From 2bd59b0e0d667dcdcb6b070596a1bf023e3e88f1 Mon Sep 17 00:00:00 2001 From: Gael Guennebaud Date: Thu, 9 Jun 2016 17:12:03 +0200 Subject: [PATCH] Take advantage that T is already diagonal in the extraction of generalized complex eigenvalues. --- .../src/Eigenvalues/GeneralizedEigenSolver.h | 25 ++++++------------- 1 file changed, 7 insertions(+), 18 deletions(-) diff --git a/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h b/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h index 08caed281..9f43fd544 100644 --- a/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h +++ b/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h @@ -327,24 +327,13 @@ GeneralizedEigenSolver::compute(const MatrixType& A, const MatrixTyp } else { - // We need to extract the generalized eigenvalues of the pair of a general 2x2 block S and a triangular 2x2 block T - // From the eigen decomposition of T = U * E * U^-1, - // we can extract the eigenvalues of (U^-1 * S * U) / E - // Here, we can take advantage that E = diag(T), and U = [ 1 T_01 ; 0 T_11-T_00], and U^-1 = [1 -T_11/(T_11-T_00) ; 0 1/(T_11-T_00)]. - // Then taking beta=T_00*T_11*(T_11-T_00), we can avoid any division, and alpha is the eigenvalues of A = (U^-1 * S * U) * diag(T_11,T_00) * (T_11-T_00): + // We need to extract the generalized eigenvalues of the pair of a general 2x2 block S and a positive diagonal 2x2 block T + // Then taking beta=T_00*T_11, we can avoid any division, and alpha is the eigenvalues of A = (U^-1 * S * U) * diag(T_11,T_00): - // T = [a b ; 0 c] - // S = [e f ; g h] - RealScalar a = m_realQZ.matrixT().coeff(i, i), b = m_realQZ.matrixT().coeff(i, i+1), c = m_realQZ.matrixT().coeff(i+1, i+1); - RealScalar e = m_matS.coeff(i, i), f = m_matS.coeff(i, i+1), g = m_matS.coeff(i+1, i), h = m_matS.coeff(i+1, i+1); - RealScalar d = c-a; - RealScalar gb = g*b; - Matrix S2; - S2 << (e*d-gb)*c, ((e*b+f*d-h*b)*d-gb*b)*a, - g*c , (gb+h*d)*a; - - // NOTE, we could also compute the SVD of T's block during the QZ factorization so that the respective T block is guaranteed to be diagonal, - // and then we could directly apply the formula below (while taking care of scaling S columns by T11,T00): + // T = [a 0] + // [0 b] + RealScalar a = m_realQZ.matrixT().coeff(i, i), b = m_realQZ.matrixT().coeff(i+1, i+1); + Matrix S2 = m_matS.template block<2,2>(i,i) * Matrix(b,a).asDiagonal(); Scalar p = Scalar(0.5) * (S2.coeff(0,0) - S2.coeff(1,1)); Scalar z = sqrt(abs(p * p + S2.coeff(1,0) * S2.coeff(0,1))); @@ -352,7 +341,7 @@ GeneralizedEigenSolver::compute(const MatrixType& A, const MatrixTyp m_alphas.coeffRef(i+1) = ComplexScalar(S2.coeff(1,1) + p, -z); m_betas.coeffRef(i) = - m_betas.coeffRef(i+1) = a*c*d; + m_betas.coeffRef(i+1) = a*b; i += 2; }