Computes the generalized eigenvalues and eigenvectors of a pair of general matrices. More...

Public Types
enum	{ RowsAtCompileTime , ColsAtCompileTime , Options , MaxRowsAtCompileTime , MaxColsAtCompileTime }

typedef std::complex< RealScalar >	ComplexScalar
	Complex scalar type for MatrixType. More...

typedef Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	ComplexVectorType
	Type for vector of complex scalar values eigenvalues as returned by alphas(). More...

typedef CwiseBinaryOp< internal::scalar_quotient_op< ComplexScalar, Scalar >, ComplexVectorType, VectorType >	EigenvalueType
	Expression type for the eigenvalues as returned by eigenvalues(). More...

typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime >	EigenvectorsType
	Type for matrix of eigenvectors as returned by eigenvectors(). More...

typedef Eigen::Index	Index

typedef MatrixType_	MatrixType
	Synonym for the template parameter `MatrixType_`. More...

typedef NumTraits< Scalar >::Real	RealScalar

typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type MatrixType. More...

typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	VectorType
	Type for vector of real scalar values eigenvalues as returned by betas(). More...

Public Member Functions
const ComplexVectorType &	alphas () const

const VectorType &	betas () const

GeneralizedEigenSolver &	compute (const MatrixType &A, const MatrixType &B, bool computeEigenvectors=true)
	Computes generalized eigendecomposition of given matrix. More...

EigenvalueType	eigenvalues () const
	Returns an expression of the computed generalized eigenvalues. More...

EigenvectorsType	eigenvectors () const

	GeneralizedEigenSolver ()
	Default constructor. More...

	GeneralizedEigenSolver (const MatrixType &A, const MatrixType &B, bool computeEigenvectors=true)
	Constructor; computes the generalized eigendecomposition of given matrix pair. More...

	GeneralizedEigenSolver (Index size)
	Default constructor with memory preallocation. More...

ComputationInfo	info () const

GeneralizedEigenSolver &	setMaxIterations (Index maxIters)

Protected Attributes
ComplexVectorType	m_alphas

VectorType	m_betas

bool	m_computeEigenvectors

EigenvectorsType	m_eivec

bool	m_isInitialized

RealQZ< MatrixType >	m_realQZ

ComplexVectorType	m_tmp

Detailed Description

template<typename MatrixType_>
class Eigen::GeneralizedEigenSolver< MatrixType_ >

Computes the generalized eigenvalues and eigenvectors of a pair of general matrices.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues>

Eigenvalues

Template Parameters

MatrixType_ the type of the matrices of which we are computing the eigen-decomposition; this is expected to be an instantiation of the Matrix class template. Currently, only real matrices are supported.

The generalized eigenvalues and eigenvectors of a matrix pair \( A \) and \( B \) are scalars \( \lambda \) and vectors \( v \) such that \( Av = \lambda Bv \). If \( D \) is a diagonal matrix with the eigenvalues on the diagonal, and \( V \) is a matrix with the eigenvectors as its columns, then \( A V = B V D \). The matrix \( V \) is almost always invertible, in which case we have \( A = B V D V^{-1} \). This is called the generalized eigen-decomposition.

The generalized eigenvalues and eigenvectors of a matrix pair may be complex, even when the matrices are real. Moreover, the generalized eigenvalue might be infinite if the matrix B is singular. To workaround this difficulty, the eigenvalues are provided as a pair of complex \( \alpha \) and real \( \beta \) such that: \( \lambda_i = \alpha_i / \beta_i \). If \( \beta_i \) is (nearly) zero, then one can consider the well defined left eigenvalue \( \mu = \beta_i / \alpha_i\) such that: \( \mu_i A v_i = B v_i \), or even \( \mu_i u_i^T A = u_i^T B \) where \( u_i \) is called the left eigenvector.

Call the function compute() to compute the generalized eigenvalues and eigenvectors of a given matrix pair. Alternatively, you can use the GeneralizedEigenSolver(const MatrixType&, const MatrixType&, bool) constructor which computes the eigenvalues and eigenvectors at construction time. Once the eigenvalue and eigenvectors are computed, they can be retrieved with the eigenvalues() and eigenvectors() functions.

