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SelfAdjointEigenSolver.h
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_SELFADJOINTEIGENSOLVER_H
#define EIGEN_SELFADJOINTEIGENSOLVER_H
#include "./Tridiagonalization.h"
namespace Eigen {
template<typename _MatrixType>
class GeneralizedSelfAdjointEigenSolver;
namespace internal {
template<typename SolverType,int Size,bool IsComplex> struct direct_selfadjoint_eigenvalues;
}
/** \eigenvalues_module \ingroup Eigenvalues_Module
*
*
* \class SelfAdjointEigenSolver
*
* \brief Computes eigenvalues and eigenvectors of selfadjoint matrices
*
* \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.
*
* 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.
*
* 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 \b lower \b triangular \b 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 \em generalized eigenvalue problem \f$ Av = \lambda Bv \f$ and
* the likes, see the class GeneralizedSelfAdjointEigenSolver.
*
* \sa MatrixBase::eigenvalues(), class EigenSolver, class ComplexEigenSolver
*/
template<typename _MatrixType> class SelfAdjointEigenSolver
{
public:
typedef _MatrixType MatrixType;
enum {
Size = MatrixType::RowsAtCompileTime,
ColsAtCompileTime = MatrixType::ColsAtCompileTime,
Options = MatrixType::Options,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
};
/** \brief Scalar type for matrices of type \p _MatrixType. */
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::Index Index;
typedef Matrix<Scalar,Size,Size,ColMajor,MaxColsAtCompileTime,MaxColsAtCompileTime> EigenvectorsType;
/** \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;
friend struct internal::direct_selfadjoint_eigenvalues<SelfAdjointEigenSolver,Size,NumTraits<Scalar>::IsComplex>;
/** \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 internal::plain_col_type<MatrixType, RealScalar>::type RealVectorType;
typedef Tridiagonalization<MatrixType> TridiagonalizationType;
/** \brief 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 \p _MatrixType is a fixed-size matrix; use
* SelfAdjointEigenSolver(Index) for dynamic-size matrices.
*
* Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver.cpp
* Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver.out
*/
SelfAdjointEigenSolver()
: m_eivec(),
m_eivalues(),
m_subdiag(),
m_isInitialized(false)
{ }
/** \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(). 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() for an example
*/
SelfAdjointEigenSolver(Index size)
: m_eivec(size, size),
m_eivalues(size),
m_subdiag(size > 1 ? size - 1 : 1),
m_isInitialized(false)
{}
/** \brief Constructor; computes eigendecomposition of given matrix.
*
* \param[in] matrix Selfadjoint matrix whose eigendecomposition is to
* be computed. Only the lower triangular part of the matrix is referenced.
* \param[in] options Can be #ComputeEigenvectors (default) or #EigenvaluesOnly.
*
* This constructor calls compute(const MatrixType&, int) to compute the
* eigenvalues of the matrix \p matrix. The eigenvectors are computed if
* \p options equals #ComputeEigenvectors.
*
* Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType.cpp
* Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType.out
*
* \sa compute(const MatrixType&, int)
*/
SelfAdjointEigenSolver(const MatrixType& matrix, int options = ComputeEigenvectors)
: m_eivec(matrix.rows(), matrix.cols()),
m_eivalues(matrix.cols()),
m_subdiag(matrix.rows() > 1 ? matrix.rows() - 1 : 1),
m_isInitialized(false)
{
compute(matrix, options);
}
/** \brief Computes eigendecomposition of given matrix.
*
* \param[in] matrix Selfadjoint matrix whose eigendecomposition is to
* be computed. Only the lower triangular part of the matrix is referenced.
* \param[in] options Can be #ComputeEigenvectors (default) or #EigenvaluesOnly.
* \returns Reference to \c *this
*
* This function computes the eigenvalues of \p matrix. The eigenvalues()
* function can be used to retrieve them. If \p 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, <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&, int)
*/
SelfAdjointEigenSolver& compute(const MatrixType& matrix, int options = ComputeEigenvectors);
/** \brief 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.
*
* \sa compute(const MatrixType&, int options)
*/
SelfAdjointEigenSolver& computeDirect(const MatrixType& matrix, int options = ComputeEigenvectors);
/** \brief Returns the eigenvectors of given matrix.
