Eigen::SelfAdjointEigenSolver< MatrixType_ > Class Template Reference

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

Inheritance diagram for Eigen::SelfAdjointEigenSolver< MatrixType_ >:

Public Types
enum	{   Size ,   ColsAtCompileTime ,   Options ,   MaxColsAtCompileTime }
typedef Matrix< Scalar, Size, Size, ColMajor, MaxColsAtCompileTime, MaxColsAtCompileTime >	EigenvectorsType
typedef Eigen::Index	Index
typedef MatrixType_	MatrixType
typedef NumTraits< Scalar >::Real	RealScalar
	Real scalar type for MatrixType_. More...
typedef internal::plain_col_type< MatrixType, RealScalar >::type	RealVectorType
typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type MatrixType_. More...
typedef TridiagonalizationType::SubDiagonalType	SubDiagonalType
typedef Tridiagonalization< MatrixType >	TridiagonalizationType
typedef internal::plain_col_type< MatrixType, Scalar >::type	VectorType
	Type for vector of eigenvalues as returned by eigenvalues(). More...

Public Member Functions
template<typename InputType >
SelfAdjointEigenSolver< MatrixType > &	compute (const EigenBase< InputType > &a_matrix, int options)
template<typename InputType >
SelfAdjointEigenSolver &	compute (const EigenBase< InputType > &matrix, int options=ComputeEigenvectors)
	Computes eigendecomposition of given matrix. More...
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...
const RealVectorType &	eigenvalues () const
	Returns the eigenvalues of given matrix. More...
const EigenvectorsType &	eigenvectors () const
	Returns the eigenvectors of given matrix. More...
ComputationInfo	info () const
	Reports whether previous computation was successful. More...
MatrixType	operatorInverseSqrt () const
	Computes the inverse square root of the matrix. More...
MatrixType	operatorSqrt () const
	Computes the positive-definite square root of the matrix. More...
	SelfAdjointEigenSolver ()
	Default constructor for fixed-size matrices. More...
template<typename InputType >
	SelfAdjointEigenSolver (const EigenBase< InputType > &matrix, int options=ComputeEigenvectors)
	Constructor; computes eigendecomposition of given matrix. More...
	SelfAdjointEigenSolver (Index size)
	Constructor, pre-allocates memory for dynamic-size matrices. More...

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

Protected Attributes
bool	m_eigenvectorsOk
RealVectorType	m_eivalues
EigenvectorsType	m_eivec
TridiagonalizationType::CoeffVectorType	m_hcoeffs
ComputationInfo	m_info
bool	m_isInitialized
TridiagonalizationType::SubDiagonalType	m_subdiag
VectorType	m_workspace

Detailed Description

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

Computes eigenvalues and eigenvectors of selfadjoint matrices.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues> 

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

Definition at line 78 of file SelfAdjointEigenSolver.h.

Member Typedef Documentation

EigenvectorsType

template<typename MatrixType_ >

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

Definition at line 94 of file SelfAdjointEigenSolver.h.

Index

template<typename MatrixType_ >

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

Deprecated:

since Eigen 3.3

Definition at line 92 of file SelfAdjointEigenSolver.h.

MatrixType

template<typename MatrixType_ >

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

Definition at line 82 of file SelfAdjointEigenSolver.h.

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.

Definition at line 102 of file SelfAdjointEigenSolver.h.

RealVectorType

template<typename MatrixType_ >

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

Definition at line 112 of file SelfAdjointEigenSolver.h.

Scalar

template<typename MatrixType_ >

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

Scalar type for matrices of type MatrixType_.

Definition at line 91 of file SelfAdjointEigenSolver.h.

SubDiagonalType

template<typename MatrixType_ >

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

Definition at line 114 of file SelfAdjointEigenSolver.h.

TridiagonalizationType

template<typename MatrixType_ >

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

Definition at line 113 of file SelfAdjointEigenSolver.h.

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_.

Definition at line 111 of file SelfAdjointEigenSolver.h.

Member Enumeration Documentation

anonymous enum

template<typename MatrixType_ >

anonymous enum

Enumerator
Size
ColsAtCompileTime
Options
MaxColsAtCompileTime

Definition at line 83 of file SelfAdjointEigenSolver.h.

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

Constructor & Destructor Documentation

SelfAdjointEigenSolver() [1/3]

template<typename MatrixType_ >

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:

The eigenvalues of A are:  -1.58 -0.473   1.32   2.46
The eigenvalues of A+I are: -0.581  0.527   2.32   3.46

Definition at line 127 of file SelfAdjointEigenSolver.h.

        : 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::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

Definition at line 151 of file SelfAdjointEigenSolver.h.

        : 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::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:

Here is a random symmetric 5x5 matrix, A:
  1.36 -0.816  0.521   1.43 -0.144
-0.816 -0.659  0.794 -0.173 -0.406
 0.521  0.794 -0.541  0.461  0.179
  1.43 -0.173  0.461  -1.43  0.822
-0.144 -0.406  0.179  0.822  -1.37

The eigenvalues of A are:
 -2.65
 -1.77
-0.745
 0.227
  2.29
The matrix of eigenvectors, V, is:
 -0.326 -0.0984   0.347 -0.0109   0.874
 -0.207  -0.642   0.228   0.662  -0.232
 0.0495   0.629  -0.164    0.74   0.164
  0.721  -0.397  -0.402   0.115   0.385
 -0.573  -0.156  -0.799 -0.0256  0.0858

Consider the first eigenvalue, lambda = -2.65
If v is the corresponding eigenvector, then lambda * v = 
 0.865
  0.55
-0.131
 -1.91
  1.52
... and A * v = 
 0.865
  0.55
-0.131
 -1.91
  1.52

Finally, V * D * V^(-1) = 
  1.36 -0.816  0.521   1.43 -0.144
-0.816 -0.659  0.794 -0.173 -0.406
 0.521  0.794 -0.541  0.461  0.179
  1.43 -0.173  0.461  -1.43  0.822
-0.144 -0.406  0.179  0.822  -1.37

See also

compute(const MatrixType&, int)

Definition at line 178 of file SelfAdjointEigenSolver.h.