Here is an usage example of this class: Example:

GeneralizedEigenSolver<MatrixXf> ges;
MatrixXf A = MatrixXf::Random(4,4);
MatrixXf B = MatrixXf::Random(4,4);
ges.compute(A, B);
cout << "The (complex) numerators of the generalzied eigenvalues are: " << ges.alphas().transpose() << endl;
cout << "The (real) denominatore of the generalzied eigenvalues are: " << ges.betas().transpose() << endl;
cout << "The (complex) generalzied eigenvalues are (alphas./beta): " << ges.eigenvalues().transpose() << endl;

Output:

The (complex) numerators of the generalzied eigenvalues are:  (-0.126,0.569) (-0.126,-0.569)      (-0.398,0)       (-1.12,0)
The (real) denominatore of the generalzied eigenvalues are: -1.56 -1.56 -1.25 0.746
The (complex) generalzied eigenvalues are (alphas./beta): (0.081,-0.365)  (0.081,0.365)     (0.318,-0)       (-1.5,0)

See also: MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver

Definition at line 60 of file GeneralizedEigenSolver.h.

Member Typedef Documentation

◆ ComplexScalar

template<typename MatrixType_ >

typedef std::complex<RealScalar> Eigen::GeneralizedEigenSolver< MatrixType_ >::ComplexScalar

Complex scalar type for MatrixType.

This is std::complex<Scalar> if Scalar is real (e.g., float or double) and just Scalar if Scalar is complex.

Definition at line 86 of file GeneralizedEigenSolver.h.

◆ ComplexVectorType

template<typename MatrixType_ >

typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::GeneralizedEigenSolver< MatrixType_ >::ComplexVectorType

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.

Definition at line 100 of file GeneralizedEigenSolver.h.

◆ EigenvalueType

template<typename MatrixType_ >

typedef CwiseBinaryOp<internal::scalar_quotient_op<ComplexScalar,Scalar>,ComplexVectorType,VectorType> Eigen::GeneralizedEigenSolver< MatrixType_ >::EigenvalueType

Expression type for the eigenvalues as returned by eigenvalues().

Definition at line 104 of file GeneralizedEigenSolver.h.

◆ EigenvectorsType

template<typename MatrixType_ >

typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::GeneralizedEigenSolver< MatrixType_ >::EigenvectorsType

Type for matrix of eigenvectors as returned by eigenvectors().

This is a square matrix with entries of type ComplexScalar. The size is the same as the size of MatrixType.

Definition at line 111 of file GeneralizedEigenSolver.h.

◆ Index

template<typename MatrixType_ >

typedef Eigen::Index Eigen::GeneralizedEigenSolver< MatrixType_ >::Index

Deprecated:: since Eigen 3.3

Definition at line 78 of file GeneralizedEigenSolver.h.

◆ MatrixType

template<typename MatrixType_ >

typedef MatrixType_ Eigen::GeneralizedEigenSolver< MatrixType_ >::MatrixType

Synonym for the template parameter MatrixType_.

Definition at line 65 of file GeneralizedEigenSolver.h.

◆ RealScalar

template<typename MatrixType_ >

typedef NumTraits<Scalar>::Real Eigen::GeneralizedEigenSolver< MatrixType_ >::RealScalar

Definition at line 77 of file GeneralizedEigenSolver.h.

◆ Scalar

template<typename MatrixType_ >

typedef MatrixType::Scalar Eigen::GeneralizedEigenSolver< MatrixType_ >::Scalar

Scalar type for matrices of type MatrixType.

Definition at line 76 of file GeneralizedEigenSolver.h.

◆ VectorType

template<typename MatrixType_ >

typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::GeneralizedEigenSolver< MatrixType_ >::VectorType

Type for vector of real scalar values eigenvalues as returned by betas().

This is a column vector with entries of type Scalar. The length of the vector is the size of MatrixType.

Definition at line 93 of file GeneralizedEigenSolver.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ >

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 67 of file GeneralizedEigenSolver.h.

