MOAB  4.9.3pre
Public Types | Public Member Functions | Protected Types | Static Protected Member Functions | Protected Attributes | Private Member Functions
Eigen::EigenSolver< _MatrixType > Class Template Reference

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

#include <EigenSolver.h>

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Public Types

enum  {
  RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
}
typedef _MatrixType MatrixType
 Synonym for the template parameter _MatrixType.
typedef MatrixType::Scalar Scalar
 Scalar type for matrices of type MatrixType.
typedef NumTraits< Scalar >::Real RealScalar
typedef Eigen::Index Index
typedef std::complex< RealScalar > ComplexScalar
 Complex scalar type for MatrixType.
typedef Matrix< ComplexScalar,
ColsAtCompileTime, 1, Options
&~RowMajor,
MaxColsAtCompileTime, 1 > 		EigenvalueType
 Type for vector of eigenvalues as returned by eigenvalues().
typedef Matrix< ComplexScalar,
RowsAtCompileTime,
ColsAtCompileTime, Options,
MaxRowsAtCompileTime,
MaxColsAtCompileTime > 		EigenvectorsType
 Type for matrix of eigenvectors as returned by eigenvectors().

Public Member Functions

 EigenSolver ()
 Default constructor.
 EigenSolver (Index size)
 Default constructor with memory preallocation.
template<typename InputType >
 EigenSolver (const EigenBase< InputType > &matrix, bool computeEigenvectors=true)
 Constructor; computes eigendecomposition of given matrix.
EigenvectorsType eigenvectors () const
 Returns the eigenvectors of given matrix.
const MatrixType & pseudoEigenvectors () const
 Returns the pseudo-eigenvectors of given matrix.
MatrixType pseudoEigenvalueMatrix () const
 Returns the block-diagonal matrix in the pseudo-eigendecomposition.
const EigenvalueType & eigenvalues () const
 Returns the eigenvalues of given matrix.
template<typename InputType >
EigenSolver & compute (const EigenBase< InputType > &matrix, bool computeEigenvectors=true)
 Computes eigendecomposition of given matrix.
ComputationInfo info () const
EigenSolver & setMaxIterations (Index maxIters)
 Sets the maximum number of iterations allowed.
Index getMaxIterations ()
 Returns the maximum number of iterations.

Protected Types

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

Static Protected Member Functions

static void check_template_parameters ()

Protected Attributes

MatrixType m_eivec
EigenvalueType m_eivalues
bool m_isInitialized
bool m_eigenvectorsOk
ComputationInfo m_info
RealSchur< MatrixType > m_realSchur
MatrixType m_matT
ColumnVectorType m_tmp

Private Member Functions

void doComputeEigenvectors ()

Detailed Description

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

Computes eigenvalues and eigenvectors of general matrices.

Template Parameters:
_MatrixTypethe 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 64 of file EigenSolver.h.

Member Typedef Documentation

template<typename _MatrixType>
typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::EigenSolver< _MatrixType >::ColumnVectorType [protected]

Definition at line 319 of file EigenSolver.h.

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 90 of file EigenSolver.h.

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 97 of file EigenSolver.h.

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 104 of file EigenSolver.h.

template<typename _MatrixType>
typedef Eigen::Index Eigen::EigenSolver< _MatrixType >::Index
Deprecated:
since Eigen 3.3

Definition at line 82 of file EigenSolver.h.

template<typename _MatrixType>
typedef _MatrixType Eigen::EigenSolver< _MatrixType >::MatrixType

Synonym for the template parameter _MatrixType.

Definition at line 69 of file EigenSolver.h.

template<typename _MatrixType>
typedef NumTraits<Scalar>::Real Eigen::EigenSolver< _MatrixType >::RealScalar

Definition at line 81 of file EigenSolver.h.

template<typename _MatrixType>
typedef MatrixType::Scalar Eigen::EigenSolver< _MatrixType >::Scalar

Scalar type for matrices of type MatrixType.

Definition at line 80 of file EigenSolver.h.

Member Enumeration Documentation

template<typename _MatrixType>
anonymous enum
Enumerator:
RowsAtCompileTime 
ColsAtCompileTime 
Options 
MaxRowsAtCompileTime 
MaxColsAtCompileTime 

Definition at line 71 of file EigenSolver.h.

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

Constructor & Destructor Documentation

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 113 of file EigenSolver.h.

: m_eivec(), m_eivalues(), m_isInitialized(false), m_realSchur(), m_matT(), m_tmp() {}
template<typename _MatrixType>
Eigen::EigenSolver< _MatrixType >::EigenSolver ( Index  size) [inline, explicit]

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 121 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)
    {}
template<typename _MatrixType>
template<typename InputType >
Eigen::EigenSolver< _MatrixType >::EigenSolver ( const EigenBase< InputType > &  matrix,
bool  computeEigenvectors = true 
) [inline, explicit]

Constructor; computes eigendecomposition of given matrix.

