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

#include <SelfAdjointEigenSolver.h>

Inheritance diagram for Eigen::SelfAdjointEigenSolver< MatrixType_ >:

Public Types
enum	{ Size = MatrixType::RowsAtCompileTime , ColsAtCompileTime = MatrixType::ColsAtCompileTime , Options = internal::traits<MatrixType>::Options , MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }

typedef MatrixType_	MatrixType

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

typedef Eigen::Index	Index

typedef Matrix< Scalar, Size, Size, ColMajor, MaxColsAtCompileTime, MaxColsAtCompileTime >	EigenvectorsType

typedef NumTraits< Scalar >::Real	RealScalar
	Real scalar type for `MatrixType_`. More...

typedef internal::plain_col_type< MatrixType, Scalar >::type	VectorType
	Type for vector of eigenvalues as returned by eigenvalues(). More...

typedef internal::plain_col_type< MatrixType, RealScalar >::type	RealVectorType

typedef Tridiagonalization< MatrixType >	TridiagonalizationType

typedef TridiagonalizationType::SubDiagonalType	SubDiagonalType

Public Member Functions
EIGEN_DEVICE_FUNC	SelfAdjointEigenSolver ()
	Default constructor for fixed-size matrices. More...

EIGEN_DEVICE_FUNC	SelfAdjointEigenSolver (Index size)
	Constructor, pre-allocates memory for dynamic-size matrices. More...

template<typename InputType >
EIGEN_DEVICE_FUNC	SelfAdjointEigenSolver (const EigenBase< InputType > &matrix, int options=ComputeEigenvectors)
	Constructor; computes eigendecomposition of given matrix. More...

template<typename InputType >
EIGEN_DEVICE_FUNC SelfAdjointEigenSolver &	compute (const EigenBase< InputType > &matrix, int options=ComputeEigenvectors)
	Computes eigendecomposition of given matrix. More...

EIGEN_DEVICE_FUNC SelfAdjointEigenSolver &	computeDirect (const MatrixType &matrix, int options=ComputeEigenvectors)
	Computes eigendecomposition of given matrix using a closed-form algorithm. More...

SelfAdjointEigenSolver &	computeFromTridiagonal (const RealVectorType &diag, const SubDiagonalType &subdiag, int options=ComputeEigenvectors)
	Computes the eigen decomposition from a tridiagonal symmetric matrix. More...

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

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

EIGEN_DEVICE_FUNC MatrixType	operatorSqrt () const
	Computes the positive-definite square root of the matrix. More...

EIGEN_DEVICE_FUNC MatrixType	operatorInverseSqrt () const
	Computes the inverse square root of the matrix. More...

EIGEN_DEVICE_FUNC ComputationInfo	info () const
	Reports whether previous computation was successful. More...

template<typename InputType >
EIGEN_DEVICE_FUNC SelfAdjointEigenSolver< MatrixType > &	compute (const EigenBase< InputType > &a_matrix, int options)

Static Public Attributes
static const int	m_maxIterations = 30
	Maximum number of iterations. More...

Protected Attributes
EigenvectorsType	m_eivec

VectorType	m_workspace

RealVectorType	m_eivalues

TridiagonalizationType::SubDiagonalType	m_subdiag

TridiagonalizationType::CoeffVectorType	m_hcoeffs

ComputationInfo	m_info

bool	m_isInitialized

bool	m_eigenvectorsOk

Friends
struct	internal::direct_selfadjoint_eigenvalues< SelfAdjointEigenSolver, Size, NumTraits< Scalar >::IsComplex >

Detailed Description

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

Computes eigenvalues and eigenvectors of selfadjoint matrices.

\eigenvalues_module

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.

A matrix \( A \) is selfadjoint if it equals its adjoint. For real matrices, this means that the matrix is symmetric: it equals its transpose. This class computes the eigenvalues and eigenvectors of a selfadjoint matrix. These are the scalars \( \lambda \) and vectors \( v \) such that \( Av = \lambda v \). The eigenvalues of a selfadjoint matrix are always real. If \( D \) is a diagonal matrix with the eigenvalues on the diagonal, and \( V \) is a matrix with the eigenvectors as its columns, then \( A = V D V^{-1} \). This is called the eigendecomposition.

