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-rw-r--r--Eigen/src/Eigenvalues/CMakeLists.txt6
-rw-r--r--Eigen/src/Eigenvalues/ComplexEigenSolver.h148
-rw-r--r--Eigen/src/Eigenvalues/ComplexSchur.h237
-rw-r--r--Eigen/src/Eigenvalues/EigenSolver.h723
-rw-r--r--Eigen/src/Eigenvalues/HessenbergDecomposition.h206
-rw-r--r--Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h366
-rw-r--r--Eigen/src/Eigenvalues/Tridiagonalization.h317
7 files changed, 2003 insertions, 0 deletions
diff --git a/Eigen/src/Eigenvalues/CMakeLists.txt b/Eigen/src/Eigenvalues/CMakeLists.txt
new file mode 100644
index 000000000..193e02685
--- /dev/null
+++ b/Eigen/src/Eigenvalues/CMakeLists.txt
@@ -0,0 +1,6 @@
+FILE(GLOB Eigen_EIGENVALUES_SRCS "*.h")
+
+INSTALL(FILES
+ ${Eigen_EIGENVALUES_SRCS}
+ DESTINATION ${INCLUDE_INSTALL_DIR}/Eigen/src/Eigenvalues COMPONENT Devel
+ )
diff --git a/Eigen/src/Eigenvalues/ComplexEigenSolver.h b/Eigen/src/Eigenvalues/ComplexEigenSolver.h
new file mode 100644
index 000000000..666381949
--- /dev/null
+++ b/Eigen/src/Eigenvalues/ComplexEigenSolver.h
@@ -0,0 +1,148 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2009 Claire Maurice
+// Copyright (C) 2009 Gael Guennebaud <g.gael@free.fr>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_COMPLEX_EIGEN_SOLVER_H
+#define EIGEN_COMPLEX_EIGEN_SOLVER_H
+
+/** \eigenvalues_module \ingroup Eigenvalues_Module
+ * \nonstableyet
+ *
+ * \class ComplexEigenSolver
+ *
+ * \brief Eigen values/vectors solver for general complex matrices
+ *
+ * \param MatrixType the type of the matrix of which we are computing the eigen decomposition
+ *
+ * \sa class EigenSolver, class SelfAdjointEigenSolver
+ */
+template<typename _MatrixType> class ComplexEigenSolver
+{
+ public:
+ typedef _MatrixType MatrixType;
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ typedef std::complex<RealScalar> Complex;
+ typedef Matrix<Complex, MatrixType::ColsAtCompileTime,1> EigenvalueType;
+ typedef Matrix<Complex, MatrixType::RowsAtCompileTime,MatrixType::ColsAtCompileTime> EigenvectorType;
+
+ /**
+ * \brief Default Constructor.
+ *
+ * The default constructor is useful in cases in which the user intends to
+ * perform decompositions via ComplexEigenSolver::compute(const MatrixType&).
+ */
+ ComplexEigenSolver() : m_eivec(), m_eivalues(), m_isInitialized(false)
+ {}
+
+ ComplexEigenSolver(const MatrixType& matrix)
+ : m_eivec(matrix.rows(),matrix.cols()),
+ m_eivalues(matrix.cols()),
+ m_isInitialized(false)
+ {
+ compute(matrix);
+ }
+
+ EigenvectorType eigenvectors(void) const
+ {
+ ei_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
+ return m_eivec;
+ }
+
+ EigenvalueType eigenvalues() const
+ {
+ ei_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
+ return m_eivalues;
+ }
+
+ void compute(const MatrixType& matrix);
+
+ protected:
+ MatrixType m_eivec;
+ EigenvalueType m_eivalues;
+ bool m_isInitialized;
+};
+
+
+template<typename MatrixType>
+void ComplexEigenSolver<MatrixType>::compute(const MatrixType& matrix)
+{
+ // this code is inspired from Jampack
+ assert(matrix.cols() == matrix.rows());
+ int n = matrix.cols();
+ m_eivalues.resize(n,1);
+
+ RealScalar eps = epsilon<RealScalar>();
+
+ // Reduce to complex Schur form
+ ComplexSchur<MatrixType> schur(matrix);
+
+ m_eivalues = schur.matrixT().diagonal();
+
+ m_eivec.setZero();
+
+ Scalar d2, z;
+ RealScalar norm = matrix.norm();
+
+ // compute the (normalized) eigenvectors
+ for(int k=n-1 ; k>=0 ; k--)
+ {
+ d2 = schur.matrixT().coeff(k,k);
+ m_eivec.coeffRef(k,k) = Scalar(1.0,0.0);
+ for(int i=k-1 ; i>=0 ; i--)
+ {
+ m_eivec.coeffRef(i,k) = -schur.matrixT().coeff(i,k);
+ if(k-i-1>0)
+ m_eivec.coeffRef(i,k) -= (schur.matrixT().row(i).segment(i+1,k-i-1) * m_eivec.col(k).segment(i+1,k-i-1)).value();
+ z = schur.matrixT().coeff(i,i) - d2;
+ if(z==Scalar(0))
+ ei_real_ref(z) = eps * norm;
+ m_eivec.coeffRef(i,k) = m_eivec.coeff(i,k) / z;
+
+ }
+ m_eivec.col(k).normalize();
+ }
+
+ m_eivec = schur.matrixU() * m_eivec;
+ m_isInitialized = true;
+
+ // sort the eigenvalues
+ {
+ for (int i=0; i<n; i++)
+ {
+ int k;
+ m_eivalues.cwise().abs().end(n-i).minCoeff(&k);
+ if (k != 0)
+ {
+ k += i;
+ std::swap(m_eivalues[k],m_eivalues[i]);
+ m_eivec.col(i).swap(m_eivec.col(k));
+ }
+ }
+ }
+}
+
+
+
+#endif // EIGEN_COMPLEX_EIGEN_SOLVER_H
diff --git a/Eigen/src/Eigenvalues/ComplexSchur.h b/Eigen/src/Eigenvalues/ComplexSchur.h
new file mode 100644
index 000000000..58e2ea440
--- /dev/null
+++ b/Eigen/src/Eigenvalues/ComplexSchur.h
@@ -0,0 +1,237 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2009 Claire Maurice
+// Copyright (C) 2009 Gael Guennebaud <g.gael@free.fr>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_COMPLEX_SCHUR_H
+#define EIGEN_COMPLEX_SCHUR_H
+
+/** \eigenvalues_module \ingroup Eigenvalues_Module
+ * \nonstableyet
+ *
+ * \class ComplexShur
+ *
+ * \brief Performs a complex Shur decomposition of a real or complex square matrix
+ *
+ */
+template<typename _MatrixType> class ComplexSchur
+{
+ public:
+ typedef _MatrixType MatrixType;
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ typedef std::complex<RealScalar> Complex;
+ typedef Matrix<Complex, MatrixType::RowsAtCompileTime,MatrixType::ColsAtCompileTime> ComplexMatrixType;
+
+ /**
+ * \brief Default Constructor.
