Computes eigenvalues and eigenvectors of general matrices. More...

#include <EigenSolver.h>

Collaboration diagram for Eigen::EigenSolver< _MatrixType >:

Public Types
enum	{ RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }

typedef _MatrixType	MatrixType
	Synonym for the template parameter `_MatrixType`.

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

typedef NumTraits< Scalar >::Real	RealScalar

typedef MatrixType::Index	Index

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

typedef Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	EigenvalueType
	Type for vector of 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...

Public Member Functions
	EigenSolver ()
	Default constructor. More...

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

	EigenSolver (const MatrixType &matrix, bool computeEigenvectors=true)
	Constructor; computes eigendecomposition of given matrix. More...

EigenvectorsType	eigenvectors () const
	Returns the eigenvectors of given matrix. More...

const MatrixType &	pseudoEigenvectors () const
	Returns the pseudo-eigenvectors of given matrix. More...

MatrixType	pseudoEigenvalueMatrix () const
	Returns the block-diagonal matrix in the pseudo-eigendecomposition. More...

const EigenvalueType &	eigenvalues () const
	Returns the eigenvalues of given matrix. More...

EigenSolver &	compute (const MatrixType &matrix, bool computeEigenvectors=true)
	Computes eigendecomposition of given matrix. More...

ComputationInfo	info () const

EigenSolver &	setMaxIterations (Index maxIters)
	Sets the maximum number of iterations allowed.

Index	getMaxIterations ()
	Returns the maximum number of iterations.

Protected Types
typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	ColumnVectorType

Protected Attributes
MatrixType	m_eivec

EigenvalueType	m_eivalues

bool	m_isInitialized

bool	m_eigenvectorsOk

RealSchur< MatrixType >	m_realSchur

MatrixType	m_matT

ColumnVectorType	m_tmp

Detailed Description

template<typename _MatrixType>
class Eigen::EigenSolver< _MatrixType >

Computes eigenvalues and eigenvectors of general matrices.

Template Parameters

_MatrixType the type of the matrix of which we are computing the eigendecomposition; this is expected to be an instantiation of the Matrix class template. Currently, only real matrices are supported.

The eigenvalues and eigenvectors of a matrix $ A $ are scalars $\lambda$ and vectors $ v $ such that $Av = \lambda v$ . 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 = V D $ . The matrix $ V $ is almost always invertible, in which case we have $A = V D V^{-1}$ . This is called the eigendecomposition.

The eigenvalues and eigenvectors of a matrix may be complex, even when the matrix is real. However, we can choose real matrices $ V $ and $ D $ satisfying $ A V = V D $ , just like the eigendecomposition, if the matrix $ D $ is not required to be diagonal, but if it is allowed to have blocks of the form

$\begin{bmatrix} u & v \\ -v & u \end{bmatrix}$

(where $ u $ and $ v $ are real numbers) on the diagonal. These blocks correspond to complex eigenvalue pairs $u \pm iv$ . We call this variant of the eigendecomposition the pseudo-eigendecomposition.

Call the function compute() to compute the eigenvalues and eigenvectors of a given matrix. Alternatively, you can use the EigenSolver(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. The pseudoEigenvalueMatrix() and pseudoEigenvectors() methods allow the construction of the pseudo-eigendecomposition.

The documentation for EigenSolver(const MatrixType&, bool) contains an example of the typical use of this class.

Note: The implementation is adapted from JAMA (public domain). Their code is based on EISPACK.

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

Definition at line 64 of file EigenSolver.h.

Member Typedef Documentation

template<typename _MatrixType>

typedef std::complex<RealScalar> Eigen::EigenSolver< _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 90 of file EigenSolver.h.

template<typename _MatrixType>

typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::EigenSolver< _MatrixType >::EigenvalueType

Type for vector of eigenvalues as returned by eigenvalues().

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

Definition at line 97 of file EigenSolver.h.

template<typename _MatrixType>

typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::EigenSolver< _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 104 of file EigenSolver.h.

