bug #645: patch from Tobias Wood implementing the extraction of eigenvectors in GeneralizedEigenSolver

Eigen/src/Eigenvalues/GeneralizedEigenSolver.h

@@ -1,8 +1,9 @@
 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
-// Copyright (C) 2012 Gael Guennebaud <gael.guennebaud@inria.fr>
+// Copyright (C) 2012-2016 Gael Guennebaud <gael.guennebaud@inria.fr>
 // Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
+// Copyright (C) 2016 Tobias Wood <tobias@spinicist.org.uk>
 //
 // This Source Code Form is subject to the terms of the Mozilla
 // Public License v. 2.0. If a copy of the MPL was not distributed
@@ -89,7 +90,7 @@ template<typename _MatrixType> class GeneralizedEigenSolver
       */
     typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> VectorType;
 
-    /** \brief Type for vector of complex scalar values eigenvalues as returned by betas().
+    /** \brief Type for vector of complex scalar values eigenvalues as returned by alphas().
       *
       * This is a column vector with entries of type #ComplexScalar.
       * The length of the vector is the size of #MatrixType.
@@ -114,7 +115,14 @@ template<typename _MatrixType> class GeneralizedEigenSolver
       *
       * \sa compute() for an example.
       */
-    GeneralizedEigenSolver() : m_eivec(), m_alphas(), m_betas(), m_isInitialized(false), m_realQZ(), m_matS(), m_tmp() {}
+    GeneralizedEigenSolver()
+      : m_eivec(),
+        m_alphas(),
+        m_betas(),
+        m_valuesOkay(false),
+        m_vectorsOkay(false),
+        m_realQZ()
+    {}
 
     /** \brief Default constructor with memory preallocation
       *
@@ -126,11 +134,9 @@ template<typename _MatrixType> class GeneralizedEigenSolver
       : m_eivec(size, size),
         m_alphas(size),
         m_betas(size),
-        m_isInitialized(false),
+        m_valuesOkay(false),
-        m_eigenvectorsOk(false),
+        m_vectorsOkay(false),
-        m_realQZ(size),
+        m_realQZ(size)
-        m_matS(size, size),
-        m_tmp(size)
     {}
 
     /** \brief Constructor; computes the generalized eigendecomposition of given matrix pair.
@@ -149,33 +155,29 @@ template<typename _MatrixType> class GeneralizedEigenSolver
       : m_eivec(A.rows(), A.cols()),
         m_alphas(A.cols()),
         m_betas(A.cols()),
-        m_isInitialized(false),
+        m_valuesOkay(false),
-        m_eigenvectorsOk(false),
+        m_vectorsOkay(false),
-        m_realQZ(A.cols()),
+        m_realQZ(A.cols())
-        m_matS(A.rows(), A.cols()),
-        m_tmp(A.cols())
     {
       compute(A, B, computeEigenvectors);
     }
 
     /* \brief Returns the computed generalized eigenvectors.
       *
-      * \returns  %Matrix whose columns are the (possibly complex) eigenvectors.
+      * \returns  %Matrix whose columns are the (possibly complex) right eigenvectors.
+      * i.e. the eigenvectors that solve (A - l*B)x = 0. The ordering matches the eigenvalues.
       *
       * \pre Either the constructor 
       * GeneralizedEigenSolver(const MatrixType&,const MatrixType&, bool) or the member function
       * compute(const MatrixType&, const MatrixType& bool) has been called before, and
       * \p computeEigenvectors was set to true (the default).
       *
-      * Column \f$ k \f$ of the returned matrix is an eigenvector corresponding
-      * to eigenvalue number \f$ k \f$ as returned by eigenvalues().  The
-      * eigenvectors are normalized to have (Euclidean) norm equal to one. The
-      * matrix returned by this function is the matrix \f$ V \f$ in the
-      * generalized eigendecomposition \f$ A = B V D V^{-1} \f$, if it exists.
-      *
       * \sa eigenvalues()
       */
-//    EigenvectorsType eigenvectors() const;
+    EigenvectorsType eigenvectors() const {
+      eigen_assert(m_vectorsOkay && "Eigenvectors for GeneralizedEigenSolver were not calculated.")
+      return m_eivec;
+    }
 