+ *
+ * The default constructor is useful in cases in which the user intends to
+ * perform decompositions via ComplexSchur::compute(const MatrixType&).
+ */
+ ComplexSchur() : m_matT(), m_matU(), m_isInitialized(false)
+ {}
+
+ ComplexSchur(const MatrixType& matrix)
+ : m_matT(matrix.rows(),matrix.cols()),
+ m_matU(matrix.rows(),matrix.cols()),
+ m_isInitialized(false)
+ {
+ compute(matrix);
+ }
+
+ ComplexMatrixType matrixU() const
+ {
+ ei_assert(m_isInitialized && "ComplexSchur is not initialized.");
+ return m_matU;
+ }
+
+ ComplexMatrixType matrixT() const
+ {
+ ei_assert(m_isInitialized && "ComplexShur is not initialized.");
+ return m_matT;
+ }
+
+ void compute(const MatrixType& matrix);
+
+ protected:
+ ComplexMatrixType m_matT, m_matU;
+ bool m_isInitialized;
+};
+
+/** Computes the principal value of the square root of the complex \a z. */
+template<typename RealScalar>
+std::complex<RealScalar> ei_sqrt(const std::complex<RealScalar> &z)
+{
+ RealScalar t, tre, tim;
+
+ t = ei_abs(z);
+
+ if (ei_abs(ei_real(z)) <= ei_abs(ei_imag(z)))
+ {
+ // No cancellation in these formulas
+ tre = ei_sqrt(0.5*(t + ei_real(z)));
+ tim = ei_sqrt(0.5*(t - ei_real(z)));
+ }
+ else
+ {
+ // Stable computation of the above formulas
+ if (z.real() > 0)
+ {
+ tre = t + z.real();
+ tim = ei_abs(ei_imag(z))*ei_sqrt(0.5/tre);
+ tre = ei_sqrt(0.5*tre);
+ }
+ else
+ {
+ tim = t - z.real();
+ tre = ei_abs(ei_imag(z))*ei_sqrt(0.5/tim);
+ tim = ei_sqrt(0.5*tim);
+ }
+ }
+ if(z.imag() < 0)
+ tim = -tim;
+
+ return (std::complex<RealScalar>(tre,tim));
+
+}
+
+template<typename MatrixType>
+void ComplexSchur<MatrixType>::compute(const MatrixType& matrix)
+{
+ // this code is inspired from Jampack
+ assert(matrix.cols() == matrix.rows());
+ int n = matrix.cols();
+
+ // Reduce to Hessenberg form
+ HessenbergDecomposition<MatrixType> hess(matrix);
+
+ m_matT = hess.matrixH();
+ m_matU = hess.matrixQ();
+
+ int iu = m_matT.cols() - 1;
+ int il;
+ RealScalar d,sd,sf;
+ Complex c,b,disc,r1,r2,kappa;
+
+ RealScalar eps = epsilon<RealScalar>();
+
+ int iter = 0;
+ while(true)
+ {
+ //locate the range in which to iterate
+ while(iu > 0)
+ {
+ d = ei_norm1(m_matT.coeffRef(iu,iu)) + ei_norm1(m_matT.coeffRef(iu-1,iu-1));
+ sd = ei_norm1(m_matT.coeffRef(iu,iu-1));
+
+ if(sd >= eps * d) break; // FIXME : precision criterion ??
+
+ m_matT.coeffRef(iu,iu-1) = Complex(0);
+ iter = 0;
+ --iu;
+ }
+ if(iu==0) break;
+ iter++;
+
+ if(iter >= 30)
+ {
+ // FIXME : what to do when iter==MAXITER ??
+ std::cerr << "MAXITER" << std::endl;
+ return;
+ }
+
+ il = iu-1;
+ while( il > 0 )
+ {
+ // check if the current 2x2 block on the diagonal is upper triangular
+ d = ei_norm1(m_matT.coeffRef(il,il)) + ei_norm1(m_matT.coeffRef(il-1,il-1));
+ sd = ei_norm1(m_matT.coeffRef(il,il-1));
+
+ if(sd < eps * d) break; // FIXME : precision criterion ??
+
+ --il;
+ }
+
+ if( il != 0 ) m_matT.coeffRef(il,il-1) = Complex(0);
+
+ // compute the shift (the normalization by sf is to avoid under/overflow)
+ Matrix<Scalar,2,2> t = m_matT.template block<2,2>(iu-1,iu-1);
+ sf = t.cwise().abs().sum();
+ t /= sf;
+
+ c = t.determinant();
+ b = t.diagonal().sum();
+
+ disc = ei_sqrt(b*b - RealScalar(4)*c);
+
+ r1 = (b+disc)/RealScalar(2);
+ r2 = (b-disc)/RealScalar(2);
+
+ if(ei_norm1(r1) > ei_norm1(r2))
+ r2 = c/r1;
+ else
+ r1 = c/r2;
+
+ if(ei_norm1(r1-t.coeff(1,1)) < ei_norm1(r2-t.coeff(1,1)))
+ kappa = sf * r1;
+ else
+ kappa = sf * r2;
+
+ // perform the QR step using Givens rotations
+ PlanarRotation<Complex> rot;
+ rot.makeGivens(m_matT.coeff(il,il) - kappa, m_matT.coeff(il+1,il));
+
+ for(int i=il ; i<iu ; i++)
+ {
+ m_matT.block(0,i,n,n-i).applyOnTheLeft(i, i+1, rot.adjoint());
+ m_matT.block(0,0,std::min(i+2,iu)+1,n).applyOnTheRight(i, i+1, rot);
+ m_matU.applyOnTheRight(i, i+1, rot);
+
+ if(i != iu-1)
+ {
+ int i1 = i+1;
+ int i2 = i+2;
+
+ rot.makeGivens(m_matT.coeffRef(i1,i), m_matT.coeffRef(i2,i), &m_matT.coeffRef(i1,i));
+ m_matT.coeffRef(i2,i) = Complex(0);
+ }
+ }
+ }
+
+ // FIXME : is it necessary ?