Constructor & Destructor Documentation

template<typename _MatrixType>

Eigen::EigenSolver< _MatrixType >::EigenSolver ( )

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 113 of file EigenSolver.h.

113 : m_eivec(), m_eivalues(), m_isInitialized(false), m_realSchur(), m_matT(), m_tmp() {}

template<typename _MatrixType>

Eigen::EigenSolver< _MatrixType >::EigenSolver ( Index size )

inline

Default constructor with memory preallocation.

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

See also: EigenSolver()

Definition at line 121 of file EigenSolver.h.

       : m_eivec(size, size),
         m_eivalues(size),
         m_isInitialized(false),
         m_eigenvectorsOk(false),
         m_realSchur(size),
         m_matT(size, size), 
         m_tmp(size)
     {}

template<typename _MatrixType>

Eigen::EigenSolver< _MatrixType >::EigenSolver	(	const MatrixType &	matrix,
		bool	computeEigenvectors = `true`
	)

inline

Constructor; computes eigendecomposition of given matrix.

Parameters

[in]	matrix	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 eigenvalues and eigenvectors.

Example:

Output:

See also: compute()

Definition at line 146 of file EigenSolver.h.

       : m_eivec(matrix.rows(), matrix.cols()),
         m_eivalues(matrix.cols()),
         m_isInitialized(false),
         m_eigenvectorsOk(false),
         m_realSchur(matrix.cols()),
         m_matT(matrix.rows(), matrix.cols()), 
         m_tmp(matrix.cols())
     {
       compute(matrix, computeEigenvectors);
     }

Member Function Documentation

template<typename MatrixType >

EigenSolver< MatrixType > & Eigen::EigenSolver< MatrixType >::compute	(	const MatrixType &	matrix,
		bool	computeEigenvectors = `true`
	)

Computes eigendecomposition of given matrix.

Parameters

[in]	matrix	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 Schur form using the RealSchur class. The Schur decomposition is then used to compute the eigenvalues and eigenvectors.

The cost of the computation is dominated by the cost of the Schur decomposition, which is very approximately $ 25n^3 $ (where $ n $ is the size of the matrix) if computeEigenvectors is true, and $ 10n^3 $ if computeEigenvectors is false.

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

Example:

Output:

Definition at line 365 of file EigenSolver.h.

 {
   using std::sqrt;
   using std::abs;
   eigen_assert(matrix.cols() == matrix.rows());
 
   // Reduce to real Schur form.
   m_realSchur.compute(matrix, computeEigenvectors);
 
   if (m_realSchur.info() == Success)
   {
     m_matT = m_realSchur.matrixT();
     if (computeEigenvectors)
       m_eivec = m_realSchur.matrixU();
   
     // Compute eigenvalues from matT
     m_eivalues.resize(matrix.cols());
     Index i = 0;
     while (i < matrix.cols()) 
     {
       if (i == matrix.cols() - 1 || m_matT.coeff(i+1, i) == Scalar(0)) 
       {
         m_eivalues.coeffRef(i) = m_matT.coeff(i, i);
         ++i;
       }
       else
       {
         Scalar p = Scalar(0.5) * (m_matT.coeff(i, i) - m_matT.coeff(i+1, i+1));
         Scalar z = sqrt(abs(p * p + m_matT.coeff(i+1, i) * m_matT.coeff(i, i+1)));
         m_eivalues.coeffRef(i)   = ComplexScalar(m_matT.coeff(i+1, i+1) + p, z);
         m_eivalues.coeffRef(i+1) = ComplexScalar(m_matT.coeff(i+1, i+1) + p, -z);
         i += 2;
       }
     }
     
     // Compute eigenvectors.
     if (computeEigenvectors)
       doComputeEigenvectors();
   }
 
   m_isInitialized = true;
   m_eigenvectorsOk = computeEigenvectors;
 
   return *this;
 }

template<typename _MatrixType>

const EigenvalueType& Eigen::EigenSolver< _MatrixType >::eigenvalues ( ) const

inline

Returns the eigenvalues of given matrix.

Returns: A const reference to the column vector containing the eigenvalues.

Precondition: Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(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.

