Eigen::GeneralizedSelfAdjointEigenSolver< MatrixType_ > Class Template Reference

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

Inheritance diagram for Eigen::GeneralizedSelfAdjointEigenSolver< MatrixType_ >:

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

Private Types
typedef SelfAdjointEigenSolver< MatrixType_ >	Base

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

Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen problem.

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.

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

Definition at line 50 of file GeneralizedSelfAdjointEigenSolver.h.

Member Typedef Documentation

Base

template<typename MatrixType_ >

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

private

Definition at line 52 of file GeneralizedSelfAdjointEigenSolver.h.

MatrixType

template<typename MatrixType_ >

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

Definition at line 55 of file GeneralizedSelfAdjointEigenSolver.h.

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.

Definition at line 64 of file GeneralizedSelfAdjointEigenSolver.h.

: 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

Definition at line 78 of file GeneralizedSelfAdjointEigenSolver.h.

        : 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:

Here is a random symmetric 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
and a random positive-definite matrix, B:
  0.132  0.0109 -0.0512  0.0674  -0.143
 0.0109    1.68    1.13   -1.12   0.916
-0.0512    1.13     2.3   -2.14    1.86
 0.0674   -1.12   -2.14    2.69   -2.01
 -0.143   0.916    1.86   -2.01    1.68

The eigenvalues of the pencil (A,B) are:
  -227
  -3.9
-0.837
 0.101
  54.2
The matrix of eigenvectors, V, is:
   14.2   -1.03  0.0766 -0.0273   -8.36
 0.0546  -0.115   0.729   0.478   0.374
  -9.23   0.624 -0.0165   0.499    3.01
   7.88     1.3   0.225   0.109   -3.85
   20.8   0.805  -0.567 -0.0828   -8.73

Consider the first eigenvalue, lambda = -227
If v is the corresponding eigenvector, then A * v = 
 22.8
-28.8
 19.8
 21.9
-25.9
... and lambda * B * v = 
 22.8
-28.8
 19.8
 21.9
-25.9

See also

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

Definition at line 108 of file GeneralizedSelfAdjointEigenSolver.h.

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

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:

The eigenvalues of the pencil (A,B) are:
  0.0289
   0.299
    2.11
    8.64
2.08e+03
The eigenvalues of the pencil (B,A) are:
0.000481
   0.116
   0.473
    3.34
    34.6

See also

GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)

Definition at line 164 of file GeneralizedSelfAdjointEigenSolver.h.

{
  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;
}

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

• GeneralizedSelfAdjointEigenSolver.h