+ /*
+ for(int i=0 ; i<n ; i++)
+ for(int j=0 ; j<n ; j++)
+ {
+ if(ei_abs(ei_real(m_matT.coeff(i,j))) < eps)
+ ei_real_ref(m_matT.coeffRef(i,j)) = 0;
+ if(ei_imag(ei_abs(m_matT.coeff(i,j))) < eps)
+ ei_imag_ref(m_matT.coeffRef(i,j)) = 0;
+ }
+ */
+
+ m_isInitialized = true;
+}
+
+#endif // EIGEN_COMPLEX_SCHUR_H
diff --git a/Eigen/src/Eigenvalues/EigenSolver.h b/Eigen/src/Eigenvalues/EigenSolver.h
new file mode 100644
index 000000000..3fc36c080
--- /dev/null
+++ b/Eigen/src/Eigenvalues/EigenSolver.h
@@ -0,0 +1,723 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_EIGENSOLVER_H
+#define EIGEN_EIGENSOLVER_H
+
+/** \eigenvalues_module \ingroup Eigenvalues_Module
+ * \nonstableyet
+ *
+ * \class EigenSolver
+ *
+ * \brief Eigen values/vectors solver for non selfadjoint matrices
+ *
+ * \param MatrixType the type of the matrix of which we are computing the eigen decomposition
+ *
+ * Currently it only support real matrices.
+ *
+ * \note this code was adapted from JAMA (public domain)
+ *
+ * \sa MatrixBase::eigenvalues(), SelfAdjointEigenSolver
+ */
+template<typename _MatrixType> class EigenSolver
+{
+ public:
+
+ typedef _MatrixType MatrixType;
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ typedef std::complex<RealScalar> Complex;
+ typedef Matrix<Complex, MatrixType::ColsAtCompileTime, 1> EigenvalueType;
+ typedef Matrix<Complex, MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime> EigenvectorType;
+ typedef Matrix<RealScalar, MatrixType::ColsAtCompileTime, 1> RealVectorType;
+ typedef Matrix<RealScalar, Dynamic, 1> RealVectorTypeX;
+
+ /**
+ * \brief Default Constructor.
+ *
+ * The default constructor is useful in cases in which the user intends to
+ * perform decompositions via EigenSolver::compute(const MatrixType&).
+ */
+ EigenSolver() : m_eivec(), m_eivalues(), m_isInitialized(false) {}
+
+ EigenSolver(const MatrixType& matrix)
+ : m_eivec(matrix.rows(), matrix.cols()),
+ m_eivalues(matrix.cols()),
+ m_isInitialized(false)
+ {
+ compute(matrix);
+ }
+
+
+ EigenvectorType eigenvectors(void) const;
+
+ /** \returns a real matrix V of pseudo eigenvectors.
+ *
+ * Let D be the block diagonal matrix with the real eigenvalues in 1x1 blocks,
+ * and any complex values u+iv in 2x2 blocks [u v ; -v u]. Then, the matrices D
+ * and V satisfy A*V = V*D.
+ *
+ * More precisely, if the diagonal matrix of the eigen values is:\n
+ * \f$
+ * \left[ \begin{array}{cccccc}
+ * u+iv & & & & & \\
+ * & u-iv & & & & \\
+ * & & a+ib & & & \\
+ * & & & a-ib & & \\
+ * & & & & x & \\
+ * & & & & & y \\
+ * \end{array} \right]
+ * \f$ \n
+ * then, we have:\n
+ * \f$
+ * D =\left[ \begin{array}{cccccc}
+ * u & v & & & & \\
+ * -v & u & & & & \\
+ * & & a & b & & \\
+ * & & -b & a & & \\
+ * & & & & x & \\
+ * & & & & & y \\
+ * \end{array} \right]
+ * \f$
+ *
+ * \sa pseudoEigenvalueMatrix()
+ */
+ const MatrixType& pseudoEigenvectors() const
+ {
+ ei_assert(m_isInitialized && "EigenSolver is not initialized.");
+ return m_eivec;
+ }
+
+ MatrixType pseudoEigenvalueMatrix() const;
+
+ /** \returns the eigenvalues as a column vector */
+ EigenvalueType eigenvalues() const
+ {
+ ei_assert(m_isInitialized && "EigenSolver is not initialized.");
+ return m_eivalues;
+ }
+
+ EigenSolver& compute(const MatrixType& matrix);
+
+ private:
+
+ void orthes(MatrixType& matH, RealVectorType& ort);
+ void hqr2(MatrixType& matH);
+
+ protected:
+ MatrixType m_eivec;
+ EigenvalueType m_eivalues;
+ bool m_isInitialized;
+};
+
+/** \returns the real block diagonal matrix D of the eigenvalues.
+ *
+ * See pseudoEigenvectors() for the details.
+ */
+template<typename MatrixType>
+MatrixType EigenSolver<MatrixType>::pseudoEigenvalueMatrix() const
+{
+ ei_assert(m_isInitialized && "EigenSolver is not initialized.");
+ int n = m_eivec.cols();
+ MatrixType matD = MatrixType::Zero(n,n);
+ for (int i=0; i<n; ++i)
+ {
+ if (ei_isMuchSmallerThan(ei_imag(m_eivalues.coeff(i)), ei_real(m_eivalues.coeff(i))))
+ matD.coeffRef(i,i) = ei_real(m_eivalues.coeff(i));
+ else
+ {
+ matD.template block<2,2>(i,i) << ei_real(m_eivalues.coeff(i)), ei_imag(m_eivalues.coeff(i)),
+ -ei_imag(m_eivalues.coeff(i)), ei_real(m_eivalues.coeff(i));
+ ++i;
+ }
+ }
+ return matD;
+}
+
+/** \returns the normalized complex eigenvectors as a matrix of column vectors.