Example:

Output:

See also: eigenvectors(), pseudoEigenvalueMatrix(), MatrixBase::eigenvalues()

Definition at line 243 of file EigenSolver.h.

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

template<typename MatrixType >

EigenSolver< MatrixType >::EigenvectorsType Eigen::EigenSolver< MatrixType >::eigenvectors ( ) const

Returns the eigenvectors of given matrix.

Returns: Matrix whose columns are the (possibly complex) eigenvectors.

Precondition: Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before, and computeEigenvectors was set to true (the default).

Column $ k $ of the returned matrix is an eigenvector corresponding to eigenvalue number $ k $ as returned by eigenvalues(). The eigenvectors are normalized to have (Euclidean) norm equal to one. The matrix returned by this function is the matrix $ V $ in the eigendecomposition $A = V D V^{-1}$ , if it exists.

Example:

Output:

See also: eigenvalues(), pseudoEigenvectors()

Definition at line 333 of file EigenSolver.h.

 {
   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);
   for (Index j=0; j<n; ++j)
   {
     if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(j)), numext::real(m_eivalues.coeff(j))) || j+1==n)
     {
       // we have a real eigen value
       matV.col(j) = m_eivec.col(j).template cast<ComplexScalar>();
       matV.col(j).normalize();
     }
     else
     {
       // we have a pair of complex eigen values
       for (Index i=0; i<n; ++i)
       {
         matV.coeffRef(i,j)   = ComplexScalar(m_eivec.coeff(i,j),  m_eivec.coeff(i,j+1));
         matV.coeffRef(i,j+1) = ComplexScalar(m_eivec.coeff(i,j), -m_eivec.coeff(i,j+1));
       }
       matV.col(j).normalize();
       matV.col(j+1).normalize();
       ++j;
     }
   }
   return matV;
 }

template<typename MatrixType >

MatrixType Eigen::EigenSolver< MatrixType >::pseudoEigenvalueMatrix ( ) const

Returns the block-diagonal matrix in the pseudo-eigendecomposition.

Returns: A block-diagonal matrix.

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

The matrix $ D $ returned by this function is real and block-diagonal. The blocks on the diagonal are either 1-by-1 or 2-by-2 blocks of the form $\begin{bmatrix} u & v \\ -v & u \end{bmatrix}$ . These blocks are not sorted in any particular order. The matrix $ D $ and the matrix $ V $ returned by pseudoEigenvectors() satisfy $ AV = VD $ .

See also: pseudoEigenvectors() for an example, eigenvalues()

Definition at line 313 of file EigenSolver.h.

 {
   eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
   Index n = m_eivalues.rows();
   MatrixType matD = MatrixType::Zero(n,n);
   for (Index i=0; i<n; ++i)
   {
     if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i))))
       matD.coeffRef(i,i) = numext::real(m_eivalues.coeff(i));
     else
     {
       matD.template block<2,2>(i,i) <<  numext::real(m_eivalues.coeff(i)), numext::imag(m_eivalues.coeff(i)),
                                        -numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i));
       ++i;
     }
   }
   return matD;
 }

template<typename _MatrixType>

const MatrixType& Eigen::EigenSolver< _MatrixType >::pseudoEigenvectors ( ) const

inline

Returns the pseudo-eigenvectors of given matrix.

Returns: Const reference to matrix whose columns are the pseudo-eigenvectors.

Precondition: Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before, and computeEigenvectors was set to true (the default).

The real matrix $ V $ returned by this function and the block-diagonal matrix $ D $ returned by pseudoEigenvalueMatrix() satisfy $ AV = VD $ .

Example:

Output:

See also: pseudoEigenvalueMatrix(), eigenvectors()

Definition at line 198 of file EigenSolver.h.

     {
       eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
       eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
       return m_eivec;
     }

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

C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/Eigenvalues/EigenSolver.h

Public Types

Public Member Functions

Protected Types

Protected Attributes

Detailed Description

template<typename _MatrixType> class Eigen::EigenSolver< _MatrixType >

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

template<typename _MatrixType>
class Eigen::EigenSolver< _MatrixType >