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

Public Types
enum	{ RowsAtCompileTime , ColsAtCompileTime , Options , MaxRowsAtCompileTime , MaxColsAtCompileTime }

typedef std::complex< RealScalar >	ComplexScalar
	Complex scalar type for MatrixType. More...

typedef Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	EigenvalueType
	Type for vector of eigenvalues as returned by eigenvalues(). More...

typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime >	EigenvectorsType
	Type for matrix of eigenvectors as returned by eigenvectors(). More...

typedef Eigen::Index	Index

typedef MatrixType_	MatrixType
	Synonym for the template parameter `MatrixType_`. More...

typedef NumTraits< Scalar >::Real	RealScalar

typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type MatrixType. More...

Public Member Functions
template<typename InputType >
EigenSolver< MatrixType > &	compute (const EigenBase< InputType > &matrix, bool computeEigenvectors)

template<typename InputType >
EigenSolver &	compute (const EigenBase< InputType > &matrix, bool computeEigenvectors=true)
	Computes eigendecomposition of given matrix. More...

	EigenSolver ()
	Default constructor. More...

template<typename InputType >
	EigenSolver (const EigenBase< InputType > &matrix, bool computeEigenvectors=true)
	Constructor; computes eigendecomposition of given matrix. More...

	EigenSolver (Index size)
	Default constructor with memory preallocation. More...

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

EigenvectorsType	eigenvectors () const
	Returns the eigenvectors of given matrix. More...

Index	getMaxIterations ()
	Returns the maximum number of iterations. More...

ComputationInfo	info () const

MatrixType	pseudoEigenvalueMatrix () const
	Returns the block-diagonal matrix in the pseudo-eigendecomposition. More...

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

EigenSolver &	setMaxIterations (Index maxIters)
	Sets the maximum number of iterations allowed. More...

Protected Types
typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	ColumnVectorType

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
bool	m_eigenvectorsOk

EigenvalueType	m_eivalues

MatrixType	m_eivec

ComputationInfo	m_info

bool	m_isInitialized

MatrixType	m_matT

RealSchur< MatrixType >	m_realSchur

ColumnVectorType	m_tmp

Private Member Functions
void	doComputeEigenvectors ()

Detailed Description

template<typename MatrixType_>
class Eigen::EigenSolver< MatrixType_ >

Computes eigenvalues and eigenvectors of general matrices.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues>

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. Currently, only real matrices are supported.

The eigenvalues and eigenvectors of a matrix \( A \) are scalars \( \lambda \) and vectors \( v \) such that \( Av = \lambda v \). If \( D \) is a diagonal matrix with the eigenvalues on the diagonal, and \( V \) is a matrix with the eigenvectors as its columns, then \( A V = V D \). The matrix \( V \) is almost always invertible, in which case we have \( A = V D V^{-1} \). This is called the eigendecomposition.

The eigenvalues and eigenvectors of a matrix may be complex, even when the matrix is real. However, we can choose real matrices \( V \) and \( D \) satisfying \( A V = V D \), just like the eigendecomposition, if the matrix \( D \) is not required to be diagonal, but if it is allowed to have blocks of the form

\[ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \]

(where \( u \) and \( v \) are real numbers) on the diagonal. These blocks correspond to complex eigenvalue pairs \( u \pm iv \). We call this variant of the eigendecomposition the pseudo-eigendecomposition.

Call the function compute() to compute the eigenvalues and eigenvectors of a given matrix. Alternatively, you can use the EigenSolver(const MatrixType&, bool) 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 pseudoEigenvalueMatrix() and pseudoEigenvectors() methods allow the construction of the pseudo-eigendecomposition.

The documentation for EigenSolver(const MatrixType&, bool) contains an example of the typical use of this class.

Note: The implementation is adapted from JAMA (public domain). Their code is based on EISPACK.

See also: MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver

Definition at line 66 of file EigenSolver.h.

Member Typedef Documentation

◆ ColumnVectorType

template<typename MatrixType_ >

typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::EigenSolver< MatrixType_ >::ColumnVectorType

protected

Definition at line 321 of file EigenSolver.h.

◆ ComplexScalar

template<typename MatrixType_ >

typedef std::complex<RealScalar> Eigen::EigenSolver< MatrixType_ >::ComplexScalar

Complex scalar type for MatrixType.

This is std::complex<Scalar> if Scalar is real (e.g., float or double) and just Scalar if Scalar is complex.