*
* \returns A const reference to the 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$.
*
* Example: \include SelfAdjointEigenSolver_eigenvectors.cpp
* Output: \verbinclude SelfAdjointEigenSolver_eigenvectors.out
*
* \sa eigenvalues()
*/
const EigenvectorsType& eigenvectors() const
{
eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
return m_eivec;
}
/** \brief Returns the eigenvalues of given matrix.
*
* \returns A const reference to the column vector containing the eigenvalues.
*
* \pre The eigenvalues have been computed before.
*
* The eigenvalues are repeated according to their algebraic multiplicity,
* so there are as many eigenvalues as rows in the matrix. The eigenvalues
* are sorted in increasing order.
*
* Example: \include SelfAdjointEigenSolver_eigenvalues.cpp
* Output: \verbinclude SelfAdjointEigenSolver_eigenvalues.out
*
* \sa eigenvectors(), MatrixBase::eigenvalues()
*/
const RealVectorType& eigenvalues() const
{
eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
return m_eivalues;
}
/** \brief Computes the positive-definite square root of the matrix.
*
* \returns the positive-definite square root of the matrix
*
* \pre The eigenvalues and eigenvectors of a positive-definite matrix
* have been computed before.
*
* The square root of a positive-definite matrix \f$ 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$.
*
* Example: \include SelfAdjointEigenSolver_operatorSqrt.cpp
* Output: \verbinclude SelfAdjointEigenSolver_operatorSqrt.out
*
* \sa operatorInverseSqrt(),
* \ref MatrixFunctions_Module "MatrixFunctions Module"
*/
MatrixType operatorSqrt() const
{
eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
return m_eivec * m_eivalues.cwiseSqrt().asDiagonal() * m_eivec.adjoint();
}
/** \brief Computes the inverse square root of the matrix.
*
* \returns the inverse positive-definite square root of the matrix
*
* \pre The eigenvalues and eigenvectors of a positive-definite matrix
* have been computed before.
*
* This function uses the eigendecomposition \f$ 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
{
eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
return m_eivec * m_eivalues.cwiseInverse().cwiseSqrt().asDiagonal() * m_eivec.adjoint();
}
/** \brief Reports whether previous computation was successful.
*
* \returns \c Success if computation was succesful, \c NoConvergence otherwise.
*/
ComputationInfo info() const
{
eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
return m_info;
}
/** \brief 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).
*/
static const int m_maxIterations = 30;
#ifdef EIGEN2_SUPPORT
SelfAdjointEigenSolver(const MatrixType& matrix, bool computeEigenvectors)
: m_eivec(matrix.rows(), matrix.cols()),
m_eivalues(matrix.cols()),
m_subdiag(matrix.rows() > 1 ? matrix.rows() - 1 : 1),
m_isInitialized(false)
{
compute(matrix, computeEigenvectors);
}
SelfAdjointEigenSolver(const MatrixType& matA, const MatrixType& matB, bool computeEigenvectors = true)
: m_eivec(matA.cols(), matA.cols()),
m_eivalues(matA.cols()),
m_subdiag(matA.cols() > 1 ? matA.cols() - 1 : 1),
m_isInitialized(false)
{
static_cast<GeneralizedSelfAdjointEigenSolver<MatrixType>*>(this)->compute(matA, matB, computeEigenvectors ? ComputeEigenvectors : EigenvaluesOnly);
}
void compute(const MatrixType& matrix, bool computeEigenvectors)
{
compute(matrix, computeEigenvectors ? ComputeEigenvectors : EigenvaluesOnly);
}
void compute(const MatrixType& matA, const MatrixType& matB, bool computeEigenvectors = true)
{
compute(matA, matB, computeEigenvectors ? ComputeEigenvectors : EigenvaluesOnly);
}
#endif // EIGEN2_SUPPORT
protected:
static void check_template_parameters()
{
EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
}
EigenvectorsType m_eivec;
RealVectorType m_eivalues;
typename TridiagonalizationType::SubDiagonalType m_subdiag;
ComputationInfo m_info;
bool m_isInitialized;
bool m_eigenvectorsOk;
};
/** \internal
*
* \eigenvalues_module \ingroup Eigenvalues_Module
*
* Performs a QR step on a tridiagonal symmetric matrix represented as a
* pair of two vectors \a diag and \a subdiag.