     /** \brief Returns an expression of the computed generalized eigenvalues.
       *
@@ -197,7 +199,7 @@ template<typename _MatrixType> class GeneralizedEigenSolver
       */
     EigenvalueType eigenvalues() const
     {
-      eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized.");
+      eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized.");
       return EigenvalueType(m_alphas,m_betas);
     }
 
@@ -208,7 +210,7 @@ template<typename _MatrixType> class GeneralizedEigenSolver
       * \sa betas(), eigenvalues() */
     ComplexVectorType alphas() const
     {
-      eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized.");
+      eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized.");
       return m_alphas;
     }
 
@@ -219,7 +221,7 @@ template<typename _MatrixType> class GeneralizedEigenSolver
       * \sa alphas(), eigenvalues() */
     VectorType betas() const
     {
-      eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized.");
+      eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized.");
       return m_betas;
     }
 
@@ -250,7 +252,7 @@ template<typename _MatrixType> class GeneralizedEigenSolver
 
     ComputationInfo info() const
     {
-      eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
+      eigen_assert(m_valuesOkay && "EigenSolver is not initialized.");
       return m_realQZ.info();
     }
 
@@ -270,29 +272,13 @@ template<typename _MatrixType> class GeneralizedEigenSolver
       EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL);
     }
     
-    MatrixType m_eivec;
+    EigenvectorsType m_eivec;
     ComplexVectorType m_alphas;
     VectorType m_betas;
-    bool m_isInitialized;
+    bool m_valuesOkay, m_vectorsOkay;
-    bool m_eigenvectorsOk;
     RealQZ<MatrixType> m_realQZ;
-    MatrixType m_matS;
-
-    typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
-    ColumnVectorType m_tmp;
 };
 