Definition at line 92 of file EigenSolver.h.

◆ EigenvalueType

template<typename MatrixType_ >

typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::EigenSolver< MatrixType_ >::EigenvalueType

Type for vector of eigenvalues as returned by eigenvalues().

This is a column vector with entries of type ComplexScalar. The length of the vector is the size of MatrixType.

Definition at line 99 of file EigenSolver.h.

◆ EigenvectorsType

template<typename MatrixType_ >

typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::EigenSolver< MatrixType_ >::EigenvectorsType

Type for matrix of eigenvectors as returned by eigenvectors().

This is a square matrix with entries of type ComplexScalar. The size is the same as the size of MatrixType.

Definition at line 106 of file EigenSolver.h.

◆ Index

template<typename MatrixType_ >

typedef Eigen::Index Eigen::EigenSolver< MatrixType_ >::Index

Deprecated:: since Eigen 3.3

Definition at line 84 of file EigenSolver.h.

◆ MatrixType

template<typename MatrixType_ >

typedef MatrixType_ Eigen::EigenSolver< MatrixType_ >::MatrixType

Synonym for the template parameter MatrixType_.

Definition at line 71 of file EigenSolver.h.

◆ RealScalar

template<typename MatrixType_ >

typedef NumTraits<Scalar>::Real Eigen::EigenSolver< MatrixType_ >::RealScalar

Definition at line 83 of file EigenSolver.h.

◆ Scalar

template<typename MatrixType_ >

typedef MatrixType::Scalar Eigen::EigenSolver< MatrixType_ >::Scalar

Scalar type for matrices of type MatrixType.

Definition at line 82 of file EigenSolver.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ >

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 73 of file EigenSolver.h.

          {
       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
       Options = MatrixType::Options,
       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
     };

Constructor & Destructor Documentation

◆ EigenSolver() [1/3]

template<typename MatrixType_ >

Eigen::EigenSolver< MatrixType_ >::EigenSolver ( )

inline

Default constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via EigenSolver::compute(const MatrixType&, bool).

See also: compute() for an example.

Definition at line 115 of file EigenSolver.h.

115 : m_eivec(), m_eivalues(), m_isInitialized(false), m_eigenvectorsOk(false), m_realSchur(), m_matT(), m_tmp() {}

Eigen::EigenSolver::m_eivec

MatrixType m_eivec

Definition: EigenSolver.h:313

Eigen::EigenSolver::m_realSchur

RealSchur< MatrixType > m_realSchur

Definition: EigenSolver.h:318

Eigen::EigenSolver::m_eigenvectorsOk

bool m_eigenvectorsOk

Definition: EigenSolver.h:316

Eigen::EigenSolver::m_isInitialized

bool m_isInitialized

Definition: EigenSolver.h:315

Eigen::EigenSolver::m_matT

MatrixType m_matT

Definition: EigenSolver.h:319

Eigen::EigenSolver::m_eivalues

EigenvalueType m_eivalues

Definition: EigenSolver.h:314

Eigen::EigenSolver::m_tmp

ColumnVectorType m_tmp

Definition: EigenSolver.h:322

◆ EigenSolver() [2/3]

template<typename MatrixType_ >

Eigen::EigenSolver< MatrixType_ >::EigenSolver ( Index size )

inlineexplicit

Default constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also: EigenSolver()

Definition at line 123 of file EigenSolver.h.

       : m_eivec(size, size),
         m_eivalues(size),
         m_isInitialized(false),
         m_eigenvectorsOk(false),
         m_realSchur(size),
         m_matT(size, size), 
         m_tmp(size)
     {}

◆ EigenSolver() [3/3]

template<typename MatrixType_ >

template<typename InputType >

Eigen::EigenSolver< MatrixType_ >::EigenSolver	(	const EigenBase< InputType > &	matrix,
		bool	computeEigenvectors = `true`
	)

inlineexplicit

Constructor; computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

This constructor calls compute() to compute the eigenvalues and eigenvectors.

