Eigen::GeneralizedSelfAdjointEigenSolver< MatrixType_ > Class Template Reference

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

#include <GeneralizedSelfAdjointEigenSolver.h>

Inheritance diagram for Eigen::GeneralizedSelfAdjointEigenSolver< MatrixType_ >:

Public Types
typedef MatrixType_	MatrixType
Public Types inherited from Eigen::SelfAdjointEigenSolver< MatrixType_ >
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
	GeneralizedSelfAdjointEigenSolver ()
	Default constructor for fixed-size matrices. More...
	GeneralizedSelfAdjointEigenSolver (Index size)
	Constructor, pre-allocates memory for dynamic-size matrices. More...
	GeneralizedSelfAdjointEigenSolver (const MatrixType &matA, const MatrixType &matB, int options=ComputeEigenvectors|Ax_lBx)
	Constructor; computes generalized eigendecomposition of given matrix pencil. More...
GeneralizedSelfAdjointEigenSolver &	compute (const MatrixType &matA, const MatrixType &matB, int options=ComputeEigenvectors|Ax_lBx)
	Computes generalized eigendecomposition of given matrix pencil. More...
Public Member Functions inherited from Eigen::SelfAdjointEigenSolver< MatrixType_ >
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)

Private Types
typedef SelfAdjointEigenSolver< MatrixType_ >	Base

Additional Inherited Members
Static Public Attributes inherited from Eigen::SelfAdjointEigenSolver< MatrixType_ >
static const int	m_maxIterations = 30
	Maximum number of iterations. More...
Protected Attributes inherited from Eigen::SelfAdjointEigenSolver< MatrixType_ >
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

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

MatrixType

template<typename MatrixType_ >

typedef MatrixType_ Eigen::GeneralizedSelfAdjointEigenSolver< MatrixType_ >::MatrixType

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.

: Base() {}

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

: Base(size) {}

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:

MatrixXd X = MatrixXd::Random(5, 5);
MatrixXd A = X + X.transpose();
cout << "Here is a random symmetric matrix, A:" << endl << A << endl;
X = MatrixXd::Random(5, 5);
MatrixXd B = X * X.transpose();
cout << "and a random positive-definite matrix, B:" << endl << B << endl << endl;
 
GeneralizedSelfAdjointEigenSolver<MatrixXd> es(A, B);
cout << "The eigenvalues of the pencil (A,B) 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 A * v = " << endl << A * v << endl;
cout << "... and lambda * B * v = " << endl << lambda * B * v << endl << endl;

Output:

See also

compute(const MatrixType&, const MatrixType&, int)

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

References Eigen::GeneralizedSelfAdjointEigenSolver< MatrixType_ >::compute(), matA(), and matB().

Member Function Documentation

compute()

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

According 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 \( A \) and matB the positive definite matrix \( B \). In addition, each eigenvector \( x \) satisfies the property \( x^* B x = 1 \).

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:

MatrixXd X = MatrixXd::Random(5, 5);
MatrixXd A = X * X.transpose();
X = MatrixXd::Random(5, 5);
MatrixXd B = X * X.transpose();
 
GeneralizedSelfAdjointEigenSolver<MatrixXd> es(A, B, EigenvaluesOnly);
cout << "The eigenvalues of the pencil (A,B) are:" << endl << es.eigenvalues() << endl;
es.compute(B, A, false);
cout << "The eigenvalues of the pencil (B,A) are:" << endl << es.eigenvalues() << endl;

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::PlainObjectBase< Derived >::cols(), compute(), Eigen::ComputeEigenvectors, eigen_assert, Eigen::EigenvaluesOnly, Eigen::EigVecMask, Eigen::GenEigMask, matA(), matB(), Eigen::LLT< MatrixType_, UpLo_ >::matrixL(), Eigen::LLT< MatrixType_, UpLo_ >::matrixU(), Eigen::PlainObjectBase< Derived >::rows(), and compute_granudrum_aor::type.

Referenced by generalized_eigensolver_real(), Eigen::GeneralizedSelfAdjointEigenSolver< MatrixType_ >::GeneralizedSelfAdjointEigenSolver(), and selfadjointeigensolver().

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

• GeneralizedSelfAdjointEigenSolver.h