Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen problem. More...

#include <src/eigen/Eigen/src/Eigenvalues/GeneralizedSelfAdjointEigenSolver.h>

Inheritance diagram for Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >:

Collaboration diagram for Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >:

Public Types
typedef _MatrixType	MatrixType

enum	{ Size = MatrixType::RowsAtCompileTime , ColsAtCompileTime = MatrixType::ColsAtCompileTime , Options = MatrixType::Options , MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }

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

typedef Eigen::Index	Index

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

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

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

typedef Tridiagonalization< MatrixType >	TridiagonalizationType

typedef TridiagonalizationType::SubDiagonalType	SubDiagonalType

Public Member Functions
	GeneralizedSelfAdjointEigenSolver ()
	Default constructor for fixed-size matrices.

	GeneralizedSelfAdjointEigenSolver (Index size)
	Constructor, pre-allocates memory for dynamic-size matrices.

	GeneralizedSelfAdjointEigenSolver (const MatrixType &matA, const MatrixType &matB, int options=ComputeEigenvectors\|Ax_lBx)
	Constructor; computes generalized eigendecomposition of given matrix pencil.

GeneralizedSelfAdjointEigenSolver &	compute (const MatrixType &matA, const MatrixType &matB, int options=ComputeEigenvectors\|Ax_lBx)
	Computes generalized eigendecomposition of given matrix pencil.

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

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

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

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

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

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

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

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

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

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
EigenvectorsType	m_eivec

RealVectorType	m_eivalues

TridiagonalizationType::SubDiagonalType	m_subdiag

ComputationInfo	m_info

bool	m_isInitialized

bool	m_eigenvectorsOk

Private Types
typedef SelfAdjointEigenSolver< _MatrixType >	Base

Detailed Description

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

Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen problem.

\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.

This class solves the generalized eigenvalue problem $Av = \lambda Bv$ . In this case, the matrix $ A $ should be selfadjoint and the matrix $ B $ should be positive definite.

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 GeneralizedSelfAdjointEigenSolver(const MatrixType&, 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 GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int) contains an example of the typical use of this class.

See also: class SelfAdjointEigenSolver, class EigenSolver, class ComplexEigenSolver

Member Typedef Documentation

◆ Base

template<typename _MatrixType >

typedef SelfAdjointEigenSolver<_MatrixType> Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::Base

private

◆ EigenvectorsType

template<typename _MatrixType >

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

inherited

◆ Index

template<typename _MatrixType >

typedef Eigen::Index Eigen::SelfAdjointEigenSolver< _MatrixType >::Index

inherited

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::MatrixType

◆ RealScalar

template<typename _MatrixType >

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

inherited

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

inherited

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.

◆ Scalar

template<typename _MatrixType >

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

inherited

Scalar type for matrices of type _MatrixType.

◆ SubDiagonalType

template<typename _MatrixType >

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

inherited

◆ TridiagonalizationType

template<typename _MatrixType >

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

inherited

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType >

anonymous enum

inherited

Enumerator
Size
ColsAtCompileTime
Options
MaxColsAtCompileTime

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

Constructor & Destructor Documentation

◆ GeneralizedSelfAdjointEigenSolver() [1/3]

template<typename _MatrixType >

Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::GeneralizedSelfAdjointEigenSolver ( )

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 GeneralizedSelfAdjointEigenSolver(Index) for dynamic-size matrices.

62: Base() {}

Eigen::GeneralizedSelfAdjointEigenSolver::Base

SelfAdjointEigenSolver< _MatrixType > Base

Definition GeneralizedSelfAdjointEigenSolver.h:50

◆ GeneralizedSelfAdjointEigenSolver() [2/3]

template<typename _MatrixType >

Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::GeneralizedSelfAdjointEigenSolver ( 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

77 : Base(size)

78 {}

◆ GeneralizedSelfAdjointEigenSolver() [3/3]

template<typename _MatrixType >

Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::GeneralizedSelfAdjointEigenSolver	(	const MatrixType &	matA,
		const MatrixType &	matB,
		int	options = `ComputeEigenvectors\|Ax_lBx`
	)

inline

Constructor; computes generalized eigendecomposition of given matrix pencil.

Parameters

[in]	matA	Selfadjoint matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]	matB	Positive-definite matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]	options	A or-ed set of flags {ComputeEigenvectors,EigenvaluesOnly} \| {Ax_lBx,ABx_lx,BAx_lx}. Default is ComputeEigenvectors\|Ax_lBx.

This constructor calls compute(const MatrixType&, const MatrixType&, int) to compute the eigenvalues and (if requested) the eigenvectors of the generalized eigenproblem $Ax = \lambda B x$ with matA the selfadjoint matrix $ A $ and matB the positive definite matrix $ B $ . Each eigenvector $ x $ satisfies the property $ x^* B x = 1 $ . The eigenvectors are computed if options contains ComputeEigenvectors.