Example:

MatrixXd A = MatrixXd::Random(6,6);
cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;
 
EigenSolver<MatrixXd> es(A);
cout << "The eigenvalues of A are:" << endl << es.eigenvalues() << endl;
cout << "The matrix of eigenvectors, V, is:" << endl << es.eigenvectors() << endl << endl;
 
complex<double> lambda = es.eigenvalues()[0];
cout << "Consider the first eigenvalue, lambda = " << lambda << endl;
VectorXcd v = es.eigenvectors().col(0);
cout << "If v is the corresponding eigenvector, then lambda * v = " << endl << lambda * v << endl;
cout << "... and A * v = " << endl << A.cast<complex<double> >() * v << endl << endl;
 
MatrixXcd D = es.eigenvalues().asDiagonal();
MatrixXcd V = es.eigenvectors();
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output:

Here is a random 6x6 matrix, A:
   0.68   -0.33   -0.27  -0.717  -0.687  0.0259
 -0.211   0.536  0.0268   0.214  -0.198   0.678
  0.566  -0.444   0.904  -0.967   -0.74   0.225
  0.597   0.108   0.832  -0.514  -0.782  -0.408
  0.823 -0.0452   0.271  -0.726   0.998   0.275
 -0.605   0.258   0.435   0.608  -0.563  0.0486

The eigenvalues of A are:
  (0.049,1.06)
 (0.049,-1.06)
     (0.967,0)
     (0.353,0)
 (0.618,0.129)
(0.618,-0.129)
The matrix of eigenvectors, V, is:
 (-0.292,-0.454)   (-0.292,0.454)      (-0.0607,0)       (-0.733,0)    (0.59,-0.121)     (0.59,0.121)
  (0.134,-0.104)    (0.134,0.104)       (-0.799,0)        (0.136,0)    (0.334,0.368)   (0.334,-0.368)
  (-0.422,-0.18)    (-0.422,0.18)        (0.192,0)       (0.0563,0)  (-0.335,-0.143)   (-0.335,0.143)
 (-0.589,0.0274) (-0.589,-0.0274)      (-0.0788,0)       (-0.627,0)   (0.322,-0.155)    (0.322,0.155)
  (-0.248,0.132)  (-0.248,-0.132)        (0.401,0)        (0.218,0) (-0.335,-0.0761)  (-0.335,0.0761)
    (0.105,0.18)    (0.105,-0.18)       (-0.392,0)     (-0.00564,0)  (-0.0324,0.103) (-0.0324,-0.103)

Consider the first eigenvalue, lambda = (0.049,1.06)
If v is the corresponding eigenvector, then lambda * v = 
  (0.466,-0.331)
   (0.117,0.137)
   (0.17,-0.456)
(-0.0578,-0.622)
 (-0.152,-0.256)
   (-0.186,0.12)
... and A * v = 
  (0.466,-0.331)
   (0.117,0.137)
   (0.17,-0.456)
(-0.0578,-0.622)
 (-0.152,-0.256)
   (-0.186,0.12)

Finally, V * D * V^(-1) = 
          (0.68,0)  (-0.33,-5.55e-17)   (-0.27,6.66e-16)         (-0.717,0) (-0.687,-8.88e-16) (0.0259,-4.44e-16)
 (-0.211,2.22e-16)   (0.536,3.21e-17)         (0.0268,0)          (0.214,0)         (-0.198,0)  (0.678,-4.44e-16)
  (0.566,4.44e-16)  (-0.444,5.55e-17)   (0.904,1.11e-16) (-0.967,-3.33e-16)   (-0.74,4.44e-16)      (0.225,2e-15)
 (0.597,-4.44e-16)  (0.108,-2.78e-17)   (0.832,3.33e-16) (-0.514,-1.11e-16) (-0.782,-1.33e-15)  (-0.408,1.33e-15)
  (0.823,8.88e-16) (-0.0452,4.16e-17)  (0.271,-3.89e-16) (-0.726,-6.66e-16)   (0.998,1.33e-15)   (0.275,2.22e-15)
(-0.605,-9.71e-17)   (0.258,4.16e-17)  (0.435,-8.33e-17)   (0.608,1.18e-16) (-0.563,-2.78e-16) (0.0486,-2.95e-16)

See also: compute()

Definition at line 149 of file EigenSolver.h.

       : m_eivec(matrix.rows(), matrix.cols()),
         m_eivalues(matrix.cols()),
         m_isInitialized(false),
         m_eigenvectorsOk(false),
         m_realSchur(matrix.cols()),
         m_matT(matrix.rows(), matrix.cols()), 
         m_tmp(matrix.cols())
     {
       compute(matrix.derived(), computeEigenvectors);
     }

Member Function Documentation

◆ check_template_parameters()

template<typename MatrixType_ >

static void Eigen::EigenSolver< MatrixType_ >::check_template_parameters ( )

inlinestaticprotected

Definition at line 307 of file EigenSolver.h.