*
* \param matA the input selfadjoint matrix
* \param hCoeffs returned Householder coefficients
*
* For compilation efficiency reasons, this procedure does not use eigen expression
* for its arguments.
*
* Implemented from Golub's "Matrix Computations", algorithm 8.3.2:
* "implicit symmetric QR step with Wilkinson shift"
*/
namespace internal {
template<typename RealScalar, typename Scalar, typename Index>
static void tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, Index start, Index end, Scalar* matrixQ, Index n);
}
template<typename MatrixType>
SelfAdjointEigenSolver<MatrixType>& SelfAdjointEigenSolver<MatrixType>
::compute(const MatrixType& matrix, int options)
{
check_template_parameters();
using std::abs;
eigen_assert(matrix.cols() == matrix.rows());
eigen_assert((options&~(EigVecMask|GenEigMask))==0
&& (options&EigVecMask)!=EigVecMask
&& "invalid option parameter");
bool computeEigenvectors = (options&ComputeEigenvectors)==ComputeEigenvectors;
Index n = matrix.cols();
m_eivalues.resize(n,1);
if(n==1)
{
m_eivalues.coeffRef(0,0) = numext::real(matrix.coeff(0,0));
if(computeEigenvectors)
m_eivec.setOnes(n,n);
m_info = Success;
m_isInitialized = true;
m_eigenvectorsOk = computeEigenvectors;
return *this;
}
// declare some aliases
RealVectorType& diag = m_eivalues;
EigenvectorsType& mat = m_eivec;
// map the matrix coefficients to [-1:1] to avoid over- and underflow.
mat = matrix.template triangularView<Lower>();
RealScalar scale = mat.cwiseAbs().maxCoeff();
if(scale==RealScalar(0)) scale = RealScalar(1);
mat.template triangularView<Lower>() /= scale;
m_subdiag.resize(n-1);
internal::tridiagonalization_inplace(mat, diag, m_subdiag, computeEigenvectors);
Index end = n-1;
Index start = 0;
Index iter = 0; // total number of iterations
while (end>0)
{
for (Index i = start; i<end; ++i)
if (internal::isMuchSmallerThan(abs(m_subdiag[i]),(abs(diag[i])+abs(diag[i+1]))))
m_subdiag[i] = 0;
// find the largest unreduced block
while (end>0 && m_subdiag[end-1]==0)
{
end--;
}
if (end<=0)
break;
// if we spent too many iterations, we give up
iter++;
if(iter > m_maxIterations * n) break;
start = end - 1;
while (start>0 && m_subdiag[start-1]!=0)
start--;
internal::tridiagonal_qr_step(diag.data(), m_subdiag.data(), start, end, computeEigenvectors ? m_eivec.data() : (Scalar*)0, n);
}
if (iter <= m_maxIterations * n)
m_info = Success;
else
m_info = NoConvergence;
// Sort eigenvalues and corresponding vectors.
// TODO make the sort optional ?
// TODO use a better sort algorithm !!
if (m_info == Success)
{
for (Index i = 0; i < n-1; ++i)
{
Index k;
m_eivalues.segment(i,n-i).minCoeff(&k);
if (k > 0)
{
std::swap(m_eivalues[i], m_eivalues[k+i]);
if(computeEigenvectors)
m_eivec.col(i).swap(m_eivec.col(k+i));
}
}
}
// scale back the eigen values
m_eivalues *= scale;
m_isInitialized = true;
m_eigenvectorsOk = computeEigenvectors;
return *this;
}
namespace internal {
template<typename SolverType,int Size,bool IsComplex> struct direct_selfadjoint_eigenvalues
{
static inline void run(SolverType& eig, const typename SolverType::MatrixType& A, int options)
{ eig.compute(A,options); }
};
template<typename SolverType> struct direct_selfadjoint_eigenvalues<SolverType,3,false>
{
typedef typename SolverType::MatrixType MatrixType;
typedef typename SolverType::RealVectorType VectorType;
typedef typename SolverType::Scalar Scalar;
typedef typename MatrixType::Index Index;
typedef typename SolverType::EigenvectorsType EigenvectorsType;
/** \internal
* Computes the roots of the characteristic polynomial of \a m.
