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Improve numerical robustness of RealSchur: add scaling and compare sub-diag entries to largest diagonal entry instead of the 2 neighbors.

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This commit is contained in:
Gael Guennebaud 2016-07-06 13:45:30 +02:00
parent d2b5a19e0f
commit 5b3a6f51d3
2 changed files with 42 additions and 16 deletions
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25
Eigen/src/Eigenvalues/RealSchur.h
View File

@ -236,7 +236,7 @@ template<typename _MatrixType> class RealSchur
typedef Matrix<Scalar,3,1> Vector3s;
Scalar computeNormOfT();
Index findSmallSubdiagEntry(Index iu);
Index findSmallSubdiagEntry(Index iu, const Scalar& maxDiagEntry);
void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift);
void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
@ -253,18 +253,24 @@ RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const EigenBase<InputType>
if (maxIters == -1)
maxIters = m_maxIterationsPerRow * matrix.rows();
Scalar scale = matrix.derived().cwiseAbs().maxCoeff();
// Step 1. Reduce to Hessenberg form
m_hess.compute(matrix.derived());
m_hess.compute(matrix.derived()/scale);
// Step 2. Reduce to real Schur form
computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);
m_matT *= scale;
return *this;
}
template<typename MatrixType>
template<typename HessMatrixType, typename OrthMatrixType>
RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU)
{
using std::abs;
m_matT = matrixH;
if(computeU)
m_matU = matrixQ;
@ -287,14 +293,18 @@ RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMa
if(norm!=0)
{
Scalar maxDiagEntry = m_matT.cwiseAbs().diagonal().maxCoeff();
while (iu >= 0)
{
Index il = findSmallSubdiagEntry(iu);
Index il = findSmallSubdiagEntry(iu,maxDiagEntry);
// Check for convergence
if (il == iu) // One root found
{
m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
// keep track of the largest diagonal coefficient
maxDiagEntry = numext::maxi(maxDiagEntry,abs(m_matT.coeffRef(iu,iu)));
if (iu > 0)
m_matT.coeffRef(iu, iu-1) = Scalar(0);
iu--;
@ -303,6 +313,8 @@ RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMa
else if (il == iu-1) // Two roots found
{
splitOffTwoRows(iu, computeU, exshift);
// keep track of the largest diagonal coefficient
maxDiagEntry = numext::maxi(maxDiagEntry,numext::maxi(abs(m_matT.coeff(iu,iu)), abs(m_matT.coeff(iu-1,iu-1))));
iu -= 2;
iter = 0;
}
@ -317,6 +329,8 @@ RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMa
Index im;
initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
// keep track of the largest diagonal coefficient
maxDiagEntry = numext::maxi(maxDiagEntry,m_matT.cwiseAbs().diagonal().segment(im,iu-im).maxCoeff());
}
}
}
@ -346,14 +360,13 @@ inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
/** \internal Look for single small sub-diagonal element and returns its index */
template<typename MatrixType>
inline Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu)
inline Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, const Scalar& maxDiagEntry)
{
using std::abs;
Index res = iu;
while (res > 0)
{
Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));
if (abs(m_matT.coeff(res,res-1)) <= NumTraits<Scalar>::epsilon() * s)
if (abs(m_matT.coeff(res,res-1)) <= NumTraits<Scalar>::epsilon() * maxDiagEntry)
break;
res--;
}

19
test/eigensolver_generic.cpp
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@ -127,16 +127,29 @@ void test_eigensolver_generic()
}
);
// regression test for bug 793
#ifdef EIGEN_TEST_PART_2
{
// regression test for bug 793
MatrixXd a(3,3);
a << 0, 0, 1,
1, 1, 1,
1, 1e+200, 1;
Eigen::EigenSolver<MatrixXd> eig(a);
VERIFY_IS_APPROX(a * eig.pseudoEigenvectors(), eig.pseudoEigenvectors() * eig.pseudoEigenvalueMatrix());
VERIFY_IS_APPROX(a * eig.eigenvectors(), eig.eigenvectors() * eig.eigenvalues().asDiagonal());
double scale = 1e-200; // scale to avoid overflow during the comparisons
VERIFY_IS_APPROX(a * eig.pseudoEigenvectors()*scale, eig.pseudoEigenvectors() * eig.pseudoEigenvalueMatrix()*scale);
VERIFY_IS_APPROX(a * eig.eigenvectors()*scale, eig.eigenvectors() * eig.eigenvalues().asDiagonal()*scale);
}
{
// check a case where all eigenvalues are null.
MatrixXd a(2,2);
a << 1, 1,
-1, -1;
Eigen::EigenSolver<MatrixXd> eig(a);
VERIFY_IS_APPROX(eig.pseudoEigenvectors().squaredNorm(), 2.);
VERIFY_IS_APPROX((a * eig.pseudoEigenvectors()).norm()+1., 1.);
VERIFY_IS_APPROX((eig.pseudoEigenvectors() * eig.pseudoEigenvalueMatrix()).norm()+1., 1.);
VERIFY_IS_APPROX((a * eig.eigenvectors()).norm()+1., 1.);
VERIFY_IS_APPROX((eig.eigenvectors() * eig.eigenvalues().asDiagonal()).norm()+1., 1.);
}
#endif