     {
       EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
       EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL);
     }

◆ compute() [1/2]

template<typename MatrixType_ >

template<typename InputType >

EigenSolver<MatrixType>& Eigen::EigenSolver< MatrixType_ >::compute	(	const EigenBase< InputType > &	matrix,
		bool	computeEigenvectors
	)

Definition at line 381 of file EigenSolver.h.

 {
   check_template_parameters();
   
   using std::sqrt;
   using std::abs;
   using numext::isfinite;
   eigen_assert(matrix.cols() == matrix.rows());
  
   // Reduce to real Schur form.
   m_realSchur.compute(matrix.derived(), computeEigenvectors);
   
   m_info = m_realSchur.info();
  
   if (m_info == Success)
   {
     m_matT = m_realSchur.matrixT();
     if (computeEigenvectors)
       m_eivec = m_realSchur.matrixU();
   
     // Compute eigenvalues from matT
     m_eivalues.resize(matrix.cols());
     Index i = 0;
     while (i < matrix.cols()) 
     {
       if (i == matrix.cols() - 1 || m_matT.coeff(i+1, i) == Scalar(0)) 
       {
         m_eivalues.coeffRef(i) = m_matT.coeff(i, i);
         if(!(isfinite)(m_eivalues.coeffRef(i)))
         {
           m_isInitialized = true;
           m_eigenvectorsOk = false;
           m_info = NumericalIssue;
           return *this;
         }
         ++i;
       }
       else
       {
         Scalar p = Scalar(0.5) * (m_matT.coeff(i, i) - m_matT.coeff(i+1, i+1));
         Scalar z;
         // Compute z = sqrt(abs(p * p + m_matT.coeff(i+1, i) * m_matT.coeff(i, i+1)));
         // without overflow
         {
           Scalar t0 = m_matT.coeff(i+1, i);
           Scalar t1 = m_matT.coeff(i, i+1);
           Scalar maxval = numext::maxi<Scalar>(abs(p),numext::maxi<Scalar>(abs(t0),abs(t1)));
           t0 /= maxval;
           t1 /= maxval;
           Scalar p0 = p/maxval;
           z = maxval * sqrt(abs(p0 * p0 + t0 * t1));
         }
         
         m_eivalues.coeffRef(i)   = ComplexScalar(m_matT.coeff(i+1, i+1) + p, z);
         m_eivalues.coeffRef(i+1) = ComplexScalar(m_matT.coeff(i+1, i+1) + p, -z);
         if(!((isfinite)(m_eivalues.coeffRef(i)) && (isfinite)(m_eivalues.coeffRef(i+1))))
         {
           m_isInitialized = true;
           m_eigenvectorsOk = false;
           m_info = NumericalIssue;
           return *this;
         }
         i += 2;
       }
     }
     
     // Compute eigenvectors.
     if (computeEigenvectors)
       doComputeEigenvectors();
   }
  
   m_isInitialized = true;
   m_eigenvectorsOk = computeEigenvectors;
  
   return *this;
 }

◆ compute() [2/2]

template<typename MatrixType_ >

template<typename InputType >

EigenSolver& Eigen::EigenSolver< MatrixType_ >::compute	(	const EigenBase< InputType > &	matrix,
		bool	computeEigenvectors = `true`
	)

Computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

Returns: Reference to *this

This function computes the eigenvalues of the real matrix matrix. The eigenvalues() function can be used to retrieve them. If computeEigenvectors is true, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().

The matrix is first reduced to real Schur form using the RealSchur class. The Schur decomposition is then used to compute the eigenvalues and eigenvectors.

The cost of the computation is dominated by the cost of the Schur decomposition, which is very approximately \( 25n^3 \) (where \( n \) is the size of the matrix) if computeEigenvectors is true, and \( 10n^3 \) if computeEigenvectors is false.

This method reuses of the allocated data in the EigenSolver object.

Example:

EigenSolver<MatrixXf> es;
MatrixXf A = MatrixXf::Random(4,4);
es.compute(A, /* computeEigenvectors = */ false);
cout << "The eigenvalues of A are: " << es.eigenvalues().transpose() << endl;
es.compute(A + MatrixXf::Identity(4,4), false); // re-use es to compute eigenvalues of A+I
cout << "The eigenvalues of A+I are: " << es.eigenvalues().transpose() << endl;

Output:

The eigenvalues of A are:    (0.755,0.528)   (0.755,-0.528)  (-0.323,0.0965) (-0.323,-0.0965)
The eigenvalues of A+I are:    (1.75,0.528)   (1.75,-0.528)  (0.677,0.0965) (0.677,-0.0965)

◆ doComputeEigenvectors()

template<typename MatrixType >

void Eigen::EigenSolver< MatrixType >::doComputeEigenvectors

private

Definition at line 460 of file EigenSolver.h.

