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Gael Guennebaud
2016-06-14 15:33:47 +02:00
parent 1004c4df99 4c61f00838
commit 76236cdea4
28 changed files with 376 additions and 156 deletions
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35
Eigen/src/Eigenvalues/GeneralizedEigenSolver.h
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@@ -327,33 +327,22 @@ GeneralizedEigenSolver<MatrixType>::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<RealScalar,2,2> A;
A << (e*d-gb)*c, ((e*b+f*d-h*b)*d-gb*b)*a,
g*c , (gb+h*d)*a;
// T = [a 0]
// [0 b]
RealScalar a = m_realQZ.matrixT().coeff(i, i), b = m_realQZ.matrixT().coeff(i+1, i+1);
Matrix<RealScalar,2,2> S2 = m_matS.template block<2,2>(i,i) * Matrix<Scalar,2,1>(b,a).asDiagonal();
// 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):
Scalar p = Scalar(0.5) * (A.coeff(i, i) - A.coeff(i+1, i+1));
Scalar z = sqrt(abs(p * p + A.coeff(i+1, i) * A.coeff(i, i+1)));
m_alphas.coeffRef(i) = ComplexScalar(A.coeff(i+1, i+1) + p, z);
m_alphas.coeffRef(i+1) = ComplexScalar(A.coeff(i+1, i+1) + p, -z);
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)));
m_alphas.coeffRef(i) = ComplexScalar(S2.coeff(1,1) + p, z);
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;
}
}

32
Eigen/src/Eigenvalues/RealQZ.h
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@@ -552,7 +552,6 @@ namespace Eigen {
m_T.coeffRef(l,l-1) = Scalar(0.0);
}
template<typename MatrixType>
RealQZ<MatrixType>& RealQZ<MatrixType>::compute(const MatrixType& A_in, const MatrixType& B_in, bool computeQZ)
{
@@ -616,6 +615,37 @@ namespace Eigen {
}
// check if we converged before reaching iterations limit
m_info = (local_iter<m_maxIters) ? Success : NoConvergence;
// For each non triangular 2x2 diagonal block of S,
// reduce the respective 2x2 diagonal block of T to positive diagonal form using 2x2 SVD.
// This step is not mandatory for QZ, but it does help further extraction of eigenvalues/eigenvectors,
// and is in par with Lapack/Matlab QZ.
if(m_info==Success)
{
for(Index i=0; i<dim-1; ++i)
{
if(m_S.coeff(i+1, i) != Scalar(0))
{
JacobiRotation<Scalar> j_left, j_right;
internal::real_2x2_jacobi_svd(m_T, i, i+1, &j_left, &j_right);
// Apply resulting Jacobi rotations
m_S.applyOnTheLeft(i,i+1,j_left);
m_S.applyOnTheRight(i,i+1,j_right);
m_T.applyOnTheLeft(i,i+1,j_left);
m_T.applyOnTheRight(i,i+1,j_right);
m_T(i+1,i) = m_T(i,i+1) = Scalar(0);
if(m_computeQZ) {
m_Q.applyOnTheRight(i,i+1,j_left.transpose());
m_Z.applyOnTheLeft(i,i+1,j_right.transpose());
}
i++;
}
}
}
return *this;
} // end compute

4
Eigen/src/Eigenvalues/Tridiagonalization.h
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@@ -367,10 +367,10 @@ void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
hCoeffs.tail(n-i-1).noalias() = (matA.bottomRightCorner(remainingSize,remainingSize).template selfadjointView<Lower>()
* (conj(h) * matA.col(i).tail(remainingSize)));
hCoeffs.tail(n-i-1) += (conj(h)*Scalar(-0.5)*(hCoeffs.tail(remainingSize).dot(matA.col(i).tail(remainingSize)))) * matA.col(i).tail(n-i-1);
hCoeffs.tail(n-i-1) += (conj(h)*RealScalar(-0.5)*(hCoeffs.tail(remainingSize).dot(matA.col(i).tail(remainingSize)))) * matA.col(i).tail(n-i-1);
matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>()
.rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), -1);
.rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), Scalar(-1));
matA.col(i).coeffRef(i+1) = beta;
hCoeffs.coeffRef(i) = h;