* For numerical stability m.trace() should be near zero and to avoid over- or underflow m should be normalized.
*/
static inline void computeRoots(const MatrixType& m, VectorType& roots)
{
using std::sqrt;
using std::atan2;
using std::cos;
using std::sin;
const Scalar s_inv3 = Scalar(1.0)/Scalar(3.0);
const Scalar s_sqrt3 = sqrt(Scalar(3.0));
// The characteristic equation is x^3 - c2*x^2 + c1*x - c0 = 0. The
// eigenvalues are the roots to this equation, all guaranteed to be
// real-valued, because the matrix is symmetric.
Scalar c0 = m(0,0)*m(1,1)*m(2,2) + Scalar(2)*m(1,0)*m(2,0)*m(2,1) - m(0,0)*m(2,1)*m(2,1) - m(1,1)*m(2,0)*m(2,0) - m(2,2)*m(1,0)*m(1,0);
Scalar c1 = m(0,0)*m(1,1) - m(1,0)*m(1,0) + m(0,0)*m(2,2) - m(2,0)*m(2,0) + m(1,1)*m(2,2) - m(2,1)*m(2,1);
Scalar c2 = m(0,0) + m(1,1) + m(2,2);
// Construct the parameters used in classifying the roots of the equation
// and in solving the equation for the roots in closed form.
Scalar c2_over_3 = c2*s_inv3;
Scalar a_over_3 = (c2*c2_over_3 - c1)*s_inv3;
if(a_over_3<Scalar(0))
a_over_3 = Scalar(0);
Scalar half_b = Scalar(0.5)*(c0 + c2_over_3*(Scalar(2)*c2_over_3*c2_over_3 - c1));
Scalar q = a_over_3*a_over_3*a_over_3 - half_b*half_b;
if(q<Scalar(0))
q = Scalar(0);
// Compute the eigenvalues by solving for the roots of the polynomial.
Scalar rho = sqrt(a_over_3);
Scalar theta = atan2(sqrt(q),half_b)*s_inv3; // since sqrt(q) > 0, atan2 is in [0, pi] and theta is in [0, pi/3]
Scalar cos_theta = cos(theta);
Scalar sin_theta = sin(theta);
// roots are already sorted, since cos is monotonically decreasing on [0, pi]
roots(0) = c2_over_3 - rho*(cos_theta + s_sqrt3*sin_theta); // == 2*rho*cos(theta+2pi/3)
roots(1) = c2_over_3 - rho*(cos_theta - s_sqrt3*sin_theta); // == 2*rho*cos(theta+ pi/3)
roots(2) = c2_over_3 + Scalar(2)*rho*cos_theta;
}
static inline bool extract_kernel(MatrixType& mat, Ref<VectorType> res, Ref<VectorType> representative)
{
using std::abs;
Index i0;
// Find non-zero column i0 (by construction, there must exist a non zero coefficient on the diagonal):
mat.diagonal().cwiseAbs().maxCoeff(&i0);
// mat.col(i0) is a good candidate for an orthogonal vector to the current eigenvector,
// so let's save it:
representative = mat.col(i0);
Scalar n0, n1;
VectorType c0, c1;
n0 = (c0 = representative.cross(mat.col((i0+1)%3))).squaredNorm();
n1 = (c1 = representative.cross(mat.col((i0+2)%3))).squaredNorm();
if(n0>n1) res = c0/std::sqrt(n0);
else res = c1/std::sqrt(n1);
return true;
}
static inline void run(SolverType& solver, const MatrixType& mat, int options)
{
eigen_assert(mat.cols() == 3 && mat.cols() == mat.rows());
eigen_assert((options&~(EigVecMask|GenEigMask))==0
&& (options&EigVecMask)!=EigVecMask
&& "invalid option parameter");
bool computeEigenvectors = (options&ComputeEigenvectors)==ComputeEigenvectors;
EigenvectorsType& eivecs = solver.m_eivec;
VectorType& eivals = solver.m_eivalues;
// Shift the matrix to the mean eigenvalue and map the matrix coefficients to [-1:1] to avoid over- and underflow.