In addition, the two following variants can be solved via options:

ABx_lx: $ABx = \lambda x$
BAx_lx: $BAx = \lambda x$

Example:

Output:

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

      : Base(matA.cols())
    {
      compute(matA, matB, options);
    }

References Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::compute().

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Member Function Documentation

◆ check_template_parameters()

template<typename _MatrixType >

static void Eigen::SelfAdjointEigenSolver< _MatrixType >::check_template_parameters ( )

inlinestaticprotectedinherited

    {
      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
    }

References EIGEN_STATIC_ASSERT_NON_INTEGER.

◆ compute() [1/2]

template<typename _MatrixType >

template<typename InputType >

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

inherited

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:

Output:

See also: SelfAdjointEigenSolver(const MatrixType&, int)

Referenced by Eigen::SelfAdjointEigenSolver< _MatrixType >::SelfAdjointEigenSolver().

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◆ compute() [2/2]

template<typename MatrixType >

GeneralizedSelfAdjointEigenSolver< MatrixType > & Eigen::GeneralizedSelfAdjointEigenSolver< MatrixType >::compute	(	const MatrixType &	matA,
		const MatrixType &	matB,
		int	options = `ComputeEigenvectors\|Ax_lBx`
	)

Computes generalized eigendecomposition of given matrix pencil.

Parameters

[in]	matA	Selfadjoint matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]	matB	Positive-definite matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]	options	A or-ed set of flags {ComputeEigenvectors,EigenvaluesOnly} \| {Ax_lBx,ABx_lx,BAx_lx}. Default is ComputeEigenvectors\|Ax_lBx.

Returns: Reference to *this

Accoring to options, this function computes eigenvalues and (if requested) the eigenvectors of one of the following three generalized eigenproblems:

Ax_lBx: $Ax = \lambda B x$
ABx_lx: $ABx = \lambda x$
BAx_lx: $BAx = \lambda x$ with matA the selfadjoint matrix and matB the positive definite matrix . In addition, each eigenvector satisfies the property .

The eigenvalues() function can be used to retrieve the eigenvalues. If options contains ComputeEigenvectors, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().

The implementation uses LLT to compute the Cholesky decomposition $ B = LL^* $ and computes the classical eigendecomposition of the selfadjoint matrix $L^{-1} A (L^*)^{-1}$ if options contains Ax_lBx and of $L^{*} A L$ otherwise. This solves the generalized eigenproblem, because any solution of the generalized eigenproblem $Ax = \lambda B x$ corresponds to a solution $L^{-1} A (L^*)^{-1} (L^* x) = \lambda (L^* x)$ of the eigenproblem for $L^{-1} A (L^*)^{-1}$ . Similar statements can be made for the two other variants.

Example:

Output:

See also: GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)

{
  eigen_assert(matA.cols()==matA.rows() && matB.rows()==matA.rows() && matB.cols()==matB.rows());
  eigen_assert((options&~(EigVecMask|GenEigMask))==0
          && (options&EigVecMask)!=EigVecMask
          && ((options&GenEigMask)==0 || (options&GenEigMask)==Ax_lBx
           || (options&GenEigMask)==ABx_lx || (options&GenEigMask)==BAx_lx)
          && "invalid option parameter");
 
  bool computeEigVecs = ((options&EigVecMask)==0) || ((options&EigVecMask)==ComputeEigenvectors);
 
  // Compute the cholesky decomposition of matB = L L' = U'U
  LLT<MatrixType> cholB(matB);
 
  int type = (options&GenEigMask);
  if(type==0)
    type = Ax_lBx;
 
  if(type==Ax_lBx)
  {
    // compute C = inv(L) A inv(L')
    MatrixType matC = matA.template selfadjointView<Lower>();
    cholB.matrixL().template solveInPlace<OnTheLeft>(matC);
    cholB.matrixU().template solveInPlace<OnTheRight>(matC);
 
    Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly );
 
    // transform back the eigen vectors: evecs = inv(U) * evecs
    if(computeEigVecs)
      cholB.matrixU().solveInPlace(Base::m_eivec);
  }
  else if(type==ABx_lx)
  {
    // compute C = L' A L
    MatrixType matC = matA.template selfadjointView<Lower>();
    matC = matC * cholB.matrixL();
    matC = cholB.matrixU() * matC;
 
    Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly);
 
    // transform back the eigen vectors: evecs = inv(U) * evecs
    if(computeEigVecs)
      cholB.matrixU().solveInPlace(Base::m_eivec);
  }
  else if(type==BAx_lx)
  {
    // compute C = L' A L
    MatrixType matC = matA.template selfadjointView<Lower>();
    matC = matC * cholB.matrixL();
    matC = cholB.matrixU() * matC;
 
    Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly);
 
    // transform back the eigen vectors: evecs = L * evecs
    if(computeEigVecs)
      Base::m_eivec = cholB.matrixL() * Base::m_eivec;
  }
 
  return *this;
}

References Eigen::ABx_lx, Eigen::Ax_lBx, Eigen::BAx_lx, Eigen::ComputeEigenvectors, eigen_assert, Eigen::EigenvaluesOnly, Eigen::EigVecMask, Eigen::GenEigMask, Eigen::LLT< _MatrixType, _UpLo >::matrixL(), and Eigen::LLT< _MatrixType, _UpLo >::matrixU().

