File EigenvaluesProblems.h

Defines

SPECTRA_EIGENVALUES_SOLVER

Spectra standard eigenvalue problem.

Uncomment the solver the function minPosGeneralizedEigenvalue uses Eigen solver for generalized eigenvalue problem

template<typename NT, typename MT, typename VT>
class EigenvaluesProblems
#include <EigenvaluesProblems.h>

ARPACK++ standard eigenvalues solver.

Solve eigenvalues problems

Template Parameters:

  • NT – Numeric Type

  • MT – Matrix Type

  • VT – Vector Type

template<typename NT>
class EigenvaluesProblems<NT, Eigen::Matrix<NT, Eigen::Dynamic, Eigen::Dynamic>, Eigen::Matrix<NT, Eigen::Dynamic, 1>>
#include <EigenvaluesProblems.h>

A specialization of the template class EigenvaluesProblems for dense Eigen matrices and vectors.

Template Parameters:

NT – Numer Type

Public Types

typedef Eigen::Matrix<NT, Eigen::Dynamic, Eigen::Dynamic> MT

The type for Eigen Matrix.

typedef Eigen::Matrix<NT, Eigen::Dynamic, 1> VT

The type for Eigen vector.

typedef Eigen::GeneralizedEigenSolver<MT>::ComplexVectorType CVT

The type of a complex Eigen Vector for handling eigenvectors.

typedef std::pair<NT, NT> NTpair

The type of a pair of NT.

Public Functions

inline NT findSymEigenvalue(MT const &M)

Find the smallest eigenvalue of M

Parameters:

M – a symmetric matrix

Returns:

smallest eigenvalue

inline NTpair symGeneralizedProblem(MT const &A, MT const &B) const

Find the minimum positive and maximum negative eigenvalues of the generalized eigenvalue problem A + lB, where A, B symmetric and A negative definite.

Parameters:

  • A – [in] Input matrix

  • B – [in] Input matrix

Returns:

The pair (minimum positive, maximum negative) of eigenvalues

inline NT minPosLinearEigenvalue(MT const &A, MT const &B, VT &eigvec)
inline NT minPosGeneralizedEigenvalue(MT const &A, MT const &B, CVT &eigenvector)

Finds the minimum positive real eigenvalue of the generalized eigenvalue problem A + lB and the corresponding eigenvector. If the macro EIGEN_EIGENVALUES_SOLVER is defined, the Generalized Solver of Eigen is used. Otherwise, we transform the generalized to a standard eigenvalue problem and use Spectra. Warning: With Spectra we might get a value smaller than the minimum positive real eigenvalue (the real part of a complex eigenvalue). No restriction on the matrices!

Parameters:

  • A – [in] Input matrix

  • B – [in] Input matrix

  • eigenvector – [out] The eigenvector corresponding to the minimum positive eigenvalue

Returns:

The minimum positive eigenvalue

inline NT minPosLinearEigenvalue(MT const &A, MT const &B, VT &eigvec) const

Find the minimum positive and maximum negative eigenvalues of the generalized eigenvalue problem A + lB, where A, B symmetric and A negative definite.

Parameters:

  • A – [in] Input matrix

  • B – [in] Input matrix

Returns:

The pair (minimum positive, maximum negative) of eigenvalues

inline void linearization(const MT &A, const MT &B, const MT &C, MT &X, MT &Y, bool &updateOnly)

Transform the quadratic eigenvalue problem [At^2 + Bt + c] to the generalized eigenvalue problem X+lY. If the updateOnly flag is false, compute matrices X,Y from scratch; otherwise update them.

Parameters:

  • A – [in]

  • B – [in]

  • C – [in]

  • X – [inout]

  • Y – [inout]

  • updateOnly – [inout] True if X,Y were previously computed and only B,C changed

inline NT minPosQuadraticEigenvalue(MT const &A, MT const &B, MT const &C, MT &X, MT &Y, VT &eigenvector, bool &updateOnly)

Find the minimum positive real eigenvalue of the quadratic eigenvalue problem [At^2 + Bt + c]. First transform it to the generalized eigenvalue problem X+lY. If the updateOnly flag is false, compute matrices X,Y from scratch; otherwise only update them.

Parameters:

  • A – [in] Input matrix

  • B – [in] Input matrix

  • C – [in] Input matrix

  • X – [inout]

  • Y – [inout]

  • eigenvector – [out] The eigenvector corresponding to the minimum positive eigenvalue

  • updateOnly – [inout] True if X,Y were previously computed and only B,C changed

Returns:

Minimum positive eigenvalue

inline bool isPositiveSemidefinite(MT const &A) const
inline bool is_correlation_matrix(const MT &matrix, const double tol = 1e-8)

Check if a matrix is indeed a correlation matrix return true if input matrix is found to be a correlation matrix |param[in] matrix

inline NT minPosLinearEigenvalue_EigenSymSolver(MT const &A, MT const &B, VT &eigvec) const

Minimum positive eigenvalue of the generalized eigenvalue problem A - lB Use Eigen::GeneralizedSelfAdjointEigenSolver<MT> ges(B,A) (faster)

Parameters:

  • A – [in] symmetric positive definite matrix

  • B – [in] symmetric matrix

Returns:

The minimum positive eigenvalue and the corresponding eigenvector