+ *
+ * \sa eigenvalues(), pseudoEigenvectors()
+ */
+template<typename MatrixType>
+typename EigenSolver<MatrixType>::EigenvectorType EigenSolver<MatrixType>::eigenvectors(void) const
+{
+ ei_assert(m_isInitialized && "EigenSolver is not initialized.");
+ int n = m_eivec.cols();
+ EigenvectorType matV(n,n);
+ for (int j=0; j<n; ++j)
+ {
+ if (ei_isMuchSmallerThan(ei_abs(ei_imag(m_eivalues.coeff(j))), ei_abs(ei_real(m_eivalues.coeff(j)))))
+ {
+ // we have a real eigen value
+ matV.col(j) = m_eivec.col(j).template cast<Complex>();
+ }
+ else
+ {
+ // we have a pair of complex eigen values
+ for (int i=0; i<n; ++i)
+ {
+ matV.coeffRef(i,j) = Complex(m_eivec.coeff(i,j), m_eivec.coeff(i,j+1));
+ matV.coeffRef(i,j+1) = Complex(m_eivec.coeff(i,j), -m_eivec.coeff(i,j+1));
+ }
+ matV.col(j).normalize();
+ matV.col(j+1).normalize();
+ ++j;
+ }
+ }
+ return matV;
+}
+
+template<typename MatrixType>
+EigenSolver<MatrixType>& EigenSolver<MatrixType>::compute(const MatrixType& matrix)
+{
+ assert(matrix.cols() == matrix.rows());
+ int n = matrix.cols();
+ m_eivalues.resize(n,1);
+
+ MatrixType matH = matrix;
+ RealVectorType ort(n);
+
+ // Reduce to Hessenberg form.
+ orthes(matH, ort);
+
+ // Reduce Hessenberg to real Schur form.
+ hqr2(matH);
+
+ m_isInitialized = true;
+ return *this;
+}
+
+// Nonsymmetric reduction to Hessenberg form.
+template<typename MatrixType>
+void EigenSolver<MatrixType>::orthes(MatrixType& matH, RealVectorType& ort)
+{
+ // This is derived from the Algol procedures orthes and ortran,
+ // by Martin and Wilkinson, Handbook for Auto. Comp.,
+ // Vol.ii-Linear Algebra, and the corresponding
+ // Fortran subroutines in EISPACK.
+
+ int n = m_eivec.cols();
+ int low = 0;
+ int high = n-1;
+
+ for (int m = low+1; m <= high-1; ++m)
+ {
+ // Scale column.
+ RealScalar scale = matH.block(m, m-1, high-m+1, 1).cwise().abs().sum();
+ if (scale != 0.0)
+ {
+ // Compute Householder transformation.
+ RealScalar h = 0.0;
+ // FIXME could be rewritten, but this one looks better wrt cache
+ for (int i = high; i >= m; i--)
+ {
+ ort.coeffRef(i) = matH.coeff(i,m-1)/scale;
+ h += ort.coeff(i) * ort.coeff(i);
+ }
+ RealScalar g = ei_sqrt(h);
+ if (ort.coeff(m) > 0)
+ g = -g;
+ h = h - ort.coeff(m) * g;
+ ort.coeffRef(m) = ort.coeff(m) - g;
+
+ // Apply Householder similarity transformation
+ // H = (I-u*u'/h)*H*(I-u*u')/h)
+ int bSize = high-m+1;
+ matH.block(m, m, bSize, n-m).noalias() -= ((ort.segment(m, bSize)/h)
+ * (ort.segment(m, bSize).transpose() * matH.block(m, m, bSize, n-m)));
+
+ matH.block(0, m, high+1, bSize).noalias() -= ((matH.block(0, m, high+1, bSize) * ort.segment(m, bSize))
+ * (ort.segment(m, bSize)/h).transpose());
+
+ ort.coeffRef(m) = scale*ort.coeff(m);
+ matH.coeffRef(m,m-1) = scale*g;
+ }
+ }
+
+ // Accumulate transformations (Algol's ortran).
+ m_eivec.setIdentity();
+
+ for (int m = high-1; m >= low+1; m--)
+ {
+ if (matH.coeff(m,m-1) != 0.0)
+ {
+ ort.segment(m+1, high-m) = matH.col(m-1).segment(m+1, high-m);
+
+ int bSize = high-m+1;
+ m_eivec.block(m, m, bSize, bSize).noalias() += ( (ort.segment(m, bSize) / (matH.coeff(m,m-1) * ort.coeff(m)))
+ * (ort.segment(m, bSize).transpose() * m_eivec.block(m, m, bSize, bSize)) );
+ }
+ }
+}
+
+// Complex scalar division.
+template<typename Scalar>
+std::complex<Scalar> cdiv(Scalar xr, Scalar xi, Scalar yr, Scalar yi)
+{
+ Scalar r,d;
+ if (ei_abs(yr) > ei_abs(yi))
+ {
+ r = yi/yr;
+ d = yr + r*yi;
+ return std::complex<Scalar>((xr + r*xi)/d, (xi - r*xr)/d);
+ }
+ else
+ {
+ r = yr/yi;
+ d = yi + r*yr;
+ return std::complex<Scalar>((r*xr + xi)/d, (r*xi - xr)/d);
+ }
+}
+
+
+// Nonsymmetric reduction from Hessenberg to real Schur form.
+template<typename MatrixType>
+void EigenSolver<MatrixType>::hqr2(MatrixType& matH)
+{
+ // This is derived from the Algol procedure hqr2,
+ // by Martin and Wilkinson, Handbook for Auto. Comp.,
+ // Vol.ii-Linear Algebra, and the corresponding
+ // Fortran subroutine in EISPACK.
