template<typename _MatrixType>

Eigen::SelfAdjointEigenSolver class

template<typename _MatrixType>

typedef Eigen::Index Eigen::SelfAdjointEigenSolver<_MatrixType>::Index

template<typename _MatrixType>

typedef NumTraits<Scalar>::Real Eigen::SelfAdjointEigenSolver<_MatrixType>::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.

template<typename _MatrixType>

typedef internal::plain_col_type<MatrixType, RealScalar>::type Eigen::SelfAdjointEigenSolver<_MatrixType>::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.

template<typename _MatrixType>

Eigen::SelfAdjointEigenSolver<_MatrixType>::SelfAdjointEigenSolver()

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

template<typename _MatrixType>

Eigen::SelfAdjointEigenSolver<_MatrixType>::SelfAdjointEigenSolver(Index size) explicit

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

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.

template<typename _MatrixType> template<typename InputType>

Eigen::SelfAdjointEigenSolver<_MatrixType>::SelfAdjointEigenSolver(const EigenBase<InputType>& matrix, int options = ComputeEigenvectors) explicit

Constructor; computes eigendecomposition of given matrix.

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

template<typename _MatrixType> template<typename InputType>

SelfAdjointEigenSolver& Eigen::SelfAdjointEigenSolver<_MatrixType>::compute(const EigenBase<InputType>& matrix, int options = ComputeEigenvectors)

Computes eigendecomposition of given matrix.

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 if the eigenvectors are required and 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 X = MatrixXf::Random(4,4);
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

template<typename _MatrixType>

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

Computes eigendecomposition of given matrix using a closed-form algorithm.

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

Currently only 2x2 and 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 iterative 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.

For the 3x3 case, we observed the following worst case relative error regarding the eigenvalues:

• double: 1e-8
• float: 1e-3

template<typename _MatrixType>

SelfAdjointEigenSolver& Eigen::SelfAdjointEigenSolver<_MatrixType>::computeFromTridiagonal(const RealVectorType& diag, const SubDiagonalType& subdiag, int options = ComputeEigenvectors)

Computes the eigen decomposition from a tridiagonal symmetric matrix.

This function assumes that the matrix has been reduced to tridiagonal form.

template<typename _MatrixType>

const RealVectorType& Eigen::SelfAdjointEigenSolver<_MatrixType>::eigenvalues() const

Returns the eigenvalues of given matrix.

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

template<typename _MatrixType>

const EigenvectorsType& Eigen::SelfAdjointEigenSolver<_MatrixType>::eigenvectors() const

Returns the eigenvectors of given matrix.

Column of the returned matrix is an eigenvector corresponding to eigenvalue number 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 , then the matrix returned by this function is the matrix in the eigendecomposition .

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

template<typename _MatrixType>

ComputationInfo Eigen::SelfAdjointEigenSolver<_MatrixType>::info() const

Reports whether previous computation was successful.

template<typename _MatrixType>

MatrixType Eigen::SelfAdjointEigenSolver<_MatrixType>::operatorInverseSqrt() const

Computes the inverse square root of the matrix.

This function uses the eigendecomposition to compute the inverse square root as . 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

template<typename _MatrixType>

MatrixType Eigen::SelfAdjointEigenSolver<_MatrixType>::operatorSqrt() const

Computes the positive-definite square root of the matrix.

The square root of a positive-definite matrix is the positive-definite matrix whose square equals . This function uses the eigendecomposition to compute the square root as .

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

template<typename _MatrixType>

static const int Eigen::SelfAdjointEigenSolver<_MatrixType>::m_maxIterations

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).