 {
   using std::abs;
   const Index size = m_eivec.cols();
   const Scalar eps = NumTraits<Scalar>::epsilon();
  
   // inefficient! this is already computed in RealSchur
   Scalar norm(0);
   for (Index j = 0; j < size; ++j)
   {
     norm += m_matT.row(j).segment((std::max)(j-1,Index(0)), size-(std::max)(j-1,Index(0))).cwiseAbs().sum();
   }
   
   // Backsubstitute to find vectors of upper triangular form
   if (norm == Scalar(0))
   {
     return;
   }
  
   for (Index n = size-1; n >= 0; n--)
   {
     Scalar p = m_eivalues.coeff(n).real();
     Scalar q = m_eivalues.coeff(n).imag();
  
     // Scalar vector
     if (q == Scalar(0))
     {
       Scalar lastr(0), lastw(0);
       Index l = n;
  
       m_matT.coeffRef(n,n) = Scalar(1);
       for (Index i = n-1; i >= 0; i--)
       {
         Scalar w = m_matT.coeff(i,i) - p;
         Scalar r = m_matT.row(i).segment(l,n-l+1).dot(m_matT.col(n).segment(l, n-l+1));
  
         if (m_eivalues.coeff(i).imag() < Scalar(0))
         {
           lastw = w;
           lastr = r;
         }
         else
         {
           l = i;
           if (m_eivalues.coeff(i).imag() == Scalar(0))
           {
             if (w != Scalar(0))
               m_matT.coeffRef(i,n) = -r / w;
             else
               m_matT.coeffRef(i,n) = -r / (eps * norm);
           }
           else // Solve real equations
           {
             Scalar x = m_matT.coeff(i,i+1);
             Scalar y = m_matT.coeff(i+1,i);
             Scalar denom = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag();
             Scalar t = (x * lastr - lastw * r) / denom;
             m_matT.coeffRef(i,n) = t;
             if (abs(x) > abs(lastw))
               m_matT.coeffRef(i+1,n) = (-r - w * t) / x;
             else
               m_matT.coeffRef(i+1,n) = (-lastr - y * t) / lastw;
           }
  
           // Overflow control
           Scalar t = abs(m_matT.coeff(i,n));
           if ((eps * t) * t > Scalar(1))
             m_matT.col(n).tail(size-i) /= t;
         }
       }
     }
     else if (q < Scalar(0) && n > 0) // Complex vector
     {
       Scalar lastra(0), lastsa(0), lastw(0);
       Index l = n-1;
  
       // Last vector component imaginary so matrix is triangular
       if (abs(m_matT.coeff(n,n-1)) > abs(m_matT.coeff(n-1,n)))
       {
         m_matT.coeffRef(n-1,n-1) = q / m_matT.coeff(n,n-1);
         m_matT.coeffRef(n-1,n) = -(m_matT.coeff(n,n) - p) / m_matT.coeff(n,n-1);
       }
       else
       {
         ComplexScalar cc = ComplexScalar(Scalar(0),-m_matT.coeff(n-1,n)) / ComplexScalar(m_matT.coeff(n-1,n-1)-p,q);
         m_matT.coeffRef(n-1,n-1) = numext::real(cc);
         m_matT.coeffRef(n-1,n) = numext::imag(cc);
       }
       m_matT.coeffRef(n,n-1) = Scalar(0);
       m_matT.coeffRef(n,n) = Scalar(1);
       for (Index i = n-2; i >= 0; i--)
       {
         Scalar ra = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n-1).segment(l, n-l+1));
         Scalar sa = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n).segment(l, n-l+1));
         Scalar w = m_matT.coeff(i,i) - p;
  
         if (m_eivalues.coeff(i).imag() < Scalar(0))
         {
           lastw = w;
           lastra = ra;
           lastsa = sa;
         }
         else
         {
           l = i;
           if (m_eivalues.coeff(i).imag() == RealScalar(0))
           {
             ComplexScalar cc = ComplexScalar(-ra,-sa) / ComplexScalar(w,q);
             m_matT.coeffRef(i,n-1) = numext::real(cc);
             m_matT.coeffRef(i,n) = numext::imag(cc);
           }
           else
           {
             // Solve complex equations
             Scalar x = m_matT.coeff(i,i+1);
             Scalar y = m_matT.coeff(i+1,i);
             Scalar vr = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag() - q * q;
             Scalar vi = (m_eivalues.coeff(i).real() - p) * Scalar(2) * q;
             if ((vr == Scalar(0)) && (vi == Scalar(0)))
               vr = eps * norm * (abs(w) + abs(q) + abs(x) + abs(y) + abs(lastw));
  