For a selfadjoint matrix, \( V \) is unitary, meaning its inverse is equal to its adjoint, \( V^{-1} = V^{\dagger} \). If \( A \) is real, then \( V \) is also real and therefore orthogonal, meaning its inverse is equal to its transpose, \( V^{-1} = V^T \).

The algorithm exploits the fact that the matrix is selfadjoint, making it faster and more accurate than the general purpose eigenvalue algorithms implemented in EigenSolver and ComplexEigenSolver.

Only the lower triangular part of the input matrix is referenced.

Call the function compute() to compute the eigenvalues and eigenvectors of a given matrix. Alternatively, you can use the SelfAdjointEigenSolver(const MatrixType&, int) 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 documentation for SelfAdjointEigenSolver(const MatrixType&, int) contains an example of the typical use of this class.

To solve the generalized eigenvalue problem \( Av = \lambda Bv \) and the likes, see the class GeneralizedSelfAdjointEigenSolver.

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

Member Typedef Documentation

◆ EigenvectorsType

template<typename MatrixType_ >

typedef Matrix<Scalar, Size, Size, ColMajor, MaxColsAtCompileTime, MaxColsAtCompileTime> Eigen::SelfAdjointEigenSolver< MatrixType_ >::EigenvectorsType

◆ Index

template<typename MatrixType_ >

typedef Eigen::Index Eigen::SelfAdjointEigenSolver< MatrixType_ >::Index

Deprecated:: since Eigen 3.3

◆ MatrixType

template<typename MatrixType_ >

typedef MatrixType_ Eigen::SelfAdjointEigenSolver< MatrixType_ >::MatrixType

◆ RealScalar

template<typename MatrixType_ >

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

Real scalar type for MatrixType_.

This is just Scalar if Scalar is real (e.g., float or double), and the type of the real part of Scalar if Scalar is complex.

◆ RealVectorType

template<typename MatrixType_ >

typedef internal::plain_col_type<MatrixType, RealScalar>::type Eigen::SelfAdjointEigenSolver< MatrixType_ >::RealVectorType

◆ Scalar

template<typename MatrixType_ >

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

Scalar type for matrices of type MatrixType_.

◆ SubDiagonalType

template<typename MatrixType_ >

typedef TridiagonalizationType::SubDiagonalType Eigen::SelfAdjointEigenSolver< MatrixType_ >::SubDiagonalType

◆ TridiagonalizationType

template<typename MatrixType_ >

typedef Tridiagonalization<MatrixType> Eigen::SelfAdjointEigenSolver< MatrixType_ >::TridiagonalizationType

◆ VectorType

template<typename MatrixType_ >

typedef internal::plain_col_type<MatrixType, Scalar>::type Eigen::SelfAdjointEigenSolver< MatrixType_ >::VectorType

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

This is a column vector with entries of type RealScalar. The length of the vector is the size of MatrixType_.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ >

anonymous enum

Enumerator
Size
ColsAtCompileTime
Options
MaxColsAtCompileTime

        {
     Size = MatrixType::RowsAtCompileTime,
     ColsAtCompileTime = MatrixType::ColsAtCompileTime,
     Options = internal::traits<MatrixType>::Options,
     MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
   };

Constructor & Destructor Documentation

◆ SelfAdjointEigenSolver() [1/3]

template<typename MatrixType_ >

EIGEN_DEVICE_FUNC Eigen::SelfAdjointEigenSolver< MatrixType_ >::SelfAdjointEigenSolver ( )

inline

Default constructor for fixed-size matrices.

The default constructor is useful in cases in which the user intends to perform decompositions via compute(). This constructor can only be used if MatrixType_ is a fixed-size matrix; use SelfAdjointEigenSolver(Index) for dynamic-size matrices.

