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Document SelfAdjointEigenSolver and add examples.

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Jitse Niesen 2010-05-04 17:11:32 +01:00
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271
Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h
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@ -30,13 +30,42 @@
*
* \class SelfAdjointEigenSolver
*
* \brief Eigen values/vectors solver for selfadjoint matrix
* \brief Computes eigenvalues and eigenvectors of selfadjoint matrices
*
* \param MatrixType the type of the matrix of which we are computing the eigen decomposition
* \tparam _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.
*
* \note MatrixType must be an actual Matrix type, it can't be an expression type.
* A matrix \f$ A \f$ 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 \f$ \lambda \f$ and vectors
* \f$ v \f$ such that \f$ Av = \lambda v \f$. The eigenvalues of a
* selfadjoint matrix are always real. If \f$ D \f$ is a diagonal matrix with
* the eigenvalues on the diagonal, and \f$ V \f$ is a matrix with the
* eigenvectors as its columns, then \f$ A = V D V^{-1} \f$ (for selfadjoint
* matrices, the matrix \f$ V \f$ is always invertible). This is called the
* eigendecomposition.
*
* \sa MatrixBase::eigenvalues(), class EigenSolver
* 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.
*
* This class can also be used to solve the generalized eigenvalue problem
* \f$ Av = \lambda Bv \f$. In this case, the matrix \f$ A \f$ should be
* selfadjoint and the matrix \f$ B \f$ should be positive definite.
*
* Call the function compute() to compute the eigenvalues and eigenvectors of
* a given matrix. Alternatively, you can use the
* SelfAdjointEigenSolver(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 documentation for SelfAdjointEigenSolver(const MatrixType&, bool)
* contains an example of the typical use of this class.
*
* \sa MatrixBase::eigenvalues(), class EigenSolver, class ComplexEigenSolver
*/
template<typename _MatrixType> class SelfAdjointEigenSolver
{
@ -49,13 +78,37 @@ template<typename _MatrixType> class SelfAdjointEigenSolver
Options = MatrixType::Options,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
};
/** \brief Scalar type for matrices of type \p _MatrixType. */
typedef typename MatrixType::Scalar Scalar;
/** \brief Real scalar type for \p _MatrixType.
*
* This is just \c Scalar if #Scalar is real (e.g., \c float or
* \c double), and the type of the real part of \c Scalar if #Scalar is
* complex.
*/
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef std::complex<RealScalar> Complex;
/** \brief 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 \p _MatrixType.
*/
typedef typename ei_plain_col_type<MatrixType, RealScalar>::type RealVectorType;
typedef Tridiagonalization<MatrixType> TridiagonalizationType;
// typedef typename TridiagonalizationType::TridiagonalMatrixType TridiagonalMatrixType;
/** \brief Default constructor for fixed-size matrices.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via compute(const MatrixType&, bool) or
* compute(const MatrixType&, const MatrixType&, bool). This constructor
* can only be used if \p _MatrixType is a fixed-size matrix; use
* SelfAdjointEigenSolver(int) for dynamic-size matrices.
*
* Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver.cpp
* Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver.out
*/
SelfAdjointEigenSolver()
: m_eivec(int(Size), int(Size)),
m_eivalues(int(Size)),
@ -64,16 +117,42 @@ template<typename _MatrixType> class SelfAdjointEigenSolver
ei_assert(Size!=Dynamic);
}
/** \brief Constructor, pre-allocates memory for dynamic-size matrices.
*
* \param [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(const MatrixType&, bool)
* or compute(const MatrixType&, const MatrixType&, bool). The \p size
* parameter is only used as a hint. It is not an error to give a wrong
* \p size, but it may impair performance.
*
* \sa compute(const MatrixType&, bool) for an example
*/
SelfAdjointEigenSolver(int size)
: m_eivec(size, size),
m_eivalues(size),
m_subdiag(TridiagonalizationType::SizeMinusOne)
{}
/** Constructors computing the eigenvalues of the selfadjoint matrix \a matrix,
* as well as the eigenvectors if \a computeEigenvectors is true.
/** \brief Constructor; computes eigendecomposition of given matrix.
*
* \param[in] matrix Selfadjoint matrix whose eigendecomposition is to
* be computed.
* \param[in] computeEigenvectors If true, both the eigenvectors and the
* eigenvalues are computed; if false, only the eigenvalues are
* computed.
*
* \sa compute(MatrixType,bool), SelfAdjointEigenSolver(MatrixType,MatrixType,bool)
* This constructor calls compute(const MatrixType&, bool) to compute the