          {
       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
       Options = MatrixType::Options,
       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
     };

Constructor & Destructor Documentation

◆ GeneralizedEigenSolver() [1/3]

template<typename MatrixType_ >

Eigen::GeneralizedEigenSolver< MatrixType_ >::GeneralizedEigenSolver ( )

inline

Default constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via EigenSolver::compute(const MatrixType&, bool).

See also: compute() for an example.

Definition at line 120 of file GeneralizedEigenSolver.h.

       : m_eivec(),
         m_alphas(),
         m_betas(),
         m_computeEigenvectors(false),
         m_isInitialized(false),
         m_realQZ()
     {}

◆ GeneralizedEigenSolver() [2/3]

template<typename MatrixType_ >

Eigen::GeneralizedEigenSolver< MatrixType_ >::GeneralizedEigenSolver ( Index size )

inlineexplicit

Default constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also: GeneralizedEigenSolver()

Definition at line 135 of file GeneralizedEigenSolver.h.

       : m_eivec(size, size),
         m_alphas(size),
         m_betas(size),
         m_computeEigenvectors(false),
         m_isInitialized(false),
         m_realQZ(size),
         m_tmp(size)
     {}

◆ GeneralizedEigenSolver() [3/3]

template<typename MatrixType_ >

Eigen::GeneralizedEigenSolver< MatrixType_ >::GeneralizedEigenSolver	(	const MatrixType &	A,
		const MatrixType &	B,
		bool	computeEigenvectors = `true`
	)

inline

Constructor; computes the generalized eigendecomposition of given matrix pair.

Parameters

[in]	A	Square matrix whose eigendecomposition is to be computed.
[in]	B	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

This constructor calls compute() to compute the generalized eigenvalues and eigenvectors.

See also: compute()

Definition at line 157 of file GeneralizedEigenSolver.h.

       : m_eivec(A.rows(), A.cols()),
         m_alphas(A.cols()),
         m_betas(A.cols()),
         m_computeEigenvectors(false),
         m_isInitialized(false),
         m_realQZ(A.cols()),
         m_tmp(A.cols())
     {
       compute(A, B, computeEigenvectors);
     }

Member Function Documentation

◆ alphas()

template<typename MatrixType_ >

const ComplexVectorType& Eigen::GeneralizedEigenSolver< MatrixType_ >::alphas ( ) const

inline

Returns: A const reference to the vectors containing the alpha values

This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).

See also: betas(), eigenvalues()

Definition at line 216 of file GeneralizedEigenSolver.h.

     {
       eigen_assert(info() == Success && "GeneralizedEigenSolver failed to compute alphas.");
       return m_alphas;
     }

◆ betas()

template<typename MatrixType_ >

const VectorType& Eigen::GeneralizedEigenSolver< MatrixType_ >::betas ( ) const

inline

Returns: A const reference to the vectors containing the beta values

This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).

See also: alphas(), eigenvalues()

Definition at line 227 of file GeneralizedEigenSolver.h.

     {
       eigen_assert(info() == Success && "GeneralizedEigenSolver failed to compute betas.");
       return m_betas;
     }

◆ compute()

template<typename MatrixType >

GeneralizedEigenSolver< MatrixType > & Eigen::GeneralizedEigenSolver< MatrixType >::compute	(	const MatrixType &	A,
		const MatrixType &	B,
		bool	computeEigenvectors = `true`
	)

Computes generalized eigendecomposition of given matrix.

Parameters

[in]	A	Square matrix whose eigendecomposition is to be computed.
[in]	B	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

Returns: Reference to *this

This function computes the eigenvalues of the real matrix matrix. The eigenvalues() function can be used to retrieve them. If computeEigenvectors is true, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().

The matrix is first reduced to real generalized Schur form using the RealQZ class. The generalized Schur decomposition is then used to compute the eigenvalues and eigenvectors.

The cost of the computation is dominated by the cost of the generalized Schur decomposition.

This method reuses of the allocated data in the GeneralizedEigenSolver object.

Definition at line 288 of file GeneralizedEigenSolver.h.

