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SelfAdjointEigenSolver< _MatrixType > Class Template Reference

Detailed Description

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

Computes eigenvalues and eigenvectors of selfadjoint matrices.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues>
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.

A matrix $ A $ is selfadjoint if it equals its adjoint. For real matrices, this means that the matrix is symmetric: it equals its transpose. This class computes the eigenvalues and eigenvectors of a selfadjoint matrix. These are the scalars $ \lambda $ and vectors $ v $ such that $ Av = \lambda v $. The eigenvalues of a selfadjoint matrix are always real. 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 D V^{-1} $ (for selfadjoint matrices, the matrix $ V $ is always invertible). This is called the eigendecomposition.

The algorithm exploits the fact that the matrix is selfadjoint, making it faster and more accurate than the general purpose eigenvalue algorithms implemented in EigenSolver and ComplexEigenSolver.

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 SelfAdjointEigenSolver(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 SelfAdjointEigenSolver(const MatrixType&, int) contains an example of the typical use of this class.

To solve the generalized eigenvalue problem $ Av = \lambda Bv $ and the likes, see the class GeneralizedSelfAdjointEigenSolver.

See Also
MatrixBase::eigenvalues(), class EigenSolver, class ComplexEigenSolver
+ Inheritance diagram for SelfAdjointEigenSolver< _MatrixType >:

Public Types

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 MatrixType::Scalar Scalar
 Scalar type for matrices of type _MatrixType.
 

Public Member Functions

SelfAdjointEigenSolvercompute (const MatrixType &matrix, int options=ComputeEigenvectors)
 Computes eigendecomposition of given matrix. More...
 
SelfAdjointEigenSolvercomputeDirect (const MatrixType &matrix, int options=ComputeEigenvectors)
 Computes eigendecomposition of given matrix using a direct algorithm. More...
 
const RealVectorTypeeigenvalues () const
 Returns the eigenvalues of given matrix. More...
 
const EigenvectorsTypeeigenvectors () 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...
 
 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...
 

Static Public Attributes

static const int m_maxIterations
 Maximum number of iterations. More...
 

Member Typedef Documentation

typedef NumTraits<Scalar>::Real RealScalar

Real scalar type for _MatrixType.

This is just Scalar if Scalar is real (e.g., float or double), and the type of the real part of Scalar if Scalar is complex.

typedef internal::plain_col_type<MatrixType, RealScalar>::type RealVectorType

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

This is a column vector with entries of type RealScalar. The length of the vector is the size of _MatrixType.

Constructor & Destructor Documentation

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 SelfAdjointEigenSolver(Index) for dynamic-size matrices.

Example:

SelfAdjointEigenSolver<Matrix4f> es;
Matrix4f X = Matrix4f::Random(4,4);
Matrix4f A = X + X.transpose();
es.compute(A);
cout << "The eigenvalues of A are: " << es.eigenvalues().transpose() << endl;
es.compute(A + Matrix4f::Identity(4,4)); // 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:  -1.58 -0.473   1.32   2.46
The eigenvalues of A+I are: -0.581  0.527   2.32   3.46
SelfAdjointEigenSolver ( Index  size)
inline

Constructor, pre-allocates memory for dynamic-size matrices.

Parameters
[in]sizePositive 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
SelfAdjointEigenSolver ( const MatrixType &  matrix,
int  options = ComputeEigenvectors 
)
inline

Constructor; computes eigendecomposition of given matrix.

Parameters
[in]matrixSelfadjoint matrix whose eigendecomposition is to be computed. Only the lower triangular part of the matrix is referenced.
[in]optionsCan be ComputeEigenvectors (default) or EigenvaluesOnly.

This constructor calls compute(const MatrixType&, int) to compute the eigenvalues of the matrix matrix. The eigenvectors are computed if options equals ComputeEigenvectors.