* eigenvalues of the matrix \p matrix. The eigenvectors are computed if
* \p computeEigenvectors is true.
*
* Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType.cpp
* Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType.out
*
* \sa compute(const MatrixType&, bool),
* SelfAdjointEigenSolver(const MatrixType&, const MatrixType&, bool)
*/
SelfAdjointEigenSolver(const MatrixType& matrix, bool computeEigenvectors = true)
: m_eivec(matrix.rows(), matrix.cols()),
@ -84,12 +163,26 @@ template<typename _MatrixType> class SelfAdjointEigenSolver
compute(matrix, computeEigenvectors);
}
/** Constructors computing the eigenvalues of the generalized eigen problem
* \f$ Ax = lambda B x \f$ with \a matA the selfadjoint matrix \f$ A \f$
* and \a matB the positive definite matrix \f$ B \f$ . The eigenvectors
* are computed if \a computeEigenvectors is true.
/** \brief Constructor; computes eigendecomposition of given matrix pencil.
*
* \param[in] matA Selfadjoint matrix in matrix pencil.
* \param[in] matB Positive-definite matrix in matrix pencil.
* \param[in] computeEigenvectors If true, both the eigenvectors and the
* eigenvalues are computed; if false, only the eigenvalues are
* computed.
*
* \sa compute(MatrixType,MatrixType,bool), SelfAdjointEigenSolver(MatrixType,bool)
* This constructor calls compute(const MatrixType&, const MatrixType&, bool)
* to compute the eigenvalues and (if requested) the eigenvectors of the
* generalized eigenproblem \f$ Ax = \lambda B x \f$ with \a matA the
* selfadjoint matrix \f$ A \f$ and \a matB the positive definite matrix
* \f$ B \f$ . The eigenvectors are computed if \a computeEigenvectors is
* true.
*
* Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType2.cpp
* Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType2.out
*
* \sa compute(const MatrixType&, const MatrixType&, bool),
* SelfAdjointEigenSolver(const MatrixType&, bool)
*/
SelfAdjointEigenSolver(const MatrixType& matA, const MatrixType& matB, bool computeEigenvectors = true)
: m_eivec(matA.rows(), matA.cols()),
@ -100,12 +193,91 @@ template<typename _MatrixType> class SelfAdjointEigenSolver
compute(matA, matB, computeEigenvectors);
}
/** \brief Computes eigendecomposition of given matrix.
*
* \param[in] matrix Selfadjoint matrix whose eigendecomposition is to
* be computed.
* \param[in] computeEigenvectors If true, both the eigenvectors and the
* eigenvalues are computed; if false, only the eigenvalues are
* computed.
* \returns Reference to \c *this
*
* This function computes the eigenvalues of \p matrix. The eigenvalues()
* function can be used to retrieve them. If \p computeEigenvectors is
* true, 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, <i>%Matrix Computations</i>.
*
* The cost of the computation is about \f$ 9n^3 \f$ if the eigenvectors
* are required and \f$ 4n^3/3 \f$ 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: \include SelfAdjointEigenSolver_compute_MatrixType.cpp
* Output: \verbinclude SelfAdjointEigenSolver_compute_MatrixType.out
*
* \sa SelfAdjointEigenSolver(const MatrixType&, bool)
*/
SelfAdjointEigenSolver& compute(const MatrixType& matrix, bool computeEigenvectors = true);
/** \brief Computes eigendecomposition of given matrix pencil.
*
* \param[in] matA Selfadjoint matrix in matrix pencil.
* \param[in] matB Positive-definite matrix in matrix pencil.
* \param[in] computeEigenvectors If true, both the eigenvectors and the
* eigenvalues are computed; if false, only the eigenvalues are
* computed.
* \returns Reference to \c *this
*
* This function computes eigenvalues and (if requested) the eigenvectors
* of the generalized eigenproblem \f$ Ax = \lambda B x \f$ with \a matA
* the selfadjoint matrix \f$ A \f$ and \a matB the positive definite
* matrix \f$ B \f$. The eigenvalues() function can be used to retrieve
* the eigenvalues. If \p computeEigenvectors is true, then the
* eigenvectors are also computed and can be retrieved by calling
* eigenvectors().
*
* The implementation uses LLT to compute the Cholesky decomposition
* \f$ B = LL^* \f$ and calls compute(const MatrixType&, bool) to compute
* the eigendecomposition \f$ L^{-1} A (L^*)^{-1} \f$. This solves the
* generalized eigenproblem, because any solution of the generalized
* eigenproblem \f$ Ax = \lambda B x \f$ corresponds to a solution
* \f$ L^{-1} A (L^*)^{-1} (L^* x) = \lambda (L^* x) \f$ of the
* eigenproblem for \f$ L^{-1} A (L^*)^{-1} \f$.
*
* Example: \include SelfAdjointEigenSolver_compute_MatrixType2.cpp
* Output: \verbinclude SelfAdjointEigenSolver_compute_MatrixType2.out
*
* \sa SelfAdjointEigenSolver(const MatrixType&, const MatrixType&, bool)
*/
SelfAdjointEigenSolver& compute(const MatrixType& matA, const MatrixType& matB, bool computeEigenvectors = true);
