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

#include <SelfAdjointEigenSolver.h>

Inheritance diagram for Eigen::SelfAdjointEigenSolver< _MatrixType >:

Collaboration diagram for Eigen::SelfAdjointEigenSolver< _MatrixType >:

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

typedef _MatrixType	MatrixType

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

typedef MatrixType::Index	Index

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

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

typedef Tridiagonalization< MatrixType >	TridiagonalizationType

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

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

	SelfAdjointEigenSolver (const MatrixType &matrix, int options=ComputeEigenvectors)
	Constructor; computes eigendecomposition of given matrix. More...

SelfAdjointEigenSolver &	compute (const MatrixType &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 direct algorithm. More...

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

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

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

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

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

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

Protected Attributes
MatrixType	m_eivec

RealVectorType	m_eivalues

TridiagonalizationType::SubDiagonalType	m_subdiag

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.

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 $A = V D V^{-1}$ (for selfadjoint matrices, the matrix $ V $ is always invertible). This is called the eigendecomposition.

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 68 of file SelfAdjointEigenSolver.h.

Member Typedef Documentation

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 90 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

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

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 99 of file SelfAdjointEigenSolver.h.

Constructor & Destructor Documentation

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:

Output:

Definition at line 112 of file SelfAdjointEigenSolver.h.

         : m_eivec(),
           m_eivalues(),
           m_subdiag(),
           m_isInitialized(false)
     { }

template<typename _MatrixType>

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

inline

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 131 of file SelfAdjointEigenSolver.h.

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

template<typename _MatrixType>

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

inline

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:

Output:

See also: compute(const MatrixType&, int)

Definition at line 153 of file SelfAdjointEigenSolver.h.

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

Member Function Documentation

template<typename MatrixType >

SelfAdjointEigenSolver< MatrixType > & Eigen::SelfAdjointEigenSolver< MatrixType >::compute	(	const MatrixType &	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:

Output:

See also: SelfAdjointEigenSolver(const MatrixType&, int)

Definition at line 385 of file SelfAdjointEigenSolver.h.

 {
   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_eivalues.coeffRef(0,0) = numext::real(matrix.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;
   MatrixType& 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(scale==RealScalar(0)) scale = RealScalar(1);
   mat.template triangularView<Lower>() /= scale;
   m_subdiag.resize(n-1);
   internal::tridiagonalization_inplace(mat, diag, m_subdiag, computeEigenvectors);
   
   Index end = n-1;
   Index start = 0;
   Index iter = 0; // total number of iterations
 
   while (end>0)
   {
     for (Index i = start; i<end; ++i)
       if (internal::isMuchSmallerThan(abs(m_subdiag[i]),(abs(diag[i])+abs(diag[i+1]))))
         m_subdiag[i] = 0;
 
     // find the largest unreduced block
     while (end>0 && m_subdiag[end-1]==0)
     {
       end--;
     }
     if (end<=0)
       break;
 
     // if we spent too many iterations, we give up
     iter++;
     if(iter > m_maxIterations * n) break;
 
     start = end - 1;
     while (start>0 && m_subdiag[start-1]!=0)
       start--;
 
     internal::tridiagonal_qr_step<MatrixType::Flags&RowMajorBit ? RowMajor : ColMajor>(diag.data(), m_subdiag.data(), start, end, computeEigenvectors ? m_eivec.data() : (Scalar*)0, n);
   }
 
   if (iter <= m_maxIterations * n)
     m_info = Success;
   else
     m_info = NoConvergence;
 
   // Sort eigenvalues and corresponding vectors.
   // TODO make the sort optional ?
   // TODO use a better sort algorithm !!
   if (m_info == Success)
   {
     for (Index i = 0; i < n-1; ++i)
     {
       Index k;
       m_eivalues.segment(i,n-i).minCoeff(&k);
       if (k > 0)
       {
         std::swap(m_eivalues[i], m_eivalues[k+i]);
         if(computeEigenvectors)
           m_eivec.col(i).swap(m_eivec.col(k+i));
       }
     }
   }
   
   // scale back the eigen values
   m_eivalues *= scale;
 
   m_isInitialized = true;
   m_eigenvectorsOk = computeEigenvectors;
   return *this;
 }

template<typename MatrixType >

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

Computes eigendecomposition of given matrix using a direct algorithm.

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

Currently only 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 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.

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

Definition at line 732 of file SelfAdjointEigenSolver.h.

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

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:

Output:

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

Definition at line 250 of file SelfAdjointEigenSolver.h.

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

template<typename _MatrixType>

const MatrixType& 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 $A = V D V^{-1}$ .

Example:

Output:

See also: eigenvalues()

Definition at line 228 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;
     }

template<typename _MatrixType>

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

inline

Reports whether previous computation was successful.

Returns: Success if computation was succesful, NoConvergence otherwise.

Definition at line 310 of file SelfAdjointEigenSolver.h.

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

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:

Output:

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

Definition at line 299 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();
     }

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:

Output:

See also: operatorInverseSqrt(), MatrixFunctions Module

Definition at line 274 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

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).

Definition at line 321 of file SelfAdjointEigenSolver.h.

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

C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/Eigenvalues/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

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

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