Example:

SelfAdjointEigenSolver<Matrix4f> es;
Matrix4f X = Matrix4f::Random(4, 4);
Matrix4f A = X + X.transpose();
es.compute(A);
cout << "The eigenvalues of A are: " << es.eigenvalues().transpose() << endl;
es.compute(A + Matrix4f::Identity(4, 4));  // re-use es to compute eigenvalues of A+I
cout << "The eigenvalues of A+I are: " << es.eigenvalues().transpose() << endl;

Output:

       : m_eivec(),
         m_workspace(),
         m_eivalues(),
         m_subdiag(),
         m_hcoeffs(),
         m_info(InvalidInput),
         m_isInitialized(false),
         m_eigenvectorsOk(false) {}

◆ SelfAdjointEigenSolver() [2/3]

template<typename MatrixType_ >

EIGEN_DEVICE_FUNC Eigen::SelfAdjointEigenSolver< MatrixType_ >::SelfAdjointEigenSolver ( Index size )

inlineexplicit

Constructor, pre-allocates memory for dynamic-size matrices.

Parameters

[in] size Positive integer, size of the matrix whose eigenvalues and eigenvectors will be computed.

This constructor is useful for dynamic-size matrices, when the user intends to perform decompositions via compute(). The size parameter is only used as a hint. It is not an error to give a wrong size, but it may impair performance.

See also: compute() for an example

       : m_eivec(size, size),
         m_workspace(size),
         m_eivalues(size),
         m_subdiag(size > 1 ? size - 1 : 1),
         m_hcoeffs(size > 1 ? size - 1 : 1),
         m_isInitialized(false),
         m_eigenvectorsOk(false) {}

◆ SelfAdjointEigenSolver() [3/3]

template<typename MatrixType_ >

template<typename InputType >

EIGEN_DEVICE_FUNC Eigen::SelfAdjointEigenSolver< MatrixType_ >::SelfAdjointEigenSolver	(	const EigenBase< InputType > &	matrix,
		int	options = `ComputeEigenvectors`
	)

inlineexplicit

Constructor; computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Selfadjoint matrix whose eigendecomposition is to be computed. Only the lower triangular part of the matrix is referenced.
[in]	options	Can be ComputeEigenvectors (default) or EigenvaluesOnly.

This constructor calls compute(const MatrixType&, int) to compute the eigenvalues of the matrix matrix. The eigenvectors are computed if options equals ComputeEigenvectors.

Example:

MatrixXd X = MatrixXd::Random(5, 5);
MatrixXd A = X + X.transpose();
cout << "Here is a random symmetric 5x5 matrix, A:" << endl << A << endl << endl;
 
SelfAdjointEigenSolver<MatrixXd> es(A);
cout << "The eigenvalues of A are:" << endl << es.eigenvalues() << endl;
cout << "The matrix of eigenvectors, V, is:" << endl << es.eigenvectors() << endl << endl;
 
double lambda = es.eigenvalues()[0];
cout << "Consider the first eigenvalue, lambda = " << lambda << endl;
VectorXd v = es.eigenvectors().col(0);
cout << "If v is the corresponding eigenvector, then lambda * v = " << endl << lambda * v << endl;
cout << "... and A * v = " << endl << A * v << endl << endl;
 
MatrixXd D = es.eigenvalues().asDiagonal();
MatrixXd V = es.eigenvectors();
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output:

See also: compute(const MatrixType&, int)

       : m_eivec(matrix.rows(), matrix.cols()),
         m_workspace(matrix.cols()),
         m_eivalues(matrix.cols()),
         m_subdiag(matrix.rows() > 1 ? matrix.rows() - 1 : 1),
         m_hcoeffs(matrix.cols() > 1 ? matrix.cols() - 1 : 1),
         m_isInitialized(false),
         m_eigenvectorsOk(false) {
     compute(matrix.derived(), options);
   }

References Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute(), and matrix().