 {
   using std::sqrt;
   using std::abs;
   eigen_assert(A.cols() == A.rows() && B.cols() == A.rows() && B.cols() == B.rows());
   Index size = A.cols();
   // 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)
   {
     // Resize storage
     m_alphas.resize(size);
     m_betas.resize(size);
     if (computeEigenvectors)
     {
       m_eivec.resize(size,size);
       m_tmp.resize(size);
     }
  
     // Aliases:
     Map<VectorType> v(reinterpret_cast<Scalar*>(m_tmp.data()), size);
     ComplexVectorType &cv = m_tmp;
     const MatrixType &mS = m_realQZ.matrixS();
     const MatrixType &mT = m_realQZ.matrixT();
  
     Index i = 0;
     while (i < size)
     {
       if (i == size - 1 || mS.coeff(i+1, i) == Scalar(0))
       {
         // Real eigenvalue
         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 = (alpha*mT.template block<2,Dynamic>(j-1,st,2,sz) - beta*mS.template block<2,Dynamic>(j-1,st,2,sz)) .lazyProduct( v.segment(st,sz) );
                 Matrix<Scalar, 2, 2> lhs = beta * mS.template block<2,2>(j-1,j-1) - alpha * mT.template block<2,2>(j-1,j-1);
                 v.template segment<2>(j-1) = 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().noalias() = m_realQZ.matrixZ().transpose() * v;
           m_eivec.col(i).real().normalize();
           m_eivec.col(i).imag().setConstant(0);
         }
         ++i;
       }
       else
       {
         // 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 0]
         //      [0 b]
         RealScalar a = mT.diagonal().coeff(i),
                    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 = 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)));
         const ComplexScalar alpha = ComplexScalar(S2.coeff(1,1) + p, (beta > 0) ? z : -z);
         m_alphas.coeffRef(i)   = conj(alpha);
         m_alphas.coeffRef(i+1) = alpha;
  
         if (computeEigenvectors) {
           // Compute eigenvector in position (i+1) and then position (i) is just the conjugate
           cv.setZero();
           cv.coeffRef(i+1) = Scalar(1.0);
           // here, the "static_cast" workaound expression template issues.
           cv.coeffRef(i) = -(static_cast<Scalar>(beta*mS.coeffRef(i,i+1)) - alpha*mT.coeffRef(i,i+1))
                           / (static_cast<Scalar>(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 = (alpha*mT.template block<2,Dynamic>(j-1,st,2,sz) - beta*mS.template block<2,Dynamic>(j-1,st,2,sz)) .lazyProduct( cv.segment(st,sz) );
               Matrix<ComplexScalar, 2, 2> lhs = beta * mS.template block<2,2>(j-1,j-1) - alpha * mT.template block<2,2>(j-1,j-1);
               cv.template segment<2>(j-1) = 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()
                               / (alpha*mT.coeffRef(j,j) - static_cast<Scalar>(beta*mS.coeffRef(j,j)));
             }
           }
           m_eivec.col(i+1).noalias() = (m_realQZ.matrixZ().transpose() * cv);
           m_eivec.col(i+1).normalize();
           m_eivec.col(i) = m_eivec.col(i+1).conjugate();
         }
         i += 2;
       }
     }
   }
   m_computeEigenvectors = computeEigenvectors;
   m_isInitialized = true;
   return *this;
 }

◆ eigenvalues()

template<typename MatrixType_ >

EigenvalueType Eigen::GeneralizedEigenSolver< MatrixType_ >::eigenvalues ( ) const

inline

Returns an expression of the computed generalized eigenvalues.

Returns: An expression of the column vector containing the eigenvalues.

It is a shortcut for

this->alphas().cwiseQuotient(this->betas());

Eigen::GeneralizedEigenSolver::betas

const VectorType & betas() const

Definition: GeneralizedEigenSolver.h:227

Eigen::GeneralizedEigenSolver::alphas

const ComplexVectorType & alphas() const

Definition: GeneralizedEigenSolver.h:216

Not that betas might contain zeros. It is therefore not recommended to use this function, but rather directly deal with the alphas and betas vectors.

Precondition: Either the constructor GeneralizedEigenSolver(const MatrixType&,const MatrixType&,bool) or the member function compute(const MatrixType&,const MatrixType&,bool) has been called before.

The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are not sorted in any particular order.

See also: alphas(), betas(), eigenvectors()

Definition at line 205 of file GeneralizedEigenSolver.h.

