecs.col(1).dot(eivecs.col(k)); - eivecs.col(1) = (eivecs.col(1) - d * eivecs.col(k)).normalized(); - } - - eivecs.col(k==2 ? 0 : 2) = eivecs.col(k).cross(eivecs.col(1)).normalized(); + // Compute last eigenvector from the other two + eivecs.col(1) = eivecs.col(2).cross(eivecs.col(0)).normalized(); } } + // Rescale back to the original size. eivals *= scale; + eivals.array() += shift; solver.m_info = Success; solver.m_isInitialized = true; @@ -732,7 +701,7 @@ struct direct_selfadjoint_eigenvalues static inline void computeRoots(const MatrixType& m, VectorType& roots) { using std::sqrt; - const Scalar t0 = Scalar(0.5) * sqrt( numext::abs2(m(0,0)-m(1,1)) + Scalar(4)*m(1,0)*m(1,0)); + const Scalar t0 = Scalar(0.5) * sqrt( numext::abs2(m(0,0)-m(1,1)) + Scalar(4)*numext::abs2(m(1,0))); const Scalar t1 = Scalar(0.5) * (m(0,0) + m(1,1)); roots(0) = t1 - t0; roots(1) = t1 + t0;