             ComplexScalar cc = ComplexScalar(x*lastra-lastw*ra+q*sa,x*lastsa-lastw*sa-q*ra) / ComplexScalar(vr,vi);
             m_matT.coeffRef(i,n-1) = numext::real(cc);
             m_matT.coeffRef(i,n) = numext::imag(cc);
             if (abs(x) > (abs(lastw) + abs(q)))
             {
               m_matT.coeffRef(i+1,n-1) = (-ra - w * m_matT.coeff(i,n-1) + q * m_matT.coeff(i,n)) / x;
               m_matT.coeffRef(i+1,n) = (-sa - w * m_matT.coeff(i,n) - q * m_matT.coeff(i,n-1)) / x;
             }
             else
             {
               cc = ComplexScalar(-lastra-y*m_matT.coeff(i,n-1),-lastsa-y*m_matT.coeff(i,n)) / ComplexScalar(lastw,q);
               m_matT.coeffRef(i+1,n-1) = numext::real(cc);
               m_matT.coeffRef(i+1,n) = numext::imag(cc);
             }
           }
  
           // Overflow control
           Scalar t = numext::maxi<Scalar>(abs(m_matT.coeff(i,n-1)),abs(m_matT.coeff(i,n)));
           if ((eps * t) * t > Scalar(1))
             m_matT.block(i, n-1, size-i, 2) /= t;
  
         }
       }
       
       // We handled a pair of complex conjugate eigenvalues, so need to skip them both
       n--;
     }
     else
     {
       eigen_assert(0 && "Internal bug in EigenSolver (INF or NaN has not been detected)"); // this should not happen
     }
   }
  
   // Back transformation to get eigenvectors of original matrix
   for (Index j = size-1; j >= 0; j--)
   {
     m_tmp.noalias() = m_eivec.leftCols(j+1) * m_matT.col(j).segment(0, j+1);
     m_eivec.col(j) = m_tmp;
   }
 }

◆ eigenvalues()

template<typename MatrixType_ >

const EigenvalueType& Eigen::EigenSolver< MatrixType_ >::eigenvalues ( ) const

inline

Returns the eigenvalues of given matrix.

Returns: A const reference to the column vector containing the eigenvalues.

Precondition: Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before.

The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are not sorted in any particular order.

Example:

MatrixXd ones = MatrixXd::Ones(3,3);
EigenSolver<MatrixXd> es(ones, false);
cout << "The eigenvalues of the 3x3 matrix of ones are:" 
     << endl << es.eigenvalues() << endl;

Output:

The eigenvalues of the 3x3 matrix of ones are:
(-5.31e-17,0)
        (3,0)
        (0,0)

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

Definition at line 246 of file EigenSolver.h.

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

◆ eigenvectors()

template<typename MatrixType >

EigenSolver< MatrixType >::EigenvectorsType Eigen::EigenSolver< MatrixType >::eigenvectors

Returns the eigenvectors of given matrix.

Returns: Matrix whose columns are the (possibly complex) eigenvectors.

Precondition: Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before, and computeEigenvectors was set to true (the default).

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. The matrix returned by this function is the matrix \( V \) in the eigendecomposition \( A = V D V^{-1} \), if it exists.

Example:

MatrixXd ones = MatrixXd::Ones(3,3);
EigenSolver<MatrixXd> es(ones);
cout << "The first eigenvector of the 3x3 matrix of ones is:"
     << endl << es.eigenvectors().col(0) << endl;

Output:

The first eigenvector of the 3x3 matrix of ones is:
(-0.816,0)
 (0.408,0)
 (0.408,0)

See also: eigenvalues(), pseudoEigenvectors()

Definition at line 347 of file EigenSolver.h.