Example:

MatrixXd X = MatrixXd::Random(5,5);
MatrixXd A = X + X.transpose();
cout << "Here is a random symmetric 5x5 matrix, A:" << endl << A << endl << endl;
SelfAdjointEigenSolver<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;
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 lambda * v = " << endl << lambda * v << endl;
cout << "... and A * v = " << endl << A * v << endl << endl;
MatrixXd D = es.eigenvalues().asDiagonal();
MatrixXd V = es.eigenvectors();
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output:

Here is a random symmetric 5x5 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

The eigenvalues of A are:
 -2.65
 -1.77
-0.745
 0.227
  2.29
The matrix of eigenvectors, V, is:
  0.326 -0.0984  -0.347  0.0109   0.874
  0.207  -0.642  -0.228  -0.662  -0.232
-0.0495   0.629   0.164   -0.74   0.164
 -0.721  -0.397   0.402  -0.115   0.385
  0.573  -0.156   0.799  0.0256  0.0858

Consider the first eigenvalue, lambda = -2.65
If v is the corresponding eigenvector, then lambda * v = 
-0.865
 -0.55
 0.131
  1.91
 -1.52
... and A * v = 
-0.865
 -0.55
 0.131
  1.91
 -1.52

Finally, V * D * V^(-1) = 
  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
See Also
compute(const MatrixType&, int)

References SelfAdjointEigenSolver< _MatrixType >::compute().

Member Function Documentation

SelfAdjointEigenSolver< MatrixType > & compute ( const MatrixType &  matrix,
int  options = ComputeEigenvectors 
)

Computes eigendecomposition of given matrix.

Parameters
[in]matrixSelfadjoint matrix whose eigendecomposition is to be computed. Only the lower triangular part of the matrix is referenced.
[in]optionsCan be ComputeEigenvectors (default) or EigenvaluesOnly.
Returns
Reference to *this

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

This implementation uses a symmetric QR algorithm. The matrix is first reduced to tridiagonal form using the Tridiagonalization class. The tridiagonal matrix is then brought to diagonal form with implicit symmetric QR steps with Wilkinson shift. Details can be found in Section 8.3 of Golub & Van Loan, Matrix Computations.

The cost of the computation is about $ 9n^3 $ if the eigenvectors are required and $ 4n^3/3 $ if they are not required.

This method reuses the memory in the SelfAdjointEigenSolver object that was allocated when the object was constructed, if the size of the matrix does not change.

Example:

SelfAdjointEigenSolver<MatrixXf> es(4);
MatrixXf A = X + X.transpose();
es.compute(A);
cout << "The eigenvalues of A are: " << es.eigenvalues().transpose() << endl;
es.compute(A + MatrixXf::Identity(4,4)); // 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:  -1.58 -0.473   1.32   2.46
The eigenvalues of A+I are: -0.581  0.527   2.32   3.46
See Also
SelfAdjointEigenSolver(const MatrixType&, int)

References Eigen::ComputeEigenvectors, Eigen::NoConvergence, PlainObjectBase< Derived >::resize(), and Eigen::Success.

Referenced by SelfAdjointEigenSolver< _MatrixType >::SelfAdjointEigenSolver().

SelfAdjointEigenSolver< MatrixType > & computeDirect ( const MatrixType &  matrix,
int  options = ComputeEigenvectors 
)

Computes eigendecomposition of given matrix using a direct algorithm.

This is a variant of compute(const MatrixType&, int options) which directly solves the underlying polynomial equation.

Currently only 3x3 matrices for which the sizes are known at compile time are supported (e.g., Matrix3d).

This method is usually significantly faster than the QR algorithm but it might also be less accurate. It is also worth noting that for 3x3 matrices it involves trigonometric operations which are not necessarily available for all scalar types.

See Also
compute(const MatrixType&, int options)
const RealVectorType& eigenvalues ( ) const
inline

Returns the eigenvalues of given matrix.

Returns
A const reference to the column vector containing the eigenvalues.
Precondition
The eigenvalues have been computed before.

The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are sorted in increasing order.

Example:

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

Output:

The eigenvalues of the 3x3 matrix of ones are:
-3.09e-16
        0
        3
See Also
eigenvectors(), MatrixBase::eigenvalues()
const EigenvectorsType& eigenvectors ( ) const
inline

Returns the eigenvectors of given matrix.

Returns
A const reference to the matrix whose columns are the eigenvectors.
Precondition
The eigenvectors have been computed before.

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. If this object was used to solve the eigenproblem for the selfadjoint matrix $ A $, then the matrix returned by this function is the matrix $ V $ in the eigendecomposition $ A = V D V^{-1} $.

