// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010,2012 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_REAL_SCHUR_H
#define EIGEN_REAL_SCHUR_H
#include "./HessenbergDecomposition.h"
namespace Eigen {
/** \eigenvalues_module \ingroup Eigenvalues_Module
*
*
* \class RealSchur
*
* \brief Performs a real Schur decomposition of a square matrix
*
* \tparam _MatrixType the type of the matrix of which we are computing the
* real Schur decomposition; this is expected to be an instantiation of the
* Matrix class template.
*
* Given a real square matrix A, this class computes the real Schur
* decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and
* T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose
* inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
* matrix is a block-triangular matrix whose diagonal consists of 1-by-1
* blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the
* blocks on the diagonal of T are the same as the eigenvalues of the matrix
* A, and thus the real Schur decomposition is used in EigenSolver to compute
* the eigendecomposition of a matrix.
*
* Call the function compute() to compute the real Schur decomposition of a
* given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool)
* constructor which computes the real Schur decomposition at construction
* time. Once the decomposition is computed, you can use the matrixU() and
* matrixT() functions to retrieve the matrices U and T in the decomposition.
*
* The documentation of RealSchur(const MatrixType&, bool) contains an example
* of the typical use of this class.
*
* \note The implementation is adapted from
* <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
* Their code is based on EISPACK.
*
* \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver
*/
template<typename _MatrixType> class RealSchur
{
public:
typedef _MatrixType MatrixType;
enum {
RowsAtCompileTime = MatrixType::RowsAtCompileTime,
ColsAtCompileTime = MatrixType::ColsAtCompileTime,
Options = MatrixType::Options,
MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
};
typedef typename MatrixType::Scalar Scalar;
typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
typedef typename MatrixType::Index Index;
typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
/** \brief Default constructor.
*
* \param [in] size Positive integer, size of the matrix whose Schur decomposition will be computed.
*
* The default constructor is useful in cases in which 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.
*/
RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
: m_matT(size, size),
m_matU(size, size),
m_workspaceVector(size),
m_hess(size),
m_isInitialized(false),
m_matUisUptodate(false),
m_maxIters(-1)
{ }
/** \brief Constructor; computes real Schur decomposition of given matrix.
*
* \param[in] matrix Square matrix whose Schur decomposition is to be computed.
* \param[in] computeU If true, both T and U are computed; if false, only T is computed.
*
* This constructor calls compute() to compute the Schur decomposition.
*
* Example: \include RealSchur_RealSchur_MatrixType.cpp
* Output: \verbinclude RealSchur_RealSchur_MatrixType.out
*/
RealSchur(const MatrixType& matrix, bool computeU = true)
: m_matT(matrix.rows(),matrix.cols()),
m_matU(matrix.rows(),matrix.cols()),
m_workspaceVector(matrix.rows()),
m_hess(matrix.rows()),
m_isInitialized(false),
m_matUisUptodate(false),
m_maxIters(-1)
{
compute(matrix, computeU);
}
/** \brief Returns the orthogonal matrix in the Schur decomposition.
*
* \returns A const reference to the matrix U.
*
* \pre Either the constructor RealSchur(const MatrixType&, bool) or the
* member function compute(const MatrixType&, bool) has been called before
* to compute the Schur decomposition of a matrix, and \p computeU was set
* to true (the default value).
*
* \sa RealSchur(const MatrixType&, bool) for an example
*/
const MatrixType& matrixU() const
{
eigen_assert(m_isInitialized && "RealSchur is not initialized.");
eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
return m_matU;
}
/** \brief Returns the quasi-triangular matrix in the Schur decomposition.
*
* \returns A const reference to the matrix T.
*
* \pre Either the constructor RealSchur(const MatrixType&, bool) or the
* member function compute(const MatrixType&, bool) has been called before
* to compute the Schur decomposition of a matrix.
*
* \sa RealSchur(const MatrixType&, bool) for an example
*/
const MatrixType& matrixT() const
{
eigen_assert(m_isInitialized && "RealSchur is not initialized.");
return m_matT;
}
/** \brief Computes Schur decomposition of given matrix.
*
* \param[in] matrix Square matrix whose Schur decomposition is to be computed.
* \param[in] computeU If true, both T and U are computed; if false, only T is computed.