 {
   eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
   eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
   const RealScalar precision = RealScalar(2)*NumTraits<RealScalar>::epsilon();
   Index n = m_eivec.cols();
   EigenvectorsType matV(n,n);
   for (Index j=0; j<n; ++j)
   {
     if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(j)), numext::real(m_eivalues.coeff(j)), precision) || j+1==n)
     {
       // we have a real eigen value
       matV.col(j) = m_eivec.col(j).template cast<ComplexScalar>();
       matV.col(j).normalize();
     }
     else
     {
       // we have a pair of complex eigen values
       for (Index i=0; i<n; ++i)
       {
         matV.coeffRef(i,j)   = ComplexScalar(m_eivec.coeff(i,j),  m_eivec.coeff(i,j+1));
         matV.coeffRef(i,j+1) = ComplexScalar(m_eivec.coeff(i,j), -m_eivec.coeff(i,j+1));
       }
       matV.col(j).normalize();
       matV.col(j+1).normalize();
       ++j;
     }
   }
   return matV;
 }

◆ getMaxIterations()

template<typename MatrixType_ >

Index Eigen::EigenSolver< MatrixType_ >::getMaxIterations ( )

inline

Returns the maximum number of iterations.

Definition at line 297 of file EigenSolver.h.

     {
       return m_realSchur.getMaxIterations();
     }

◆ info()

template<typename MatrixType_ >

ComputationInfo Eigen::EigenSolver< MatrixType_ >::info ( ) const

inline

Returns: NumericalIssue if the input contains INF or NaN values or overflow occurred. Returns Success otherwise.

Definition at line 283 of file EigenSolver.h.

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

◆ pseudoEigenvalueMatrix()

template<typename MatrixType >

MatrixType Eigen::EigenSolver< MatrixType >::pseudoEigenvalueMatrix

Returns the block-diagonal matrix in the pseudo-eigendecomposition.

Returns: A block-diagonal matrix.

Precondition: Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before.

The matrix \( D \) returned by this function is real and block-diagonal. The blocks on the diagonal are either 1-by-1 or 2-by-2 blocks of the form \( \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \). These blocks are not sorted in any particular order. The matrix \( D \) and the matrix \( V \) returned by pseudoEigenvectors() satisfy \( AV = VD \).

See also: pseudoEigenvectors() for an example, eigenvalues()

Definition at line 326 of file EigenSolver.h.

 {
   eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
   const RealScalar precision = RealScalar(2)*NumTraits<RealScalar>::epsilon();
   Index n = m_eivalues.rows();
   MatrixType matD = MatrixType::Zero(n,n);
   for (Index i=0; i<n; ++i)
   {
     if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i)), precision))
       matD.coeffRef(i,i) = numext::real(m_eivalues.coeff(i));
     else
     {
       matD.template block<2,2>(i,i) <<  numext::real(m_eivalues.coeff(i)), numext::imag(m_eivalues.coeff(i)),
                                        -numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i));
       ++i;
     }
   }
   return matD;
 }

◆ pseudoEigenvectors()

template<typename MatrixType_ >

const MatrixType& Eigen::EigenSolver< MatrixType_ >::pseudoEigenvectors ( ) const

inline

Returns the pseudo-eigenvectors of given matrix.

Returns: Const reference to matrix whose columns are the pseudo-eigenvectors.

Precondition: Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before, and computeEigenvectors was set to true (the default).

The real matrix \( V \) returned by this function and the block-diagonal matrix \( D \) returned by pseudoEigenvalueMatrix() satisfy \( AV = VD \).

Example:

MatrixXd A = MatrixXd::Random(6,6);
cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;
 
EigenSolver<MatrixXd> es(A);
MatrixXd D = es.pseudoEigenvalueMatrix();
MatrixXd V = es.pseudoEigenvectors();
cout << "The pseudo-eigenvalue matrix D is:" << endl << D << endl;
cout << "The pseudo-eigenvector matrix V is:" << endl << V << endl;
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output:

Here is a random 6x6 matrix, A:
   0.68   -0.33   -0.27  -0.717  -0.687  0.0259
 -0.211   0.536  0.0268   0.214  -0.198   0.678
  0.566  -0.444   0.904  -0.967   -0.74   0.225
  0.597   0.108   0.832  -0.514  -0.782  -0.408
  0.823 -0.0452   0.271  -0.726   0.998   0.275
 -0.605   0.258   0.435   0.608  -0.563  0.0486