-//template<typename MatrixType>
-//typename GeneralizedEigenSolver<MatrixType>::EigenvectorsType GeneralizedEigenSolver<MatrixType>::eigenvectors() const
-//{
-//  eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
-//  eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
-//  Index n = m_eivec.cols();
-//  EigenvectorsType matV(n,n);
-//  // TODO
-//  return matV;
-//}
-
 template<typename MatrixType>
 GeneralizedEigenSolver<MatrixType>&
 GeneralizedEigenSolver<MatrixType>::compute(const MatrixType& A, const MatrixType& B, bool computeEigenvectors)
@@ -302,28 +288,70 @@ GeneralizedEigenSolver<MatrixType>::compute(const MatrixType& A, const MatrixTyp
   using std::sqrt;
   using std::abs;
   eigen_assert(A.cols() == A.rows() && B.cols() == A.rows() && B.cols() == B.rows());
-
+  m_valuesOkay = false;
+  m_vectorsOkay = false;
   // Reduce to generalized real Schur form:
   // A = Q S Z and B = Q T Z
   m_realQZ.compute(A, B, computeEigenvectors);
-
   if (m_realQZ.info() == Success)
   {
-    m_matS = m_realQZ.matrixS();
+    // Temp space for the untransformed eigenvectors
-    const MatrixType &matT = m_realQZ.matrixT();
+    VectorType v;
-    if (computeEigenvectors)
+    ComplexVectorType cv;
-      m_eivec = m_realQZ.matrixZ().transpose();
+    // Resize storage
-  
-    // Compute eigenvalues from matS
     m_alphas.resize(A.cols());
     m_betas.resize(A.cols());
+    if (computeEigenvectors) {
+      m_eivec.resize(A.cols(), A.cols());
+      v.resize(A.cols());
+      cv.resize(A.cols());
+    }
+    // Grab some references
+    const MatrixType &mZT = m_realQZ.matrixZ().transpose();
+    const MatrixType &mS = m_realQZ.matrixS();
+    const MatrixType &mT = m_realQZ.matrixT();
     Index i = 0;
     while (i < A.cols())
     {
-      if (i == A.cols() - 1 || m_matS.coeff(i+1, i) == Scalar(0))
+      if (i == A.cols() - 1 || mS.coeff(i+1, i) == Scalar(0))
       {
-        m_alphas.coeffRef(i) = m_matS.coeff(i, i);
+        // Real eigenvalue
-        m_betas.coeffRef(i)  = matT.coeff(i,i);
+        m_alphas.coeffRef(i) = mS.diagonal().coeff(i);
+        m_betas.coeffRef(i)  = mT.diagonal().coeff(i);
+        if (computeEigenvectors)
+        {
+          v.setConstant(Scalar(0.0));
+          v.coeffRef(i) = Scalar(1.0);
+          // For singular eigenvalues do nothing more
+          if(abs(m_betas.coeffRef(i)) >= (std::numeric_limits<RealScalar>::min)())
+          {
+            // Non-singular eigenvalue
+            const Scalar alpha = real(m_alphas.coeffRef(i));
+            const Scalar beta = m_betas.coeffRef(i);
+            for (Index j = i-1; j >= 0; j--)
+            {
+              const Index st = j+1;
+              const Index sz = i-j;
+              if (j > 0 && mS.coeff(j, j-1) != Scalar(0))
+              { // 2x2 block
+                Matrix<Scalar, 2, 1> RHS;
+                RHS[0] = -v.segment(st,sz).transpose().cwiseProduct(beta*mS.block(j-1,st,1,sz) - alpha*mT.block(j-1,st,1,sz)).sum();
+                RHS[1] = -v.segment(st,sz).transpose().cwiseProduct(beta*mS.block(j,st,1,sz) - alpha*mT.block(j,st,1,sz)).sum();
+                Matrix<Scalar, 2, 2> LHS;
+                LHS << beta*mS.coeffRef(j-1,j-1) - alpha*mT.coeffRef(j-1,j-1), beta*mS.coeffRef(j-1,j) - alpha*mT.coeffRef(j-1,j),
+                       beta*mS.coeffRef(j,j-1) - alpha*mT.coeffRef(j,j-1), beta*mS.coeffRef(j,j) - alpha*mT.coeffRef(j,j);
+                v.segment(j-1,2) = LHS.partialPivLu().solve(RHS);
+                j--;
+              }
+              else
+              {
+                v.coeffRef(j) = -v.segment(st,sz).transpose().cwiseProduct(beta*mS.block(j,st,1,sz) - alpha*mT.block(j,st,1,sz)).sum() / (beta*mS.coeffRef(j,j) - alpha*mT.coeffRef(j,j));
+              }
+            }
+          }
+          m_eivec.col(i).real() = (mZT * v).normalized();
+          m_eivec.col(i).imag().setConstant(0);
+        }
         ++i;
       }
       else
@@ -333,27 +361,49 @@ GeneralizedEigenSolver<MatrixType>::compute(const MatrixType& A, const MatrixTyp
 
         // T =  [a 0]
         //      [0 b]
-        RealScalar a = matT.diagonal().coeff(i),
+        RealScalar a = mT.diagonal().coeff(i),
-                   b = matT.diagonal().coeff(i+1);
+                   b = mT.diagonal().coeff(i+1);
+        const RealScalar beta = m_betas.coeffRef(i) = m_betas.coeffRef(i+1) = a*b;
+
         // ^^ NOTE: using diagonal()(i) instead of coeff(i,i) workarounds a MSVC bug.
-        Matrix<RealScalar,2,2> S2 = m_matS.template block<2,2>(i,i) * Matrix<Scalar,2,1>(b,a).asDiagonal();
+        Matrix<RealScalar,2,2> S2 = mS.template block<2,2>(i,i) * Matrix<Scalar,2,1>(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)));
-        m_alphas.coeffRef(i)   = ComplexScalar(S2.coeff(1,1) + p,  z);
+        const ComplexScalar alpha = ComplexScalar(S2.coeff(1,1) + p, (beta > 0) ? z : -z);
-        m_alphas.coeffRef(i+1) = ComplexScalar(S2.coeff(1,1) + p, -z);
+        m_alphas.coeffRef(i)   = conj(alpha);
-
+        m_alphas.coeffRef(i+1) = alpha;
-        m_betas.coeffRef(i)   =
-        m_betas.coeffRef(i+1) = a*b;
 