Parameters:
[in]matrixSquare matrix whose eigendecomposition is to be computed.
[in]computeEigenvectorsIf 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: 

 Output: 


See also:
compute() 



Definition at line 147 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








template<typename _MatrixType> 

      

        

          static void Eigen::EigenSolver< _MatrixType >::check_template_parameters 
          (
          )
           [inline, static, protected]
        

      







Definition at line 305 of file EigenSolver.h.


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













template<typename MatrixType > 



template<typename InputType > 

      

        

          EigenSolver< MatrixType > & Eigen::EigenSolver< MatrixType >::compute 
          (
          const EigenBase< InputType > & 
          matrix, 
        

        

          
          
          bool 
          computeEigenvectors = true 
        

        

          
          )
          
        

      







Computes eigendecomposition of given matrix. 


Parameters:

  

    
[in]matrixSquare matrix whose eigendecomposition is to be computed. 

    
[in]computeEigenvectorsIf 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: 


 Output: 


 

Definition at line 377 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;
}













template<typename MatrixType > 

      

        

          void Eigen::EigenSolver< MatrixType >::doComputeEigenvectors 
          (
          )
           [private]
        

      







Definition at line 476 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) = 1.0;
      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
      {
        std::complex<Scalar> cc = cdiv<Scalar>(Scalar(0),-m_matT.coeff(n-1,n),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))
          {
            std::complex<Scalar> cc = cdiv(-ra,-sa,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));

            std::complex<Scalar> cc = cdiv(x*lastra-lastw*ra+q*sa,x*lastsa-lastw*sa-q*ra,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 = cdiv(-lastra-y*m_matT.coeff(i,n-1),-lastsa-y*m_matT.coeff(i,n),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;
  }
}













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: 


 Output: 


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



Definition at line 244 of file EigenSolver.h.


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













template<typename MatrixType > 

      

        

          EigenSolver< MatrixType >::EigenvectorsType Eigen::EigenSolver< MatrixType >::eigenvectors 
          (
          )
           const
        

      







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: 


 Output: 


See also:
eigenvalues(), pseudoEigenvectors() 



Definition at line 344 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.");
  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))) || 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;
}













template<typename _MatrixType> 

      

        

          Index Eigen::EigenSolver< _MatrixType >::getMaxIterations 
          (
          )
           [inline]
        

      







Returns the maximum number of iterations. 



Definition at line 295 of file EigenSolver.h.


    {
      return m_realSchur.getMaxIterations();
    }













template<typename _MatrixType> 

      

        

          ComputationInfo Eigen::EigenSolver< _MatrixType >::info 
          (
          )
           const [inline]
        

      






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



Definition at line 281 of file EigenSolver.h.


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













template<typename MatrixType > 

      

        

          MatrixType Eigen::EigenSolver< MatrixType >::pseudoEigenvalueMatrix 
          (
          )
           const
        

      







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 324 of file EigenSolver.h.


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













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: 


 Output: 


See also:
pseudoEigenvalueMatrix(), eigenvectors() 



Definition at line 199 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;
    }













template<typename _MatrixType> 

      

        

          EigenSolver& Eigen::EigenSolver< _MatrixType >::setMaxIterations 
          (
          Index 
          maxIters)
           [inline]
        

      







Sets the maximum number of iterations allowed. 



Definition at line 288 of file EigenSolver.h.


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







Member Data Documentation








template<typename _MatrixType> 

      

        

          bool Eigen::EigenSolver< _MatrixType >::m_eigenvectorsOk [protected]
        

      







Definition at line 314 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          EigenvalueType Eigen::EigenSolver< _MatrixType >::m_eivalues [protected]
        

      







Definition at line 312 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          MatrixType Eigen::EigenSolver< _MatrixType >::m_eivec [protected]
        

      







Definition at line 311 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          ComputationInfo Eigen::EigenSolver< _MatrixType >::m_info [protected]
        

      







Definition at line 315 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          bool Eigen::EigenSolver< _MatrixType >::m_isInitialized [protected]
        

      







Definition at line 313 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          MatrixType Eigen::EigenSolver< _MatrixType >::m_matT [protected]
        

      







Definition at line 317 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          RealSchur<MatrixType> Eigen::EigenSolver< _MatrixType >::m_realSchur [protected]
        

      







Definition at line 316 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          ColumnVectorType Eigen::EigenSolver< _MatrixType >::m_tmp [protected]
        

      







Definition at line 320 of file EigenSolver.h.







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







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