/** \returns the computed eigen vectors as a matrix of column vectors */
MatrixType eigenvectors(void) const
/** \brief Returns the eigenvectors of given matrix (pencil).
*
* \returns %Matrix whose columns are the eigenvectors.
*
* \pre The eigenvectors have been computed before.
*
* Column \f$ k \f$ of the returned matrix is an eigenvector corresponding
* to eigenvalue number \f$ k \f$ 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 \f$ A \f$, then the matrix returned by this function is the
* matrix \f$ V \f$ in the eigendecomposition \f$ A = V D V^{-1} \f$.
*
* Example: \include SelfAdjointEigenSolver_eigenvectors.cpp
* Output: \verbinclude SelfAdjointEigenSolver_eigenvectors.out
*
* \sa eigenvalues()
*/
MatrixType eigenvectors() const
{
#ifndef NDEBUG
ei_assert(m_eigenvectorsOk);
@ -113,21 +285,62 @@ template<typename _MatrixType> class SelfAdjointEigenSolver
return m_eivec;
}
/** \returns the computed eigen values */
RealVectorType eigenvalues(void) const { return m_eivalues; }
/** \returns the positive square root of the matrix
/** \brief Returns the eigenvalues of given matrix (pencil).
*
* \note the matrix itself must be positive in order for this to make sense.
* \returns Column vector containing the eigenvalues.
*
* \pre The eigenvalues have been computed before.
*
* The eigenvalues are repeated according to their algebraic multiplicity,
* so there are as many eigenvalues as rows in the matrix.
*
* Example: \include SelfAdjointEigenSolver_eigenvalues.cpp
* Output: \verbinclude SelfAdjointEigenSolver_eigenvalues.out
*
* \sa eigenvectors(), MatrixBase::eigenvalues()
*/
RealVectorType eigenvalues() const { return m_eivalues; }
/** \brief Computes the positive-definite square root of the matrix.
*
* \returns the positive-definite square root of the matrix
*
* \pre The eigenvalues and eigenvectors of a positive-definite matrix
* have been computed before.
*
* The square root of a positive-definite matrix \f$ A \f$ is the
* positive-definite matrix whose square equals \f$ A \f$. This function
* uses the eigendecomposition \f$ A = V D V^{-1} \f$ to compute the
* square root as \f$ A^{1/2} = V D^{1/2} V^{-1} \f$.
*
* Example: \include SelfAdjointEigenSolver_operatorSqrt.cpp
* Output: \verbinclude SelfAdjointEigenSolver_operatorSqrt.out
*
* \sa operatorInverseSqrt(),
* \ref MatrixFunctions_Module "MatrixFunctions Module"
*/
MatrixType operatorSqrt() const
{
return m_eivec * m_eivalues.cwiseSqrt().asDiagonal() * m_eivec.adjoint();
}
/** \returns the positive inverse square root of the matrix
/** \brief Computes the inverse square root of the matrix.
*
* \note the matrix itself must be positive definite in order for this to make sense.
* \returns the inverse positive-definite square root of the matrix
*
* \pre The eigenvalues and eigenvectors of a positive-definite matrix
* have been computed before.
*
* This function uses the eigendecomposition \f$ A = V D V^{-1} \f$ to
* compute the inverse square root as \f$ V D^{-1/2} V^{-1} \f$. This is
* cheaper than first computing the square root with operatorSqrt() and
* then its inverse with MatrixBase::inverse().
*
* Example: \include SelfAdjointEigenSolver_operatorInverseSqrt.cpp
* Output: \verbinclude SelfAdjointEigenSolver_operatorInverseSqrt.out
*
* \sa operatorSqrt(), MatrixBase::inverse(),
* \ref MatrixFunctions_Module "MatrixFunctions Module"
*/
MatrixType operatorInverseSqrt() const
{
@ -165,11 +378,6 @@ template<typename _MatrixType> class SelfAdjointEigenSolver
template<typename RealScalar, typename Scalar>
static void ei_tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, int start, int end, Scalar* matrixQ, int n);
/** Computes the eigenvalues of the selfadjoint matrix \a matrix,
* as well as the eigenvectors if \a computeEigenvectors is true.
*
* \sa SelfAdjointEigenSolver(MatrixType,bool), compute(MatrixType,MatrixType,bool)
*/
template<typename MatrixType>
SelfAdjointEigenSolver<MatrixType>& SelfAdjointEigenSolver<MatrixType>::compute(const MatrixType& matrix, bool computeEigenvectors)
{
@ -233,13 +441,6 @@ SelfAdjointEigenSolver<MatrixType>& SelfAdjointEigenSolver<MatrixType>::compute(
return *this;
}
/** Computes the eigenvalues of the generalized eigen problem
* \f$ Ax = lambda B x \f$ with \a matA the selfadjoint matrix \f$ A \f$
* and \a matB the positive definite matrix \f$ B \f$ . The eigenvectors
* are computed if \a computeEigenvectors is true.
*
* \sa SelfAdjointEigenSolver(MatrixType,MatrixType,bool), compute(MatrixType,bool)
*/
template<typename MatrixType>
SelfAdjointEigenSolver<MatrixType>& SelfAdjointEigenSolver<MatrixType>::
compute(const MatrixType& matA, const MatrixType& matB, bool computeEigenvectors)