+        if (computeEigenvectors) {
+          // Compute eigenvector in position (i+1) and then position (i) is just the conjugate
+          cv.setConstant(ComplexScalar(0.0, 0.0));
+          cv.coeffRef(i+1) = ComplexScalar(1.0, 0.0);
+          cv.coeffRef(i) = -(beta*mS.coeffRef(i,i+1) - alpha*mT.coeffRef(i,i+1)) / (beta*mS.coeffRef(i,i) - alpha*mT.coeffRef(i,i));
+          for (Index j = i-1; j >= 0; j--) {
+            const Index st = j+1;
+            const Index sz = i+1-j;
+            if (j > 0 && mS.coeff(j, j-1) != Scalar(0)) { // 2x2 block
+              Matrix<ComplexScalar, 2, 1> RHS;
+              RHS[0] = -cv.segment(st,sz).transpose().cwiseProduct(beta*mS.block(j-1,st,1,sz) - alpha*mT.block(j-1,st,1,sz)).sum();
+              RHS[1] = -cv.segment(st,sz).transpose().cwiseProduct(beta*mS.block(j,st,1,sz) - alpha*mT.block(j,st,1,sz)).sum();
+              Matrix<ComplexScalar, 2, 2> LHS;
+              LHS << beta*mS.coeffRef(j-1,j-1) - alpha*mT.coeffRef(j-1,j-1), beta*mS.coeffRef(j-1,j) - alpha*mT.coeffRef(j-1,j),
+                     beta*mS.coeffRef(j,j-1) - alpha*mT.coeffRef(j,j-1), beta*mS.coeffRef(j,j) - alpha*mT.coeffRef(j,j);
+              cv.segment(j-1,2) = LHS.partialPivLu().solve(RHS);
+              j--;
+            } else {
+              cv.coeffRef(j) = -cv.segment(st,sz).transpose().cwiseProduct(beta*mS.block(j,st,1,sz) - alpha*mT.block(j,st,1,sz)).sum() / (beta*mS.coeffRef(j,j) - alpha*mT.coeffRef(j,j));
+            }
+          }
+          m_eivec.col(i+1) = (mZT * cv).normalized();
+          m_eivec.col(i)   = m_eivec.col(i+1).conjugate();
+        }
         i += 2;
       }
     }
+    m_valuesOkay = true;
+    m_vectorsOkay = computeEigenvectors;
   }
-
-  m_isInitialized = true;
-  m_eigenvectorsOk = false;//computeEigenvectors;
-
   return *this;
 }
 

test/eigensolver_generalized_real.cpp

@@ -42,6 +42,11 @@ template<typename MatrixType> void generalized_eigensolver_real(const MatrixType
     VectorType realEigenvalues = eig.eigenvalues().real();
     std::sort(realEigenvalues.data(), realEigenvalues.data()+realEigenvalues.size());
     VERIFY_IS_APPROX(realEigenvalues, symmEig.eigenvalues());
+
+    // check eigenvectors
+    typename GeneralizedEigenSolver<MatrixType>::EigenvectorsType D = eig.eigenvalues().asDiagonal();
+    typename GeneralizedEigenSolver<MatrixType>::EigenvectorsType V = eig.eigenvectors();
+    VERIFY_IS_APPROX(spdA*V, spdB*V*D);
   }
 
   // non symmetric case:
@@ -54,6 +59,10 @@ template<typename MatrixType> void generalized_eigensolver_real(const MatrixType
         tmp /= tmp.norm();
       VERIFY_IS_MUCH_SMALLER_THAN( std::abs(tmp.determinant()), Scalar(1) );
     }
+    // check eigenvectors
+    typename GeneralizedEigenSolver<MatrixType>::EigenvectorsType D = eig.eigenvalues().asDiagonal();
+    typename GeneralizedEigenSolver<MatrixType>::EigenvectorsType V = eig.eigenvectors();
+    VERIFY_IS_APPROX(a*V, b*V*D);
   }
 
   // regression test for bug 1098