Example:

MatrixXd ones = MatrixXd::Ones(3,3);
SelfAdjointEigenSolver<MatrixXd> es(ones);
cout << "The first eigenvector of the 3x3 matrix of ones is:"
<< endl << es.eigenvectors().col(1) << endl;

Output:

The first eigenvector of the 3x3 matrix of ones is:
     0
-0.707
 0.707
See Also
eigenvalues()
ComputationInfo info ( ) const
inline

Reports whether previous computation was successful.

Returns
Success if computation was succesful, NoConvergence otherwise.
MatrixType operatorInverseSqrt ( ) const
inline

Computes the inverse square root of the matrix.

Returns
the inverse positive-definite square root of the matrix
Precondition
The eigenvalues and eigenvectors of a positive-definite matrix have been computed before.

This function uses the eigendecomposition $ A = V D V^{-1} $ to compute the inverse square root as $ V D^{-1/2} V^{-1} $. This is cheaper than first computing the square root with operatorSqrt() and then its inverse with MatrixBase::inverse().

Example:

MatrixXd X = MatrixXd::Random(4,4);
MatrixXd A = X * X.transpose();
cout << "Here is a random positive-definite matrix, A:" << endl << A << endl << endl;
SelfAdjointEigenSolver<MatrixXd> es(A);
cout << "The inverse square root of A is: " << endl;
cout << es.operatorInverseSqrt() << endl;
cout << "We can also compute it with operatorSqrt() and inverse(). That yields: " << endl;
cout << es.operatorSqrt().inverse() << endl;

Output:

Here is a random positive-definite matrix, A:
  1.41 -0.697 -0.111  0.508
-0.697  0.423 0.0991   -0.4
-0.111 0.0991   1.25  0.902
 0.508   -0.4  0.902    1.4

The inverse square root of A is: 
  1.88   2.78 -0.546  0.605
  2.78   8.61   -2.3   2.74
-0.546   -2.3   1.92  -1.36
 0.605   2.74  -1.36   2.18
We can also compute it with operatorSqrt() and inverse(). That yields: 
  1.88   2.78 -0.546  0.605
  2.78   8.61   -2.3   2.74
-0.546   -2.3   1.92  -1.36
 0.605   2.74  -1.36   2.18
See Also
operatorSqrt(), MatrixBase::inverse(), MatrixFunctions Module
MatrixType operatorSqrt ( ) const
inline

Computes the positive-definite square root of the matrix.

Returns
the positive-definite square root of the matrix
Precondition
The eigenvalues and eigenvectors of a positive-definite matrix have been computed before.

The square root of a positive-definite matrix $ A $ is the positive-definite matrix whose square equals $ A $. This function uses the eigendecomposition $ A = V D V^{-1} $ to compute the square root as $ A^{1/2} = V D^{1/2} V^{-1} $.

Example:

MatrixXd X = MatrixXd::Random(4,4);
MatrixXd A = X * X.transpose();
cout << "Here is a random positive-definite matrix, A:" << endl << A << endl << endl;
SelfAdjointEigenSolver<MatrixXd> es(A);
MatrixXd sqrtA = es.operatorSqrt();
cout << "The square root of A is: " << endl << sqrtA << endl;
cout << "If we square this, we get: " << endl << sqrtA*sqrtA << endl;

Output:

Here is a random positive-definite matrix, A:
  1.41 -0.697 -0.111  0.508
-0.697  0.423 0.0991   -0.4
-0.111 0.0991   1.25  0.902
 0.508   -0.4  0.902    1.4

The square root of A is: 
   1.09  -0.432 -0.0685     0.2
 -0.432   0.379   0.141  -0.269
-0.0685   0.141       1   0.468
    0.2  -0.269   0.468    1.04
If we square this, we get: 
  1.41 -0.697 -0.111  0.508
-0.697  0.423 0.0991   -0.4
-0.111 0.0991   1.25  0.902
 0.508   -0.4  0.902    1.4
See Also
operatorInverseSqrt(), MatrixFunctions Module

Member Data Documentation

const int m_maxIterations
static

Maximum number of iterations.

The algorithm terminates if it does not converge within m_maxIterations * n iterations, where n denotes the size of the matrix. This value is currently set to 30 (copied from LAPACK).


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