+
+ // Initialize
+ int nn = m_eivec.cols();
+ int n = nn-1;
+ int low = 0;
+ int high = nn-1;
+ Scalar eps = ei_pow(Scalar(2),ei_is_same_type<Scalar,float>::ret ? Scalar(-23) : Scalar(-52));
+ Scalar exshift = 0.0;
+ Scalar p=0,q=0,r=0,s=0,z=0,t,w,x,y;
+
+ // Store roots isolated by balanc and compute matrix norm
+ // FIXME to be efficient the following would requires a triangular reduxion code
+ // Scalar norm = matH.upper().cwise().abs().sum() + matH.corner(BottomLeft,n,n).diagonal().cwise().abs().sum();
+ Scalar norm = 0.0;
+ for (int j = 0; j < nn; ++j)
+ {
+ // FIXME what's the purpose of the following since the condition is always false
+ if ((j < low) || (j > high))
+ {
+ m_eivalues.coeffRef(j) = Complex(matH.coeff(j,j), 0.0);
+ }
+ norm += matH.row(j).segment(std::max(j-1,0), nn-std::max(j-1,0)).cwise().abs().sum();
+ }
+
+ // Outer loop over eigenvalue index
+ int iter = 0;
+ while (n >= low)
+ {
+ // Look for single small sub-diagonal element
+ int l = n;
+ while (l > low)
+ {
+ s = ei_abs(matH.coeff(l-1,l-1)) + ei_abs(matH.coeff(l,l));
+ if (s == 0.0)
+ s = norm;
+ if (ei_abs(matH.coeff(l,l-1)) < eps * s)
+ break;
+ l--;
+ }
+
+ // Check for convergence
+ // One root found
+ if (l == n)
+ {
+ matH.coeffRef(n,n) = matH.coeff(n,n) + exshift;
+ m_eivalues.coeffRef(n) = Complex(matH.coeff(n,n), 0.0);
+ n--;
+ iter = 0;
+ }
+ else if (l == n-1) // Two roots found
+ {
+ w = matH.coeff(n,n-1) * matH.coeff(n-1,n);
+ p = (matH.coeff(n-1,n-1) - matH.coeff(n,n)) * Scalar(0.5);
+ q = p * p + w;
+ z = ei_sqrt(ei_abs(q));
+ matH.coeffRef(n,n) = matH.coeff(n,n) + exshift;
+ matH.coeffRef(n-1,n-1) = matH.coeff(n-1,n-1) + exshift;
+ x = matH.coeff(n,n);
+
+ // Scalar pair
+ if (q >= 0)
+ {
+ if (p >= 0)
+ z = p + z;
+ else
+ z = p - z;
+
+ m_eivalues.coeffRef(n-1) = Complex(x + z, 0.0);
+ m_eivalues.coeffRef(n) = Complex(z!=0.0 ? x - w / z : m_eivalues.coeff(n-1).real(), 0.0);
+
+ x = matH.coeff(n,n-1);
+ s = ei_abs(x) + ei_abs(z);
+ p = x / s;
+ q = z / s;
+ r = ei_sqrt(p * p+q * q);
+ p = p / r;
+ q = q / r;
+
+ // Row modification
+ for (int j = n-1; j < nn; ++j)
+ {
+ z = matH.coeff(n-1,j);
+ matH.coeffRef(n-1,j) = q * z + p * matH.coeff(n,j);
+ matH.coeffRef(n,j) = q * matH.coeff(n,j) - p * z;
+ }
+
+ // Column modification
+ for (int i = 0; i <= n; ++i)
+ {
+ z = matH.coeff(i,n-1);
+ matH.coeffRef(i,n-1) = q * z + p * matH.coeff(i,n);
+ matH.coeffRef(i,n) = q * matH.coeff(i,n) - p * z;
+ }
+
+ // Accumulate transformations
+ for (int i = low; i <= high; ++i)
+ {
+ z = m_eivec.coeff(i,n-1);
+ m_eivec.coeffRef(i,n-1) = q * z + p * m_eivec.coeff(i,n);
+ m_eivec.coeffRef(i,n) = q * m_eivec.coeff(i,n) - p * z;
+ }
+ }
+ else // Complex pair
+ {
+ m_eivalues.coeffRef(n-1) = Complex(x + p, z);
+ m_eivalues.coeffRef(n) = Complex(x + p, -z);
+ }
+ n = n - 2;
+ iter = 0;
+ }
+ else // No convergence yet
+ {
+ // Form shift
+ x = matH.coeff(n,n);
+ y = 0.0;
+ w = 0.0;
+ if (l < n)
+ {
+ y = matH.coeff(n-1,n-1);
+ w = matH.coeff(n,n-1) * matH.coeff(n-1,n);
+ }
+
+ // Wilkinson's original ad hoc shift
+ if (iter == 10)
+ {
+ exshift += x;
+ for (int i = low; i <= n; ++i)
+ matH.coeffRef(i,i) -= x;
+ s = ei_abs(matH.coeff(n,n-1)) + ei_abs(matH.coeff(n-1,n-2));
+ x = y = Scalar(0.75) * s;
+ w = Scalar(-0.4375) * s * s;
+ }
+
+ // MATLAB's new ad hoc shift
+ if (iter == 30)
+ {
+ s = Scalar((y - x) / 2.0);
+ s = s * s + w;
+ if (s > 0)
+ {
+ s = ei_sqrt(s);
+ if (y < x)
+ s = -s;
+ s = Scalar(x - w / ((y - x) / 2.0 + s));
+ for (int i = low; i <= n; ++i)
+ matH.coeffRef(i,i) -= s;
+ exshift += s;
+ x = y = w = Scalar(0.964);
+ }
+ }
+
+ iter = iter + 1; // (Could check iteration count here.)