Scalar shift = mat.trace() / Scalar(3);
// TODO Avoid this copy. Currently it is necessary to suppress bogus values when determining maxCoeff and for computing the eigenvectors later
MatrixType scaledMat = mat.template selfadjointView<Lower>();
scaledMat.diagonal().array() -= shift;
Scalar scale = scaledMat.cwiseAbs().maxCoeff();
if(scale > 0) scaledMat /= scale; // TODO for scale==0 we could save the remaining operations
// compute the eigenvalues
computeRoots(scaledMat,eivals);
// compute the eigenvectors
if(computeEigenvectors)
{
if((eivals(2)-eivals(0))<=Eigen::NumTraits<Scalar>::epsilon())
{
// All three eigenvalues are numerically the same
eivecs.setIdentity();
}
else
{
MatrixType tmp;
tmp = scaledMat;
// Compute the eigenvector of the most distinct eigenvalue
Scalar d0 = eivals(2) - eivals(1);
Scalar d1 = eivals(1) - eivals(0);
Index k(0), l(2);
if(d0 > d1)
{
std::swap(k,l);
d0 = d1;
}
// Compute the eigenvector of index k
{
tmp.diagonal().array () -= eivals(k);
// By construction, 'tmp' is of rank 2, and its kernel corresponds to the respective eigenvector.
extract_kernel(tmp, eivecs.col(k), eivecs.col(l));
}
// Compute eigenvector of index l
if(d0<=2*Eigen::NumTraits<Scalar>::epsilon()*d1)
{
// If d0 is too small, then the two other eigenvalues are numerically the same,
// and thus we only have to ortho-normalize the near orthogonal vector we saved above.
eivecs.col(l) -= eivecs.col(k).dot(eivecs.col(l))*eivecs.col(l);
eivecs.col(l).normalize();
}
else
{
tmp = scaledMat;
tmp.diagonal().array () -= eivals(l);
VectorType dummy;
extract_kernel(tmp, eivecs.col(l), dummy);
}
// Compute last eigenvector from the other two
eivecs.col(1) = eivecs.col(2).cross(eivecs.col(0)).normalized();
}
}
// Rescale back to the original size.
eivals *= scale;
eivals.array() += shift;
solver.m_info = Success;
solver.m_isInitialized = true;
solver.m_eigenvectorsOk = computeEigenvectors;
}
};
// 2x2 direct eigenvalues decomposition, code from Hauke Heibel
template<typename SolverType> struct direct_selfadjoint_eigenvalues<SolverType,2,false>
{
typedef typename SolverType::MatrixType MatrixType;
typedef typename SolverType::RealVectorType VectorType;
typedef typename SolverType::Scalar Scalar;
typedef typename SolverType::EigenvectorsType EigenvectorsType;
static inline void computeRoots(const MatrixType& m, VectorType& roots)
{
using std::sqrt;
const Scalar t0 = Scalar(0.5) * sqrt( numext::abs2(m(0,0)-m(1,1)) + Scalar(4)*numext::abs2(m(1,0)));
const Scalar t1 = Scalar(0.5) * (m(0,0) + m(1,1));
roots(0) = t1 - t0;
roots(1) = t1 + t0;
}
static inline void run(SolverType& solver, const MatrixType& mat, int options)
{
using std::sqrt;
using std::abs;
eigen_assert(mat.cols() == 2 && mat.cols() == mat.rows());
eigen_assert((options&~(EigVecMask|GenEigMask))==0
&& (options&EigVecMask)!=EigVecMask
&& "invalid option parameter");
bool computeEigenvectors = (options&ComputeEigenvectors)==ComputeEigenvectors;
EigenvectorsType& eivecs = solver.m_eivec;
VectorType& eivals = solver.m_eivalues;
// map the matrix coefficients to [-1:1] to avoid over- and underflow.