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doc/snippets/SelfAdjointEigenSolver_SelfAdjointEigenSolver.cpp Normal file
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@ -0,0 +1,7 @@
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;

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doc/snippets/SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType.cpp Normal file
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@ -0,0 +1,17 @@
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;

16
doc/snippets/SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType2.cpp Normal file
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@ -0,0 +1,16 @@
MatrixXd X = MatrixXd::Random(5,5);
MatrixXd A = X + X.transpose();
cout << "Here is a random symmetric matrix, A:" << endl << A << endl;
X = MatrixXd::Random(5,5);
MatrixXd B = X * X.transpose();
cout << "and a random postive-definite matrix, B:" << endl << B << endl << endl;
SelfAdjointEigenSolver<MatrixXd> es(A,B);
cout << "The eigenvalues of the pencil (A,B) 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 A * v = " << endl << A * v << endl;
cout << "... and lambda * B * v = " << endl << lambda * B * v << endl << endl;

7
doc/snippets/SelfAdjointEigenSolver_compute_MatrixType.cpp Normal file
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@ -0,0 +1,7 @@
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;

9
doc/snippets/SelfAdjointEigenSolver_compute_MatrixType2.cpp Normal file
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@ -0,0 +1,9 @@
MatrixXd X = MatrixXd::Random(5,5);
MatrixXd A = X * X.transpose();
X = MatrixXd::Random(5,5);
MatrixXd B = X * X.transpose();
SelfAdjointEigenSolver<MatrixXd> es(A,B,false);
cout << "The eigenvalues of the pencil (A,B) are:" << endl << es.eigenvalues() << endl;
es.compute(B,A,false);
cout << "The eigenvalues of the pencil (B,A) are:" << endl << es.eigenvalues() << endl;

4
doc/snippets/SelfAdjointEigenSolver_eigenvalues.cpp Normal file
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@ -0,0 +1,4 @@
MatrixXd ones = MatrixXd::Ones(3,3);
SelfAdjointEigenSolver<MatrixXd> es(ones);
cout << "The eigenvalues of the 3x3 matrix of ones are:"
<< endl << es.eigenvalues() << endl;

4
doc/snippets/SelfAdjointEigenSolver_eigenvectors.cpp Normal file
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@ -0,0 +1,4 @@
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;

9
doc/snippets/SelfAdjointEigenSolver_operatorInverseSqrt.cpp Normal file
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@ -0,0 +1,9 @@
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;

8
doc/snippets/SelfAdjointEigenSolver_operatorSqrt.cpp Normal file
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@ -0,0 +1,8 @@
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;