+
+ // Look for two consecutive small sub-diagonal elements
+ int m = n-2;
+ while (m >= l)
+ {
+ z = matH.coeff(m,m);
+ r = x - z;
+ s = y - z;
+ p = (r * s - w) / matH.coeff(m+1,m) + matH.coeff(m,m+1);
+ q = matH.coeff(m+1,m+1) - z - r - s;
+ r = matH.coeff(m+2,m+1);
+ s = ei_abs(p) + ei_abs(q) + ei_abs(r);
+ p = p / s;
+ q = q / s;
+ r = r / s;
+ if (m == l) {
+ break;
+ }
+ if (ei_abs(matH.coeff(m,m-1)) * (ei_abs(q) + ei_abs(r)) <
+ eps * (ei_abs(p) * (ei_abs(matH.coeff(m-1,m-1)) + ei_abs(z) +
+ ei_abs(matH.coeff(m+1,m+1)))))
+ {
+ break;
+ }
+ m--;
+ }
+
+ for (int i = m+2; i <= n; ++i)
+ {
+ matH.coeffRef(i,i-2) = 0.0;
+ if (i > m+2)
+ matH.coeffRef(i,i-3) = 0.0;
+ }
+
+ // Double QR step involving rows l:n and columns m:n
+ for (int k = m; k <= n-1; ++k)
+ {
+ int notlast = (k != n-1);
+ if (k != m) {
+ p = matH.coeff(k,k-1);
+ q = matH.coeff(k+1,k-1);
+ r = notlast ? matH.coeff(k+2,k-1) : Scalar(0);
+ x = ei_abs(p) + ei_abs(q) + ei_abs(r);
+ if (x != 0.0)
+ {
+ p = p / x;
+ q = q / x;
+ r = r / x;
+ }
+ }
+
+ if (x == 0.0)
+ break;
+
+ s = ei_sqrt(p * p + q * q + r * r);
+
+ if (p < 0)
+ s = -s;
+
+ if (s != 0)
+ {
+ if (k != m)
+ matH.coeffRef(k,k-1) = -s * x;
+ else if (l != m)
+ matH.coeffRef(k,k-1) = -matH.coeff(k,k-1);
+
+ p = p + s;
+ x = p / s;
+ y = q / s;
+ z = r / s;
+ q = q / p;
+ r = r / p;
+
+ // Row modification
+ for (int j = k; j < nn; ++j)
+ {
+ p = matH.coeff(k,j) + q * matH.coeff(k+1,j);
+ if (notlast)
+ {
+ p = p + r * matH.coeff(k+2,j);
+ matH.coeffRef(k+2,j) = matH.coeff(k+2,j) - p * z;
+ }
+ matH.coeffRef(k,j) = matH.coeff(k,j) - p * x;
+ matH.coeffRef(k+1,j) = matH.coeff(k+1,j) - p * y;
+ }
+
+ // Column modification
+ for (int i = 0; i <= std::min(n,k+3); ++i)
+ {
+ p = x * matH.coeff(i,k) + y * matH.coeff(i,k+1);
+ if (notlast)
+ {
+ p = p + z * matH.coeff(i,k+2);
+ matH.coeffRef(i,k+2) = matH.coeff(i,k+2) - p * r;
+ }
+ matH.coeffRef(i,k) = matH.coeff(i,k) - p;
+ matH.coeffRef(i,k+1) = matH.coeff(i,k+1) - p * q;
+ }
+
+ // Accumulate transformations
+ for (int i = low; i <= high; ++i)
+ {
+ p = x * m_eivec.coeff(i,k) + y * m_eivec.coeff(i,k+1);
+ if (notlast)
+ {
+ p = p + z * m_eivec.coeff(i,k+2);
+ m_eivec.coeffRef(i,k+2) = m_eivec.coeff(i,k+2) - p * r;
+ }
+ m_eivec.coeffRef(i,k) = m_eivec.coeff(i,k) - p;
+ m_eivec.coeffRef(i,k+1) = m_eivec.coeff(i,k+1) - p * q;
+ }
+ } // (s != 0)
+ } // k loop
+ } // check convergence
+ } // while (n >= low)
+
+ // Backsubstitute to find vectors of upper triangular form
+ if (norm == 0.0)
+ {
+ return;
+ }
+
+ for (n = nn-1; n >= 0; n--)
+ {
+ p = m_eivalues.coeff(n).real();
+ q = m_eivalues.coeff(n).imag();
+
+ // Scalar vector
+ if (q == 0)
+ {
+ int l = n;
+ matH.coeffRef(n,n) = 1.0;
+ for (int i = n-1; i >= 0; i--)
+ {
+ w = matH.coeff(i,i) - p;
+ r = matH.row(i).segment(l,n-l+1).dot(matH.col(n).segment(l, n-l+1));
+
+ if (m_eivalues.coeff(i).imag() < 0.0)
+ {
+ z = w;
+ s = r;
+ }
+ else
+ {
+ l = i;
+ if (m_eivalues.coeff(i).imag() == 0.0)
+ {
+ if (w != 0.0)
+ matH.coeffRef(i,n) = -r / w;
+ else
+ matH.coeffRef(i,n) = -r / (eps * norm);
+ }
+ else // Solve real equations
+ {
+ x = matH.coeff(i,i+1);
+ y = matH.coeff(i+1,i);
+ q = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag();
+ t = (x * s - z * r) / q;
+ matH.coeffRef(i,n) = t;
+ if (ei_abs(x) > ei_abs(z))
+ matH.coeffRef(i+1,n) = (-r - w * t) / x;
+ else
+ matH.coeffRef(i+1,n) = (-s - y * t) / z;
+ }
+
+ // Overflow control
+ t = ei_abs(matH.coeff(i,n));
+ if ((eps * t) * t > 1)
+ matH.col(n).end(nn-i) /= t;
+ }
+ }
+ }
+ else if (q < 0) // Complex vector
+ {
+ std::complex<Scalar> cc;
+ int l = n-1;
+
+ // Last vector component imaginary so matrix is triangular
+ if (ei_abs(matH.coeff(n,n-1)) > ei_abs(matH.coeff(n-1,n)))
+ {
+ matH.coeffRef(n-1,n-1) = q / matH.coeff(n,n-1);
+ matH.coeffRef(n-1,n) = -(matH.coeff(n,n) - p) / matH.coeff(n,n-1);
+ }
+ else
+ {
+ cc = cdiv<Scalar>(0.0,-matH.coeff(n-1,n),matH.coeff(n-1,n-1)-p,q);
+ matH.coeffRef(n-1,n-1) = ei_real(cc);
+ matH.coeffRef(n-1,n) = ei_imag(cc);
+ }
+ matH.coeffRef(n,n-1) = 0.0;
+ matH.coeffRef(n,n) = 1.0;
+ for (int i = n-2; i >= 0; i--)
+ {
+ Scalar ra,sa,vr,vi;