* \returns Reference to \c *this
*
* The Schur decomposition is computed by first reducing the matrix to
* Hessenberg form using the class HessenbergDecomposition. The Hessenberg
* matrix is then reduced to triangular form by performing Francis QR
* iterations with implicit double shift. The cost of computing the Schur
* decomposition depends on the number of iterations; as a rough guide, it
* may be taken to be \f$25n^3\f$ flops if \a computeU is true and
* \f$10n^3\f$ flops if \a computeU is false.
*
* Example: \include RealSchur_compute.cpp
* Output: \verbinclude RealSchur_compute.out
*
* \sa compute(const MatrixType&, bool, Index)
*/
RealSchur& compute(const MatrixType& matrix, bool computeU = true);
/** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T
* \param[in] matrixH Matrix in Hessenberg form H
* \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
* \param computeU Computes the matriX U of the Schur vectors
* \return Reference to \c *this
*
* This routine assumes that the matrix is already reduced in Hessenberg form matrixH
* using either the class HessenbergDecomposition or another mean.
* It computes the upper quasi-triangular matrix T of the Schur decomposition of H
* When computeU is true, this routine computes the matrix U such that
* A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
*
* NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
* is not available, the user should give an identity matrix (Q.setIdentity())
*
* \sa compute(const MatrixType&, bool)
*/
template<typename HessMatrixType, typename OrthMatrixType>
RealSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU);
/** \brief Reports whether previous computation was successful.
*
* \returns \c Success if computation was succesful, \c NoConvergence otherwise.
*/
ComputationInfo info() const
{
eigen_assert(m_isInitialized && "RealSchur is not initialized.");
return m_info;
}
/** \brief Sets the maximum number of iterations allowed.
*
* If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
* of the matrix.
*/
RealSchur& setMaxIterations(Index maxIters)
{
m_maxIters = maxIters;
return *this;
}
/** \brief Returns the maximum number of iterations. */
Index getMaxIterations()
{
return m_maxIters;
}
/** \brief Maximum number of iterations per row.
*
* If not otherwise specified, the maximum number of iterations is this number times the size of the
* matrix. It is currently set to 40.
*/
static const int m_maxIterationsPerRow = 40;
private:
MatrixType m_matT;
MatrixType m_matU;
ColumnVectorType m_workspaceVector;
HessenbergDecomposition<MatrixType> m_hess;
ComputationInfo m_info;
bool m_isInitialized;
bool m_matUisUptodate;
Index m_maxIters;
typedef Matrix<Scalar,3,1> Vector3s;
Scalar computeNormOfT();
Index findSmallSubdiagEntry(Index iu, const Scalar& norm);
void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift);
void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace);
};
template<typename MatrixType>
RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
{
eigen_assert(matrix.cols() == matrix.rows());
Index maxIters = m_maxIters;
if (maxIters == -1)
maxIters = m_maxIterationsPerRow * matrix.rows();
// Step 1. Reduce to Hessenberg form
m_hess.compute(matrix);
// Step 2. Reduce to real Schur form
computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);
return *this;
}
template<typename MatrixType>
template<typename HessMatrixType, typename OrthMatrixType>
RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU)
{
m_matT = matrixH;
if(computeU)
m_matU = matrixQ;
Index maxIters = m_maxIters;
if (maxIters == -1)
maxIters = m_maxIterationsPerRow * matrixH.rows();
m_workspaceVector.resize(m_matT.cols());
Scalar* workspace = &m_workspaceVector.coeffRef(0);
// The matrix m_matT is divided in three parts.
// Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
// Rows il,...,iu is the part we are working on (the active window).
// Rows iu+1,...,end are already brought in triangular form.