Member Function Documentation

◆ compute() [1/2]

template<typename MatrixType_ >

template<typename InputType >

EIGEN_DEVICE_FUNC SelfAdjointEigenSolver<MatrixType>& Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute	(	const EigenBase< InputType > &	a_matrix,
		int	options
	)

                                                        {
   const InputType& matrix(a_matrix.derived());
  
   EIGEN_USING_STD(abs);
   eigen_assert(matrix.cols() == matrix.rows());
   eigen_assert((options & ~(EigVecMask | GenEigMask)) == 0 && (options & EigVecMask) != EigVecMask &&
                "invalid option parameter");
   bool computeEigenvectors = (options & ComputeEigenvectors) == ComputeEigenvectors;
   Index n = matrix.cols();
   m_eivalues.resize(n, 1);
  
   if (n == 1) {
     m_eivec = matrix;
     m_eivalues.coeffRef(0, 0) = numext::real(m_eivec.coeff(0, 0));
     if (computeEigenvectors) m_eivec.setOnes(n, n);
     m_info = Success;
     m_isInitialized = true;
     m_eigenvectorsOk = computeEigenvectors;
     return *this;
   }
  
   // declare some aliases
   RealVectorType& diag = m_eivalues;
   EigenvectorsType& mat = m_eivec;
  
   // map the matrix coefficients to [-1:1] to avoid over- and underflow.
   mat = matrix.template triangularView<Lower>();
   RealScalar scale = mat.cwiseAbs().maxCoeff();
   if (numext::is_exactly_zero(scale)) scale = RealScalar(1);
   mat.template triangularView<Lower>() /= scale;
   m_subdiag.resize(n - 1);
   m_hcoeffs.resize(n - 1);
   internal::tridiagonalization_inplace(mat, diag, m_subdiag, m_hcoeffs, m_workspace, computeEigenvectors);
  
   m_info = internal::computeFromTridiagonal_impl(diag, m_subdiag, m_maxIterations, computeEigenvectors, m_eivec);
  
   // scale back the eigen values
   m_eivalues *= scale;
  
   m_isInitialized = true;
   m_eigenvectorsOk = computeEigenvectors;
   return *this;
 }

◆ compute() [2/2]

template<typename MatrixType_ >

template<typename InputType >

EIGEN_DEVICE_FUNC SelfAdjointEigenSolver& Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute	(	const EigenBase< InputType > &	matrix,
		int	options = `ComputeEigenvectors`
	)

Computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Selfadjoint matrix whose eigendecomposition is to be computed. Only the lower triangular part of the matrix is referenced.
[in]	options	Can be ComputeEigenvectors (default) or EigenvaluesOnly.

Returns: Reference to *this

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

This implementation uses a symmetric QR algorithm. The matrix is first reduced to tridiagonal form using the Tridiagonalization class. The tridiagonal matrix is then brought to diagonal form with implicit symmetric QR steps with Wilkinson shift. Details can be found in Section 8.3 of Golub & Van Loan, Matrix Computations.

The cost of the computation is about \( 9n^3 \) if the eigenvectors are required and \( 4n^3/3 \) if they are not required.

This method reuses the memory in the SelfAdjointEigenSolver object that was allocated when the object was constructed, if the size of the matrix does not change.

Example:

SelfAdjointEigenSolver<MatrixXf> es(4);
MatrixXf X = MatrixXf::Random(4, 4);
MatrixXf A = X + X.transpose();
es.compute(A);
cout << "The eigenvalues of A are: " << es.eigenvalues().transpose() << endl;
es.compute(A + MatrixXf::Identity(4, 4));  // re-use es to compute eigenvalues of A+I
cout << "The eigenvalues of A+I are: " << es.eigenvalues().transpose() << endl;

Output:

See also: SelfAdjointEigenSolver(const MatrixType&, int)

Referenced by __attribute__(), ctms_decompositions(), main(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::SelfAdjointEigenSolver(), and selfadjointeigensolver().

◆ computeDirect()

template<typename MatrixType >

EIGEN_DEVICE_FUNC SelfAdjointEigenSolver< MatrixType > & Eigen::SelfAdjointEigenSolver< MatrixType >::computeDirect	(	const MatrixType &	matrix,
		int	options = `ComputeEigenvectors`
	)

Computes eigendecomposition of given matrix using a closed-form algorithm.

This is a variant of compute(const MatrixType&, int options) which directly solves the underlying polynomial equation.

Currently only 2x2 and 3x3 matrices for which the sizes are known at compile time are supported (e.g., Matrix3d).

This method is usually significantly faster than the QR iterative algorithm but it might also be less accurate. It is also worth noting that for 3x3 matrices it involves trigonometric operations which are not necessarily available for all scalar types.

