bug #1557: fix RealSchur and EigenSolver for matrices with only zeros on the diagonal.

Eigen/src/Eigenvalues/RealSchur.h

@@ -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& considerAsZero);
     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);
@@ -307,12 +307,16 @@ RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMa
   Index totalIter = 0; // iteration count for whole matrix
   Scalar exshift(0);   // sum of exceptional shifts
   Scalar norm = computeNormOfT();
+  // sub-diagonal entries smaller than considerAsZero will be treated as zero.
+  // We use eps^2 to enable more precision in small eigenvalues.
+  Scalar considerAsZero = numext::maxi( norm * numext::abs2(NumTraits<Scalar>::epsilon()),
+                                        (std::numeric_limits<Scalar>::min)() );
 
   if(norm!=Scalar(0))
   {
     while (iu >= 0)
     {
-      Index il = findSmallSubdiagEntry(iu);
+      Index il = findSmallSubdiagEntry(iu,considerAsZero);
 
       // Check for convergence
       if (il == iu) // One root found
@@ -369,14 +373,17 @@ 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& considerAsZero)
 {
   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)
+
+    s = numext::maxi(s * NumTraits<Scalar>::epsilon(), considerAsZero);
+    
+    if (abs(m_matT.coeff(res,res-1)) <= s)
       break;
     res--;
   }

test/eigensolver_generic.cpp

@@ -12,6 +12,21 @@
 #include <limits>
 #include <Eigen/Eigenvalues>
 
+template<typename EigType,typename MatType>
+void check_eigensolver_for_given_mat(const EigType &eig, const MatType& a)
+{
+  typedef typename NumTraits<typename MatType::Scalar>::Real RealScalar;
+  typedef Matrix<RealScalar, MatType::RowsAtCompileTime, 1> RealVectorType;
+  typedef typename std::complex<RealScalar> Complex;
+  Index n = a.rows();
+  VERIFY_IS_EQUAL(eig.info(), Success);
+  VERIFY_IS_APPROX(a * eig.pseudoEigenvectors(), eig.pseudoEigenvectors() * eig.pseudoEigenvalueMatrix());
+  VERIFY_IS_APPROX(a.template cast<Complex>() * eig.eigenvectors(),
+                   eig.eigenvectors() * eig.eigenvalues().asDiagonal());
+  VERIFY_IS_APPROX(eig.eigenvectors().colwise().norm(), RealVectorType::Ones(n).transpose());
+  VERIFY_IS_APPROX(a.eigenvalues(), eig.eigenvalues());
+}
+
 template<typename MatrixType> void eigensolver(const MatrixType& m)
 {
   /* this test covers the following files:
@@ -22,8 +37,7 @@ template<typename MatrixType> void eigensolver(const MatrixType& m)
 
   typedef typename MatrixType::Scalar Scalar;
   typedef typename NumTraits<Scalar>::Real RealScalar;
-  typedef Matrix<RealScalar, MatrixType::RowsAtCompileTime, 1> RealVectorType;
+  typedef typename std::complex<RealScalar> Complex;
-  typedef typename std::complex<typename NumTraits<typename MatrixType::Scalar>::Real> Complex;
 
   MatrixType a = MatrixType::Random(rows,cols);
   MatrixType a1 = MatrixType::Random(rows,cols);
@@ -36,12 +50,7 @@ template<typename MatrixType> void eigensolver(const MatrixType& m)
     (ei0.pseudoEigenvectors().template cast<Complex>()) * (ei0.eigenvalues().asDiagonal()));
 
   EigenSolver<MatrixType> ei1(a);
-  VERIFY_IS_EQUAL(ei1.info(), Success);
+  CALL_SUBTEST( check_eigensolver_for_given_mat(ei1,a) );
-  VERIFY_IS_APPROX(a * ei1.pseudoEigenvectors(), ei1.pseudoEigenvectors() * ei1.pseudoEigenvalueMatrix());
-  VERIFY_IS_APPROX(a.template cast<Complex>() * ei1.eigenvectors(),
-                   ei1.eigenvectors() * ei1.eigenvalues().asDiagonal());
-  VERIFY_IS_APPROX(ei1.eigenvectors().colwise().norm(), RealVectorType::Ones(rows).transpose());
-  VERIFY_IS_APPROX(a.eigenvalues(), ei1.eigenvalues());
 
   EigenSolver<MatrixType> ei2;
   ei2.setMaxIterations(RealSchur<MatrixType>::m_maxIterationsPerRow * rows).compute(a);
@@ -100,6 +109,19 @@ template<typename MatrixType> void eigensolver_verify_assert(const MatrixType& m
   VERIFY_RAISES_ASSERT(eig.pseudoEigenvectors());
 }
 
+
+template<typename CoeffType>
+Matrix<typename CoeffType::Scalar,Dynamic,Dynamic>
+make_companion(const CoeffType& coeffs)
+{
+  Index n = coeffs.size()-1;
+  Matrix<typename CoeffType::Scalar,Dynamic,Dynamic> res(n,n);
+  res.setZero();
+	res.row(0) = -coeffs.tail(n) / coeffs(0);
+	res.diagonal(-1).setOnes();
+  return res;
+}
+
 template<int>
 void eigensolver_generic_extra()
 {
@@ -126,6 +148,42 @@ void eigensolver_generic_extra()
     VERIFY_IS_APPROX((a * eig.eigenvectors()).norm()+1., 1.);
     VERIFY_IS_APPROX((eig.eigenvectors() * eig.eigenvalues().asDiagonal()).norm()+1., 1.);
   }
+
+  // regression test for bug 933
+  {
+    {
+      VectorXd coeffs(5); coeffs << 1, -3, -175, -225, 2250;
+      MatrixXd C = make_companion(coeffs);
+      EigenSolver<MatrixXd> eig(C);
+      CALL_SUBTEST( check_eigensolver_for_given_mat(eig,C) );
+    }
+    {
+      // this test is tricky because it requires high accuracy in smallest eigenvalues
+      VectorXd coeffs(5); coeffs << 6.154671e-15, -1.003870e-10, -9.819570e-01, 3.995715e+03, 2.211511e+08;
+      MatrixXd C = make_companion(coeffs);
+      EigenSolver<MatrixXd> eig(C);
+      CALL_SUBTEST( check_eigensolver_for_given_mat(eig,C) );
+      Index n = C.rows();
+      for(Index i=0;i<n;++i)
+      {
+        typedef std::complex<double> Complex;
+        MatrixXcd ac = C.cast<Complex>();
+        ac.diagonal().array() -= eig.eigenvalues()(i);
+        VectorXd sv = ac.jacobiSvd().singularValues();
+        // comparing to sv(0) is not enough here to catch the "bug",
+        // the hard-coded 1.0 is important!
+        VERIFY_IS_MUCH_SMALLER_THAN(sv(n-1), 1.0);
+      }
+    }
+  }
+  // regression test for bug 1557
+  {
+    // this test is interesting because it contains zeros on the diagonal.
+    MatrixXd A_bug1557(3,3);
+    A_bug1557 << 0, 0, 0, 1, 0, 0.5887907064808635127, 0, 1, 0;
+    EigenSolver<MatrixXd> eig(A_bug1557);
+    CALL_SUBTEST( check_eigensolver_for_given_mat(eig,A_bug1557) );
+  }
 }
 
 EIGEN_DECLARE_TEST(eigensolver_generic)