Referenced by Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::GeneralizedSelfAdjointEigenSolver().

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◆ computeDirect()

template<typename MatrixType >

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

inherited

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;
}

Referenced by igl::polar_dec().

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◆ computeFromTridiagonal()

template<typename MatrixType >

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

inherited

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, and Eigen::internal::computeFromTridiagonal_impl().

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◆ eigenvalues()

template<typename _MatrixType >

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

inlineinherited

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:

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 Eigen::SelfAdjointView< _MatrixType, UpLo >::eigenvalues(), CurvatureCalculator::finalEigenStuff(), and igl::polar_dec().

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◆ eigenvectors()

template<typename _MatrixType >

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

inlineinherited

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 $A = V D V^{-1}$ .

Example:

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 CurvatureCalculator::finalEigenStuff(), igl::fit_plane(), and igl::polar_dec().

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◆ info()

template<typename _MatrixType >

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

inlineinherited

Reports whether previous computation was successful.

Returns: Success if computation was succesful, 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.

◆ operatorInverseSqrt()

template<typename _MatrixType >

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

inlineinherited

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:

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.

◆ operatorSqrt()

template<typename _MatrixType >

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

inlineinherited

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:

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.

Member Data Documentation

◆ m_eigenvectorsOk

template<typename _MatrixType >

bool Eigen::SelfAdjointEigenSolver< _MatrixType >::m_eigenvectorsOk

protectedinherited

Referenced by Eigen::SelfAdjointEigenSolver< _MatrixType >::eigenvectors(), Eigen::SelfAdjointEigenSolver< _MatrixType >::operatorInverseSqrt(), and Eigen::SelfAdjointEigenSolver< _MatrixType >::operatorSqrt().

◆ m_eivalues

template<typename _MatrixType >

RealVectorType Eigen::SelfAdjointEigenSolver< _MatrixType >::m_eivalues

protectedinherited

Referenced by Eigen::SelfAdjointEigenSolver< _MatrixType >::eigenvalues(), Eigen::SelfAdjointEigenSolver< _MatrixType >::operatorInverseSqrt(), and Eigen::SelfAdjointEigenSolver< _MatrixType >::operatorSqrt().

◆ m_eivec

template<typename _MatrixType >

EigenvectorsType Eigen::SelfAdjointEigenSolver< _MatrixType >::m_eivec

protectedinherited

Referenced by Eigen::SelfAdjointEigenSolver< _MatrixType >::eigenvectors(), Eigen::SelfAdjointEigenSolver< _MatrixType >::operatorInverseSqrt(), and Eigen::SelfAdjointEigenSolver< _MatrixType >::operatorSqrt().

◆ m_info

template<typename _MatrixType >

ComputationInfo Eigen::SelfAdjointEigenSolver< _MatrixType >::m_info

protectedinherited

Referenced by Eigen::SelfAdjointEigenSolver< _MatrixType >::info().

◆ m_isInitialized

template<typename _MatrixType >

bool Eigen::SelfAdjointEigenSolver< _MatrixType >::m_isInitialized

protectedinherited

Referenced by 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

staticinherited

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).

◆ m_subdiag

template<typename _MatrixType >

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

protectedinherited

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

src/eigen/Eigen/src/Eigenvalues/GeneralizedSelfAdjointEigenSolver.h

Public Types

Public Member Functions

Static Public Attributes

Static Protected Member Functions

Protected Attributes

Private Types

Detailed Description

Member Typedef Documentation

◆ Base

◆ EigenvectorsType

◆ Index

◆ MatrixType

◆ RealScalar

◆ RealVectorType

◆ Scalar

◆ SubDiagonalType

◆ TridiagonalizationType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ GeneralizedSelfAdjointEigenSolver() [1/3]

◆ GeneralizedSelfAdjointEigenSolver() [2/3]

◆ GeneralizedSelfAdjointEigenSolver() [3/3]

Member Function Documentation

◆ check_template_parameters()

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeDirect()

◆ computeFromTridiagonal()

◆ eigenvalues()

◆ eigenvectors()

◆ info()

◆ operatorInverseSqrt()

◆ operatorSqrt()

Member Data Documentation

◆ m_eigenvectorsOk

◆ m_eivalues

◆ m_eivec

◆ m_info

◆ m_isInitialized

◆ m_maxIterations

◆ m_subdiag