For the 3x3 case, we observed the following worst case relative error regarding the eigenvalues:

double: 1e-8
float: 1e-3

See also: compute(const MatrixType&, int options)

                                            {
   internal::direct_selfadjoint_eigenvalues<SelfAdjointEigenSolver, Size, NumTraits<Scalar>::IsComplex>::run(
       *this, matrix, options);
   return *this;
 }

References matrix(), and Eigen::run().

Referenced by main(), selfadjointeigensolver_essential_check(), and test_eigenvalues_direct().

◆ computeFromTridiagonal()

template<typename MatrixType >

SelfAdjointEigenSolver< MatrixType > & Eigen::SelfAdjointEigenSolver< MatrixType >::computeFromTridiagonal	(	const RealVectorType &	diag,
		const SubDiagonalType &	subdiag,
		int	options = `ComputeEigenvectors`
	)

Computes the eigen decomposition from a tridiagonal symmetric matrix.

Parameters

[in]	diag	The vector containing the diagonal of the matrix.
[in]	subdiag	The subdiagonal of the matrix.
[in]	options	Can be ComputeEigenvectors (default) or EigenvaluesOnly.

Returns: Reference to *this

This function assumes that the matrix has been reduced to tridiagonal form.

See also: compute(const MatrixType&, int) for more information

                                                                              {
   // TODO : Add an option to scale the values beforehand
   bool computeEigenvectors = (options & ComputeEigenvectors) == ComputeEigenvectors;
  
   m_eivalues = diag;
   m_subdiag = subdiag;
   if (computeEigenvectors) {
     m_eivec.setIdentity(diag.size(), diag.size());
   }
   m_info = internal::computeFromTridiagonal_impl(m_eivalues, m_subdiag, m_maxIterations, computeEigenvectors, m_eivec);
  
   m_isInitialized = true;
   m_eigenvectorsOk = computeEigenvectors;
   return *this;
 }

References Eigen::ComputeEigenvectors, Eigen::internal::computeFromTridiagonal_impl(), diag, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eigenvectorsOk, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivalues, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivec, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_info, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_isInitialized, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_maxIterations, and Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_subdiag.

Referenced by selfadjointeigensolver().

◆ eigenvalues()

template<typename MatrixType_ >

EIGEN_DEVICE_FUNC const RealVectorType& Eigen::SelfAdjointEigenSolver< MatrixType_ >::eigenvalues ( ) const

inline

Returns the eigenvalues of given matrix.

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

Precondition: The eigenvalues have been computed before.

The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are sorted in increasing order.

Example:

MatrixXd ones = MatrixXd::Ones(3, 3);
SelfAdjointEigenSolver<MatrixXd> es(ones);
cout << "The eigenvalues of the 3x3 matrix of ones are:" << endl << es.eigenvalues() << endl;

Output:

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

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

References eigen_assert, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivalues, and Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_isInitialized.

Referenced by bug_1225(), EIGEN_DECLARE_TEST(), Eigen::SelfAdjointView< MatrixType_, UpLo >::eigenvalues(), generalized_eigensolver_real(), selfadjointeigensolver(), selfadjointeigensolver_essential_check(), and test_eigenvalues_direct().

◆ eigenvectors()

template<typename MatrixType_ >

EIGEN_DEVICE_FUNC const EigenvectorsType& Eigen::SelfAdjointEigenSolver< MatrixType_ >::eigenvectors ( ) const

inline

Returns the eigenvectors of given matrix.

Returns: A const reference to the matrix whose columns are the eigenvectors.

Precondition: The eigenvectors have been computed before.

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. If this object was used to solve the eigenproblem for the selfadjoint matrix \( A \), then the matrix returned by this function is the matrix \( V \) in the eigendecomposition \( A = V D V^{-1} \).

For a selfadjoint matrix, \( V \) is unitary, meaning its inverse is equal to its adjoint, \( V^{-1} = V^{\dagger} \). If \( A \) is real, then \( V \) is also real and therefore orthogonal, meaning its inverse is equal to its transpose, \( V^{-1} = V^T \).