Scalar scale = mat.cwiseAbs().maxCoeff();
scale = (std::max)(scale,Scalar(1));
MatrixType scaledMat = mat / scale;
// Compute the eigenvalues
computeRoots(scaledMat,eivals);
// compute the eigen vectors
if(computeEigenvectors)
{
if((eivals(1)-eivals(0))<=abs(eivals(1))*Eigen::NumTraits<Scalar>::epsilon())
{
eivecs.setIdentity();
}
else
{
scaledMat.diagonal().array () -= eivals(1);
Scalar a2 = numext::abs2(scaledMat(0,0));
Scalar c2 = numext::abs2(scaledMat(1,1));
Scalar b2 = numext::abs2(scaledMat(1,0));
if(a2>c2)
{
eivecs.col(1) << -scaledMat(1,0), scaledMat(0,0);
eivecs.col(1) /= sqrt(a2+b2);
}
else
{
eivecs.col(1) << -scaledMat(1,1), scaledMat(1,0);
eivecs.col(1) /= sqrt(c2+b2);
}
eivecs.col(0) << eivecs.col(1).unitOrthogonal();
}
}
// Rescale back to the original size.
eivals *= scale;
solver.m_info = Success;
solver.m_isInitialized = true;
solver.m_eigenvectorsOk = computeEigenvectors;
}
};
}
template<typename MatrixType>
SelfAdjointEigenSolver<MatrixType>& SelfAdjointEigenSolver<MatrixType>
::computeDirect(const MatrixType& matrix, int options)
{
internal::direct_selfadjoint_eigenvalues<SelfAdjointEigenSolver,Size,NumTraits<Scalar>::IsComplex>::run(*this,matrix,options);
return *this;
}
namespace internal {
template<typename RealScalar, typename Scalar, typename Index>
static void tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, Index start, Index end, Scalar* matrixQ, Index n)
{
using std::abs;
RealScalar td = (diag[end-1] - diag[end])*RealScalar(0.5);
RealScalar e = subdiag[end-1];
// Note that thanks to scaling, e^2 or td^2 cannot overflow, however they can still
// underflow thus leading to inf/NaN values when using the following commented code:
// RealScalar e2 = numext::abs2(subdiag[end-1]);
// RealScalar mu = diag[end] - e2 / (td + (td>0 ? 1 : -1) * sqrt(td*td + e2));
// This explain the following, somewhat more complicated, version:
RealScalar mu = diag[end];
if(td==0)
mu -= abs(e);
else
{
RealScalar e2 = numext::abs2(subdiag[end-1]);
RealScalar h = numext::hypot(td,e);
if(e2==0) mu -= (e / (td + (td>0 ? 1 : -1))) * (e / h);
else mu -= e2 / (td + (td>0 ? h : -h));
}
RealScalar x = diag[start] - mu;
RealScalar z = subdiag[start];
for (Index k = start; k < end; ++k)
{
JacobiRotation<RealScalar> rot;
rot.makeGivens(x, z);
// do T = G' T G
RealScalar sdk = rot.s() * diag[k] + rot.c() * subdiag[k];
RealScalar dkp1 = rot.s() * subdiag[k] + rot.c() * diag[k+1];
diag[k] = rot.c() * (rot.c() * diag[k] - rot.s() * subdiag[k]) - rot.s() * (rot.c() * subdiag[k] - rot.s() * diag[k+1]);
diag[k+1] = rot.s() * sdk + rot.c() * dkp1;
subdiag[k] = rot.c() * sdk - rot.s() * dkp1;
if (k > start)
subdiag[k - 1] = rot.c() * subdiag[k-1] - rot.s() * z;
x = subdiag[k];
if (k < end - 1)
{
z = -rot.s() * subdiag[k+1];
subdiag[k + 1] = rot.c() * subdiag[k+1];
}
// apply the givens rotation to the unit matrix Q = Q * G
if (matrixQ)
{
Map<Matrix<Scalar,Dynamic,Dynamic,ColMajor> > q(matrixQ,n,n);
q.applyOnTheRight(k,k+1,rot);
}
}
}
} // end namespace internal
} // end namespace Eigen
#endif // EIGEN_SELFADJOINTEIGENSOLVER_H
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