      : 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);
    }

Member Function Documentation

compute() [1/2]

template<typename MatrixType_ >

template<typename InputType >

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

Definition at line 423 of file SelfAdjointEigenSolver.h.

{
  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 >

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:

The eigenvalues of A are:  -1.58 -0.473   1.32   2.46
The eigenvalues of A+I are: -0.581  0.527   2.32   3.46

See also

SelfAdjointEigenSolver(const MatrixType&, int)

computeDirect()

template<typename MatrixType >

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)

Definition at line 829 of file SelfAdjointEigenSolver.h.

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

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

Definition at line 473 of file SelfAdjointEigenSolver.h.

{
  //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;
}

eigenvalues()

template<typename MatrixType_ >

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:

The eigenvalues of the 3x3 matrix of ones are:
-3.09e-16
        0
        3

See also

eigenvectors(), MatrixBase::eigenvalues()

Definition at line 306 of file SelfAdjointEigenSolver.h.

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

eigenvectors()

template<typename MatrixType_ >

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:

The first eigenvector of the 3x3 matrix of ones is:
-0.816
 0.408
 0.408

See also

eigenvalues()

Definition at line 283 of file SelfAdjointEigenSolver.h.

    {
      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;
    }

info()

template<typename MatrixType_ >

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

inline

Reports whether previous computation was successful.

Returns

Success if computation was successful, NoConvergence otherwise.

Definition at line 367 of file SelfAdjointEigenSolver.h.

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

operatorInverseSqrt()

template<typename MatrixType_ >

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:

Here is a random positive-definite matrix, A:
  1.41 -0.697 -0.111  0.508
-0.697  0.423 0.0991   -0.4
-0.111 0.0991   1.25  0.902
 0.508   -0.4  0.902    1.4

The inverse square root of A is: 
  1.88   2.78 -0.546  0.605
  2.78   8.61   -2.3   2.74
-0.546   -2.3   1.92  -1.36
 0.605   2.74  -1.36   2.18
We can also compute it with operatorSqrt() and inverse(). That yields: 
  1.88   2.78 -0.546  0.605
  2.78   8.61   -2.3   2.74
-0.546   -2.3   1.92  -1.36
 0.605   2.74  -1.36   2.18

See also

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

Definition at line 355 of file SelfAdjointEigenSolver.h.

    {
      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();
    }

operatorSqrt()

template<typename MatrixType_ >

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:

Here is a random positive-definite matrix, A:
  1.41 -0.697 -0.111  0.508
-0.697  0.423 0.0991   -0.4
-0.111 0.0991   1.25  0.902
 0.508   -0.4  0.902    1.4

The square root of A is: 
   1.09  -0.432 -0.0685     0.2
 -0.432   0.379   0.141  -0.269
-0.0685   0.141       1   0.468
    0.2  -0.269   0.468    1.04
If we square this, we get: 
  1.41 -0.697 -0.111  0.508
-0.697  0.423 0.0991   -0.4
-0.111 0.0991   1.25  0.902
 0.508   -0.4  0.902    1.4

See also

operatorInverseSqrt(), MatrixFunctions Module

Definition at line 330 of file SelfAdjointEigenSolver.h.

    {
      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();
    }

Member Data Documentation

m_eigenvectorsOk

template<typename MatrixType_ >

bool Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eigenvectorsOk

protected

Definition at line 390 of file SelfAdjointEigenSolver.h.

m_eivalues

template<typename MatrixType_ >

RealVectorType Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivalues

protected

Definition at line 385 of file SelfAdjointEigenSolver.h.

m_eivec

template<typename MatrixType_ >

EigenvectorsType Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_eivec

protected

Definition at line 383 of file SelfAdjointEigenSolver.h.

m_hcoeffs

template<typename MatrixType_ >

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

protected

Definition at line 387 of file SelfAdjointEigenSolver.h.

m_info

template<typename MatrixType_ >

ComputationInfo Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_info

protected

Definition at line 388 of file SelfAdjointEigenSolver.h.

m_isInitialized

template<typename MatrixType_ >

bool Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_isInitialized

protected

Definition at line 389 of file SelfAdjointEigenSolver.h.

m_maxIterations

template<typename MatrixType_ >

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

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).

Definition at line 378 of file SelfAdjointEigenSolver.h.

m_subdiag

template<typename MatrixType_ >

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

protected

Definition at line 386 of file SelfAdjointEigenSolver.h.

m_workspace

template<typename MatrixType_ >

VectorType Eigen::SelfAdjointEigenSolver< MatrixType_ >::m_workspace

protected

Definition at line 384 of file SelfAdjointEigenSolver.h.

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

• SelfAdjointEigenSolver.h