Example:

MatrixXd ones = MatrixXd::Ones(3, 3);
SelfAdjointEigenSolver<MatrixXd> es(ones);
cout << "The first eigenvector of the 3x3 matrix of ones is:" << endl << es.eigenvectors().col(0) << endl;

Output:

See also: eigenvalues()

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

References eigen_assert, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eigenvectorsOk, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivec, and Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_isInitialized.

Referenced by __attribute__(), EIGEN_DECLARE_TEST(), selfadjointeigensolver(), and selfadjointeigensolver_essential_check().

◆ info()

template<typename MatrixType_ >

EIGEN_DEVICE_FUNC ComputationInfo Eigen::SelfAdjointEigenSolver< MatrixType_ >::info ( ) const

inline

Reports whether previous computation was successful.

Returns: Success if computation was successful, NoConvergence otherwise.

                                                  {
     eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
     return m_info;
   }

References eigen_assert, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_info, and Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_isInitialized.

Referenced by Eigen::internal::computeFromTridiagonal_impl(), EIGEN_DECLARE_TEST(), selfadjointeigensolver(), and selfadjointeigensolver_essential_check().

◆ operatorInverseSqrt()

template<typename MatrixType_ >

EIGEN_DEVICE_FUNC MatrixType Eigen::SelfAdjointEigenSolver< MatrixType_ >::operatorInverseSqrt ( ) const

inline

Computes the inverse square root of the matrix.

Returns: the inverse positive-definite square root of the matrix

Precondition: The eigenvalues and eigenvectors of a positive-definite matrix have been computed before.

This function uses the eigendecomposition \( A = V D V^{-1} \) to compute the inverse square root as \( V D^{-1/2} V^{-1} \). This is cheaper than first computing the square root with operatorSqrt() and then its inverse with MatrixBase::inverse().

Example:

MatrixXd X = MatrixXd::Random(4, 4);
MatrixXd A = X * X.transpose();
cout << "Here is a random positive-definite matrix, A:" << endl << A << endl << endl;
 
SelfAdjointEigenSolver<MatrixXd> es(A);
cout << "The inverse square root of A is: " << endl;
cout << es.operatorInverseSqrt() << endl;
cout << "We can also compute it with operatorSqrt() and inverse(). That yields: " << endl;
cout << es.operatorSqrt().inverse() << endl;

Output:

See also: operatorSqrt(), MatrixBase::inverse(), MatrixFunctions Module

                                                            {
     eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
     eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
     return m_eivec * m_eivalues.cwiseInverse().cwiseSqrt().asDiagonal() * m_eivec.adjoint();
   }

References eigen_assert, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eigenvectorsOk, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivalues, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivec, and Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_isInitialized.

Referenced by selfadjointeigensolver().

◆ operatorSqrt()

template<typename MatrixType_ >

EIGEN_DEVICE_FUNC MatrixType Eigen::SelfAdjointEigenSolver< MatrixType_ >::operatorSqrt ( ) const

inline

Computes the positive-definite square root of the matrix.

Returns: the positive-definite square root of the matrix

Precondition: The eigenvalues and eigenvectors of a positive-definite matrix have been computed before.

The square root of a positive-definite matrix \( A \) is the positive-definite matrix whose square equals \( A \). This function uses the eigendecomposition \( A = V D V^{-1} \) to compute the square root as \( A^{1/2} = V D^{1/2} V^{-1} \).

Example:

MatrixXd X = MatrixXd::Random(4, 4);
MatrixXd A = X * X.transpose();
cout << "Here is a random positive-definite matrix, A:" << endl << A << endl << endl;
 
SelfAdjointEigenSolver<MatrixXd> es(A);
MatrixXd sqrtA = es.operatorSqrt();
cout << "The square root of A is: " << endl << sqrtA << endl;
cout << "If we square this, we get: " << endl << sqrtA * sqrtA << endl;

Output:

See also: operatorInverseSqrt(), MatrixFunctions Module

                                                     {
     eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
     eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
     return m_eivec * m_eivalues.cwiseSqrt().asDiagonal() * m_eivec.adjoint();
   }

References eigen_assert, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eigenvectorsOk, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivalues, Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivec, and Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_isInitialized.

Referenced by selfadjointeigensolver().