+ ra = matH.row(i).segment(l, n-l+1).dot(matH.col(n-1).segment(l, n-l+1));
+ sa = matH.row(i).segment(l, n-l+1).dot(matH.col(n).segment(l, n-l+1));
+ w = matH.coeff(i,i) - p;
+
+ if (m_eivalues.coeff(i).imag() < 0.0)
+ {
+ z = w;
+ r = ra;
+ s = sa;
+ }
+ else
+ {
+ l = i;
+ if (m_eivalues.coeff(i).imag() == 0)
+ {
+ cc = cdiv(-ra,-sa,w,q);
+ matH.coeffRef(i,n-1) = ei_real(cc);
+ matH.coeffRef(i,n) = ei_imag(cc);
+ }
+ else
+ {
+ // Solve complex equations
+ x = matH.coeff(i,i+1);
+ y = matH.coeff(i+1,i);
+ vr = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag() - q * q;
+ vi = (m_eivalues.coeff(i).real() - p) * Scalar(2) * q;
+ if ((vr == 0.0) && (vi == 0.0))
+ vr = eps * norm * (ei_abs(w) + ei_abs(q) + ei_abs(x) + ei_abs(y) + ei_abs(z));
+
+ cc= cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
+ matH.coeffRef(i,n-1) = ei_real(cc);
+ matH.coeffRef(i,n) = ei_imag(cc);
+ if (ei_abs(x) > (ei_abs(z) + ei_abs(q)))
+ {
+ matH.coeffRef(i+1,n-1) = (-ra - w * matH.coeff(i,n-1) + q * matH.coeff(i,n)) / x;
+ matH.coeffRef(i+1,n) = (-sa - w * matH.coeff(i,n) - q * matH.coeff(i,n-1)) / x;
+ }
+ else
+ {
+ cc = cdiv(-r-y*matH.coeff(i,n-1),-s-y*matH.coeff(i,n),z,q);
+ matH.coeffRef(i+1,n-1) = ei_real(cc);
+ matH.coeffRef(i+1,n) = ei_imag(cc);
+ }
+ }
+
+ // Overflow control
+ t = std::max(ei_abs(matH.coeff(i,n-1)),ei_abs(matH.coeff(i,n)));
+ if ((eps * t) * t > 1)
+ matH.block(i, n-1, nn-i, 2) /= t;
+
+ }
+ }
+ }
+ }
+
+ // Vectors of isolated roots
+ for (int i = 0; i < nn; ++i)
+ {
+ // FIXME again what's the purpose of this test ?
+ // in this algo low==0 and high==nn-1 !!
+ if (i < low || i > high)
+ {
+ m_eivec.row(i).end(nn-i) = matH.row(i).end(nn-i);
+ }
+ }
+
+ // Back transformation to get eigenvectors of original matrix
+ int bRows = high-low+1;
+ for (int j = nn-1; j >= low; j--)
+ {
+ int bSize = std::min(j,high)-low+1;
+ m_eivec.col(j).segment(low, bRows) = (m_eivec.block(low, low, bRows, bSize) * matH.col(j).segment(low, bSize));
+ }
+}
+
+#endif // EIGEN_EIGENSOLVER_H
diff --git a/Eigen/src/Eigenvalues/HessenbergDecomposition.h b/Eigen/src/Eigenvalues/HessenbergDecomposition.h
new file mode 100644
index 000000000..b1e21d4ee
--- /dev/null
+++ b/Eigen/src/Eigenvalues/HessenbergDecomposition.h
@@ -0,0 +1,206 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_HESSENBERGDECOMPOSITION_H
+#define EIGEN_HESSENBERGDECOMPOSITION_H
+
+/** \eigenvalues_module \ingroup Eigenvalues_Module
+ * \nonstableyet
+ *
+ * \class HessenbergDecomposition
+ *
+ * \brief Reduces a squared matrix to an Hessemberg form
+ *
+ * \param MatrixType the type of the matrix of which we are computing the Hessenberg decomposition
+ *
+ * This class performs an Hessenberg decomposition of a matrix \f$ A \f$ such that:
+ * \f$ A = Q H Q^* \f$ where \f$ Q \f$ is unitary and \f$ H \f$ a Hessenberg matrix.
+ *
+ * \sa class Tridiagonalization, class Qr
+ */
+template<typename _MatrixType> class HessenbergDecomposition
+{
+ public:
+
+ typedef _MatrixType MatrixType;
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+
+ enum {
+ Size = MatrixType::RowsAtCompileTime,
+ SizeMinusOne = MatrixType::RowsAtCompileTime==Dynamic
+ ? Dynamic
+ : MatrixType::RowsAtCompileTime-1
+ };
+
+ typedef Matrix<Scalar, SizeMinusOne, 1> CoeffVectorType;
+ typedef Matrix<RealScalar, Size, 1> DiagonalType;
+ typedef Matrix<RealScalar, SizeMinusOne, 1> SubDiagonalType;
+
+ typedef typename NestByValue<Diagonal<MatrixType,0> >::RealReturnType DiagonalReturnType;
+
+ typedef typename NestByValue<Diagonal<
+ NestByValue<Block<MatrixType,SizeMinusOne,SizeMinusOne> >,0 > >::RealReturnType SubDiagonalReturnType;
+
+ /** This constructor initializes a HessenbergDecomposition object for
+ * further use with HessenbergDecomposition::compute()
+ */
+ HessenbergDecomposition(int size = Size==Dynamic ? 2 : Size)
+ : m_matrix(size,size), m_hCoeffs(size-1)
+ {}
+
+ HessenbergDecomposition(const MatrixType& matrix)
+ : m_matrix(matrix),
+ m_hCoeffs(matrix.cols()-1)
+ {
+ _compute(m_matrix, m_hCoeffs);
+ }
+
+ /** Computes or re-compute the Hessenberg decomposition for the matrix \a matrix.
+ *
+ * This method allows to re-use the allocated data.