The pseudo-eigenvalue matrix D is:
 0.049   1.06      0      0      0      0
 -1.06  0.049      0      0      0      0
     0      0  0.967      0      0      0
     0      0      0  0.353      0      0
     0      0      0      0  0.618  0.129
     0      0      0      0 -0.129  0.618
The pseudo-eigenvector matrix V is:
  -0.571   -0.888   -0.066    -1.13     17.2    -3.53
   0.263   -0.204   -0.869     0.21     9.73     10.7
  -0.827   -0.352    0.209   0.0871    -9.74    -4.17
   -1.15   0.0535  -0.0857   -0.971     9.36    -4.52
  -0.485    0.258    0.436    0.337    -9.74    -2.21
   0.206    0.353   -0.426 -0.00873   -0.944     2.98
Finally, V * D * V^(-1) = 
   0.68   -0.33   -0.27  -0.717  -0.687  0.0259
 -0.211   0.536  0.0268   0.214  -0.198   0.678
  0.566  -0.444   0.904  -0.967   -0.74   0.225
  0.597   0.108   0.832  -0.514  -0.782  -0.408
  0.823 -0.0452   0.271  -0.726   0.998   0.275
 -0.605   0.258   0.435   0.608  -0.563  0.0486

See also: pseudoEigenvalueMatrix(), eigenvectors()

Definition at line 201 of file EigenSolver.h.

     {
       eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
       eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
       return m_eivec;
     }

◆ setMaxIterations()

template<typename MatrixType_ >

EigenSolver& Eigen::EigenSolver< MatrixType_ >::setMaxIterations ( Index maxIters )

inline

Sets the maximum number of iterations allowed.

Definition at line 290 of file EigenSolver.h.

     {
       m_realSchur.setMaxIterations(maxIters);
       return *this;
     }

Member Data Documentation

◆ m_eigenvectorsOk

template<typename MatrixType_ >

bool Eigen::EigenSolver< MatrixType_ >::m_eigenvectorsOk

protected

Definition at line 316 of file EigenSolver.h.

◆ m_eivalues

template<typename MatrixType_ >

EigenvalueType Eigen::EigenSolver< MatrixType_ >::m_eivalues

protected

Definition at line 314 of file EigenSolver.h.

◆ m_eivec

template<typename MatrixType_ >

MatrixType Eigen::EigenSolver< MatrixType_ >::m_eivec

protected

Definition at line 313 of file EigenSolver.h.

◆ m_info

template<typename MatrixType_ >

ComputationInfo Eigen::EigenSolver< MatrixType_ >::m_info

protected

Definition at line 317 of file EigenSolver.h.

◆ m_isInitialized

template<typename MatrixType_ >

bool Eigen::EigenSolver< MatrixType_ >::m_isInitialized

protected

Definition at line 315 of file EigenSolver.h.

◆ m_matT

template<typename MatrixType_ >

MatrixType Eigen::EigenSolver< MatrixType_ >::m_matT

protected

Definition at line 319 of file EigenSolver.h.

◆ m_realSchur

template<typename MatrixType_ >

RealSchur<MatrixType> Eigen::EigenSolver< MatrixType_ >::m_realSchur

protected

Definition at line 318 of file EigenSolver.h.

◆ m_tmp

template<typename MatrixType_ >

ColumnVectorType Eigen::EigenSolver< MatrixType_ >::m_tmp

protected

Definition at line 322 of file EigenSolver.h.

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

EigenSolver.h

Public Types

Public Member Functions

Protected Types

Static Protected Member Functions

Protected Attributes

Private Member Functions

Detailed Description

template<typename MatrixType_> class Eigen::EigenSolver< MatrixType_ >

Member Typedef Documentation

◆ ColumnVectorType

◆ ComplexScalar

◆ EigenvalueType

◆ EigenvectorsType

◆ Index

◆ MatrixType

◆ RealScalar

◆ Scalar

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ EigenSolver() [1/3]

◆ EigenSolver() [2/3]

◆ EigenSolver() [3/3]

Member Function Documentation

◆ check_template_parameters()

◆ compute() [1/2]

◆ compute() [2/2]

◆ doComputeEigenvectors()

◆ eigenvalues()

◆ eigenvectors()

◆ getMaxIterations()

◆ info()

◆ pseudoEigenvalueMatrix()

◆ pseudoEigenvectors()

◆ setMaxIterations()

Member Data Documentation

◆ m_eigenvectorsOk

◆ m_eivalues

◆ m_eivec

◆ m_info

◆ m_isInitialized

◆ m_matT

◆ m_realSchur

◆ m_tmp

template<typename MatrixType_>
class Eigen::EigenSolver< MatrixType_ >