     {
       eigen_assert(info() == Success && "GeneralizedEigenSolver failed to compute eigenvalues.");
       return EigenvalueType(m_alphas,m_betas);
     }

◆ eigenvectors()

template<typename MatrixType_ >

EigenvectorsType Eigen::GeneralizedEigenSolver< MatrixType_ >::eigenvectors ( ) const

inline

Definition at line 181 of file GeneralizedEigenSolver.h.

                                           {
       eigen_assert(info() == Success && "GeneralizedEigenSolver failed to compute eigenvectors");
       eigen_assert(m_computeEigenvectors && "Eigenvectors for GeneralizedEigenSolver were not calculated");
       return m_eivec;
     }

◆ info()

template<typename MatrixType_ >

ComputationInfo Eigen::GeneralizedEigenSolver< MatrixType_ >::info ( ) const

inline

Definition at line 258 of file GeneralizedEigenSolver.h.

     {
       eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
       return m_realQZ.info();
     }

◆ setMaxIterations()

template<typename MatrixType_ >

GeneralizedEigenSolver& Eigen::GeneralizedEigenSolver< MatrixType_ >::setMaxIterations ( Index maxIters )

inline

Sets the maximal number of iterations allowed.

Definition at line 266 of file GeneralizedEigenSolver.h.

     {
       m_realQZ.setMaxIterations(maxIters);
       return *this;
     }

Member Data Documentation

◆ m_alphas

template<typename MatrixType_ >

ComplexVectorType Eigen::GeneralizedEigenSolver< MatrixType_ >::m_alphas

protected

Definition at line 278 of file GeneralizedEigenSolver.h.

◆ m_betas

template<typename MatrixType_ >

VectorType Eigen::GeneralizedEigenSolver< MatrixType_ >::m_betas

protected

Definition at line 279 of file GeneralizedEigenSolver.h.

◆ m_computeEigenvectors

template<typename MatrixType_ >

bool Eigen::GeneralizedEigenSolver< MatrixType_ >::m_computeEigenvectors

protected

Definition at line 280 of file GeneralizedEigenSolver.h.

◆ m_eivec

template<typename MatrixType_ >

EigenvectorsType Eigen::GeneralizedEigenSolver< MatrixType_ >::m_eivec

protected

Definition at line 277 of file GeneralizedEigenSolver.h.

◆ m_isInitialized

template<typename MatrixType_ >

bool Eigen::GeneralizedEigenSolver< MatrixType_ >::m_isInitialized

protected

Definition at line 281 of file GeneralizedEigenSolver.h.

◆ m_realQZ

template<typename MatrixType_ >

RealQZ<MatrixType> Eigen::GeneralizedEigenSolver< MatrixType_ >::m_realQZ

protected

Definition at line 282 of file GeneralizedEigenSolver.h.

◆ m_tmp

template<typename MatrixType_ >

ComplexVectorType Eigen::GeneralizedEigenSolver< MatrixType_ >::m_tmp

protected

Definition at line 283 of file GeneralizedEigenSolver.h.

The documentation for this class was generated from the following file:

GeneralizedEigenSolver.h

Public Types

Public Member Functions

Protected Attributes

Detailed Description

template<typename MatrixType_> class Eigen::GeneralizedEigenSolver< MatrixType_ >

Member Typedef Documentation

◆ ComplexScalar

◆ ComplexVectorType

◆ EigenvalueType

◆ EigenvectorsType

◆ Index

◆ MatrixType

◆ RealScalar

◆ Scalar

◆ VectorType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ GeneralizedEigenSolver() [1/3]

◆ GeneralizedEigenSolver() [2/3]

◆ GeneralizedEigenSolver() [3/3]

Member Function Documentation

◆ alphas()

◆ betas()

◆ compute()

◆ eigenvalues()

◆ eigenvectors()

◆ info()

◆ setMaxIterations()

Member Data Documentation

◆ m_alphas

◆ m_betas

◆ m_computeEigenvectors

◆ m_eivec

◆ m_isInitialized

◆ m_realQZ

◆ m_tmp

template<typename MatrixType_>
class Eigen::GeneralizedEigenSolver< MatrixType_ >