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

#include <GeneralizedSelfAdjointEigenSolver.h>

Inheritance diagram for Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >:

Collaboration diagram for Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >:

Public Types
typedef Base::Index	Index

typedef _MatrixType	MatrixType

Public Types inherited from Eigen::SelfAdjointEigenSolver< _MatrixType >
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
	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 >
	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...

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 >
MatrixType	m_eivec

RealVectorType	m_eivalues

TridiagonalizationType::SubDiagonalType	m_subdiag

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.

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 48 of file GeneralizedSelfAdjointEigenSolver.h.

Constructor & Destructor Documentation

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 63 of file GeneralizedSelfAdjointEigenSolver.h.

63 : Base() {}

template<typename _MatrixType >

Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::GeneralizedSelfAdjointEigenSolver ( 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 77 of file GeneralizedSelfAdjointEigenSolver.h.

78 : Base(size)

79 {}

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:

Output:

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

Definition at line 107 of file GeneralizedSelfAdjointEigenSolver.h.

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

Member Function Documentation

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

Accoring 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 and matB the positive definite matrix . In addition, each eigenvector satisfies the property .

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:

Output:

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;
 }

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

C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/Eigenvalues/GeneralizedSelfAdjointEigenSolver.h

Public Types

Public Member Functions

Additional Inherited Members

Detailed Description

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

Constructor & Destructor Documentation

Member Function Documentation

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