Friends And Related Function Documentation

◆ internal::direct_selfadjoint_eigenvalues< SelfAdjointEigenSolver, Size, NumTraits< Scalar >::IsComplex >

template<typename MatrixType_ >

friend struct internal::direct_selfadjoint_eigenvalues< SelfAdjointEigenSolver, Size, NumTraits< Scalar >::IsComplex >

friend

Member Data Documentation

◆ m_eigenvectorsOk

template<typename MatrixType_ >

bool Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eigenvectorsOk

protected

Referenced by Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::computeFromTridiagonal(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::eigenvectors(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::operatorInverseSqrt(), and Eigen::SelfAdjointEigenSolver< MatrixType_ >::operatorSqrt().

◆ m_eivalues

template<typename MatrixType_ >

RealVectorType Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivalues

protected

Referenced by Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::computeFromTridiagonal(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::eigenvalues(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::operatorInverseSqrt(), and Eigen::SelfAdjointEigenSolver< MatrixType_ >::operatorSqrt().

◆ m_eivec

template<typename MatrixType_ >

EigenvectorsType Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivec

protected

Referenced by Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::computeFromTridiagonal(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::eigenvectors(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::operatorInverseSqrt(), and Eigen::SelfAdjointEigenSolver< MatrixType_ >::operatorSqrt().

◆ m_hcoeffs

template<typename MatrixType_ >

TridiagonalizationType::CoeffVectorType Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_hcoeffs

protected

Referenced by Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute().

◆ m_info

template<typename MatrixType_ >

ComputationInfo Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_info

protected

Referenced by Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::computeFromTridiagonal(), and Eigen::SelfAdjointEigenSolver< MatrixType_ >::info().

◆ m_isInitialized

template<typename MatrixType_ >

bool Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_isInitialized

protected

Referenced by Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::computeFromTridiagonal(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::eigenvalues(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::eigenvectors(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::info(), Eigen::SelfAdjointEigenSolver< MatrixType_ >::operatorInverseSqrt(), and Eigen::SelfAdjointEigenSolver< MatrixType_ >::operatorSqrt().

◆ m_maxIterations

template<typename MatrixType_ >

const int Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_maxIterations = 30

static

Maximum number of iterations.

The algorithm terminates if it does not converge within m_maxIterations * n iterations, where n denotes the size of the matrix. This value is currently set to 30 (copied from LAPACK).

Referenced by Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute(), and Eigen::SelfAdjointEigenSolver< MatrixType_ >::computeFromTridiagonal().

◆ m_subdiag

template<typename MatrixType_ >

TridiagonalizationType::SubDiagonalType Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_subdiag

protected

Referenced by Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute(), and Eigen::SelfAdjointEigenSolver< MatrixType_ >::computeFromTridiagonal().

◆ m_workspace

template<typename MatrixType_ >

VectorType Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_workspace

protected

Referenced by Eigen::SelfAdjointEigenSolver< MatrixType_ >::compute().

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

SelfAdjointEigenSolver.h

Public Types

Public Member Functions

Static Public Attributes

Protected Attributes

Friends

Detailed Description

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

Member Typedef Documentation

◆ EigenvectorsType

◆ Index

◆ MatrixType

◆ RealScalar

◆ RealVectorType

◆ Scalar

◆ SubDiagonalType

◆ TridiagonalizationType

◆ VectorType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ SelfAdjointEigenSolver() [1/3]

◆ SelfAdjointEigenSolver() [2/3]

◆ SelfAdjointEigenSolver() [3/3]

Member Function Documentation

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeDirect()

◆ computeFromTridiagonal()

◆ eigenvalues()

◆ eigenvectors()

◆ info()

◆ operatorInverseSqrt()

◆ operatorSqrt()

Friends And Related Function Documentation

◆ internal::direct_selfadjoint_eigenvalues< SelfAdjointEigenSolver, Size, NumTraits< Scalar >::IsComplex >

Member Data Documentation

◆ m_eigenvectorsOk

◆ m_eivalues

◆ m_eivec

◆ m_hcoeffs

◆ m_info

◆ m_isInitialized

◆ m_maxIterations

◆ m_subdiag

◆ m_workspace

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