Index iu = m_matT.cols() - 1;
Index iter = 0; // iteration count for current eigenvalue
Index totalIter = 0; // iteration count for whole matrix
Scalar exshift(0); // sum of exceptional shifts
Scalar norm = computeNormOfT();
if(norm!=0)
{
while (iu >= 0)
{
Index il = findSmallSubdiagEntry(iu, norm);
// Check for convergence
if (il == iu) // One root found
{
m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
if (iu > 0)
m_matT.coeffRef(iu, iu-1) = Scalar(0);
iu--;
iter = 0;
}
else if (il == iu-1) // Two roots found
{
splitOffTwoRows(iu, computeU, exshift);
iu -= 2;
iter = 0;
}
else // No convergence yet
{
// The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
Vector3s firstHouseholderVector(0,0,0), shiftInfo;
computeShift(iu, iter, exshift, shiftInfo);
iter = iter + 1;
totalIter = totalIter + 1;
if (totalIter > maxIters) break;
Index im;
initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
}
}
}
if(totalIter <= maxIters)
m_info = Success;
else
m_info = NoConvergence;
m_isInitialized = true;
m_matUisUptodate = computeU;
return *this;
}
/** \internal Computes and returns vector L1 norm of T */
template<typename MatrixType>
inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
{
const Index size = m_matT.cols();
// FIXME to be efficient the following would requires a triangular reduxion code
// Scalar norm = m_matT.upper().cwiseAbs().sum()
// + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
Scalar norm(0);
for (Index j = 0; j < size; ++j)
norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
return norm;
}
/** \internal Look for single small sub-diagonal element and returns its index */
template<typename MatrixType>
inline typename MatrixType::Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, const Scalar& norm)
{
using std::abs;
Index res = iu;
while (res > 0)
{
Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));
if (s == 0.0)
s = norm;
if (abs(m_matT.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s)
break;
res--;
}
return res;
}
/** \internal Update T given that rows iu-1 and iu decouple from the rest. */
template<typename MatrixType>
inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift)
{
using std::sqrt;
using std::abs;
const Index size = m_matT.cols();
// The eigenvalues of the 2x2 matrix [a b; c d] are
// trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu); // q = tr^2 / 4 - det = discr/4
m_matT.coeffRef(iu,iu) += exshift;
m_matT.coeffRef(iu-1,iu-1) += exshift;
if (q >= Scalar(0)) // Two real eigenvalues
{
Scalar z = sqrt(abs(q));
JacobiRotation<Scalar> rot;
if (p >= Scalar(0))
rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
else
rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
m_matT.coeffRef(iu, iu-1) = Scalar(0);
if (computeU)
m_matU.applyOnTheRight(iu-1, iu, rot);
}
if (iu > 1)
m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
}
/** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
template<typename MatrixType>
inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
{
using std::sqrt;
using std::abs;
shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
// Wilkinson's original ad hoc shift
if (iter == 10)
{
exshift += shiftInfo.coeff(0);
for (Index i = 0; i <= iu; ++i)
m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2));
shiftInfo.coeffRef(0) = Scalar(0.75) * s;
shiftInfo.coeffRef(1) = Scalar(0.75) * s;
shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
}
// MATLAB's new ad hoc shift
if (iter == 30)
{
Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
s = s * s + shiftInfo.coeff(2);
if (s > Scalar(0))
{
s = sqrt(s);
if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
s = -s;
s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
exshift += s;
for (Index i = 0; i <= iu; ++i)
m_matT.coeffRef(i,i) -= s;
shiftInfo.setConstant(Scalar(0.964));
}
}
}
/** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
template<typename MatrixType>
inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
{
using std::abs;
Vector3s& v = firstHouseholderVector; // alias to save typing
for (im = iu-2; im >= il; --im)
{
const Scalar Tmm = m_matT.coeff(im,im);
const Scalar r = shiftInfo.coeff(0) - Tmm;
const Scalar s = shiftInfo.coeff(1) - Tmm;
v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
v.coeffRef(2) = m_matT.coeff(im+2,im+1);
if (im == il) {
break;
}
const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1)));
if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
{
break;
}
}
}
/** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
template<typename MatrixType>
inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
{
eigen_assert(im >= il);
eigen_assert(im <= iu-2);
const Index size = m_matT.cols();
for (Index k = im; k <= iu-2; ++k)
{
bool firstIteration = (k == im);
Vector3s v;
if (firstIteration)
v = firstHouseholderVector;
else
v = m_matT.template block<3,1>(k,k-1);
Scalar tau, beta;
Matrix<Scalar, 2, 1> ess;
v.makeHouseholder(ess, tau, beta);
if (beta != Scalar(0)) // if v is not zero
{
if (firstIteration && k > il)
m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
else if (!firstIteration)
m_matT.coeffRef(k,k-1) = beta;
// These Householder transformations form the O(n^3) part of the algorithm
m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
if (computeU)
m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
}
}
Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
Scalar tau, beta;
Matrix<Scalar, 1, 1> ess;
v.makeHouseholder(ess, tau, beta);
if (beta != Scalar(0)) // if v is not zero
{
m_matT.coeffRef(iu-1, iu-2) = beta;
m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
if (computeU)
m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
}
// clean up pollution due to round-off errors
for (Index i = im+2; i <= iu; ++i)
{
m_matT.coeffRef(i,i-2) = Scalar(0);
if (i > im+2)
m_matT.coeffRef(i,i-3) = Scalar(0);
}
}
} // end namespace Eigen
#endif // EIGEN_REAL_SCHUR_H