+ */
+ void compute(const MatrixType& matrix)
+ {
+ m_matrix = matrix;
+ m_hCoeffs.resize(matrix.rows()-1,1);
+ _compute(m_matrix, m_hCoeffs);
+ }
+
+ /** \returns the householder coefficients allowing to
+ * reconstruct the matrix Q from the packed data.
+ *
+ * \sa packedMatrix()
+ */
+ CoeffVectorType householderCoefficients() const { return m_hCoeffs; }
+
+ /** \returns the internal result of the decomposition.
+ *
+ * The returned matrix contains the following information:
+ * - the upper part and lower sub-diagonal represent the Hessenberg matrix H
+ * - the rest of the lower part contains the Householder vectors that, combined with
+ * Householder coefficients returned by householderCoefficients(),
+ * allows to reconstruct the matrix Q as follow:
+ * Q = H_{N-1} ... H_1 H_0
+ * where the matrices H are the Householder transformation:
+ * H_i = (I - h_i * v_i * v_i')
+ * where h_i == householderCoefficients()[i] and v_i is a Householder vector:
+ * v_i = [ 0, ..., 0, 1, M(i+2,i), ..., M(N-1,i) ]
+ *
+ * See LAPACK for further details on this packed storage.
+ */
+ const MatrixType& packedMatrix(void) const { return m_matrix; }
+
+ MatrixType matrixQ() const;
+ MatrixType matrixH() const;
+
+ private:
+
+ static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs);
+
+ protected:
+ MatrixType m_matrix;
+ CoeffVectorType m_hCoeffs;
+};
+
+#ifndef EIGEN_HIDE_HEAVY_CODE
+
+/** \internal
+ * Performs a tridiagonal decomposition of \a matA in place.
+ *
+ * \param matA the input selfadjoint matrix
+ * \param hCoeffs returned Householder coefficients
+ *
+ * The result is written in the lower triangular part of \a matA.
+ *
+ * Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
+ *
+ * \sa packedMatrix()
+ */
+template<typename MatrixType>
+void HessenbergDecomposition<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs)
+{
+ assert(matA.rows()==matA.cols());
+ int n = matA.rows();
+ Matrix<Scalar,1,Dynamic> temp(n);
+ for (int i = 0; i<n-1; ++i)
+ {
+ // let's consider the vector v = i-th column starting at position i+1
+ int remainingSize = n-i-1;
+ RealScalar beta;
+ Scalar h;
+ matA.col(i).end(remainingSize).makeHouseholderInPlace(&h, &beta);
+ matA.col(i).coeffRef(i+1) = beta;
+ hCoeffs.coeffRef(i) = h;
+
+ // Apply similarity transformation to remaining columns,
+ // i.e., compute A = H A H'
+
+ // A = H A
+ matA.corner(BottomRight, remainingSize, remainingSize)
+ .applyHouseholderOnTheLeft(matA.col(i).end(remainingSize-1), h, &temp.coeffRef(0));
+
+ // A = A H'
+ matA.corner(BottomRight, n, remainingSize)
+ .applyHouseholderOnTheRight(matA.col(i).end(remainingSize-1).conjugate(), ei_conj(h), &temp.coeffRef(0));
+ }
+}
+
+/** reconstructs and returns the matrix Q */
+template<typename MatrixType>
+typename HessenbergDecomposition<MatrixType>::MatrixType
+HessenbergDecomposition<MatrixType>::matrixQ() const
+{
+ int n = m_matrix.rows();
+ MatrixType matQ = MatrixType::Identity(n,n);
+ Matrix<Scalar,1,MatrixType::ColsAtCompileTime> temp(n);
+ for (int i = n-2; i>=0; i--)
+ {
+ matQ.corner(BottomRight,n-i-1,n-i-1)
+ .applyHouseholderOnTheLeft(m_matrix.col(i).end(n-i-2), ei_conj(m_hCoeffs.coeff(i)), &temp.coeffRef(0,0));
+ }
+ return matQ;
+}
+
+#endif // EIGEN_HIDE_HEAVY_CODE
+
+/** constructs and returns the matrix H.
+ * Note that the matrix H is equivalent to the upper part of the packed matrix
+ * (including the lower sub-diagonal). Therefore, it might be often sufficient
+ * to directly use the packed matrix instead of creating a new one.
+ */
+template<typename MatrixType>
+typename HessenbergDecomposition<MatrixType>::MatrixType
+HessenbergDecomposition<MatrixType>::matrixH() const
+{
+ // FIXME should this function (and other similar) rather take a matrix as argument
+ // and fill it (to avoid temporaries)
+ int n = m_matrix.rows();
+ MatrixType matH = m_matrix;
+ if (n>2)
+ matH.corner(BottomLeft,n-2, n-2).template triangularView<LowerTriangular>().setZero();
+ return matH;
+}
+
+#endif // EIGEN_HESSENBERGDECOMPOSITION_H
diff --git a/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h b/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h
new file mode 100644
index 000000000..84856aa66
--- /dev/null
+++ b/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h
@@ -0,0 +1,366 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_SELFADJOINTEIGENSOLVER_H
+#define EIGEN_SELFADJOINTEIGENSOLVER_H
+
+/** \eigenvalues_module \ingroup Eigenvalues_Module
+ * \nonstableyet
+ *
+ * \class SelfAdjointEigenSolver
+ *
+ * \brief Eigen values/vectors solver for selfadjoint matrix
+ *
+ * \param MatrixType the type of the matrix of which we are computing the eigen decomposition
+ *
+ * \note MatrixType must be an actual Matrix type, it can't be an expression type.
+ *
+ * \sa MatrixBase::eigenvalues(), class EigenSolver
+ */
+template<typename _MatrixType> class SelfAdjointEigenSolver
+{
+ public:
+
+ enum {Size = _MatrixType::RowsAtCompileTime };
+ typedef _MatrixType MatrixType;
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ typedef std::complex<RealScalar> Complex;
+ typedef Matrix<RealScalar, MatrixType::ColsAtCompileTime, 1> RealVectorType;
+ typedef Matrix<RealScalar, Dynamic, 1> RealVectorTypeX;
+ typedef Tridiagonalization<MatrixType> TridiagonalizationType;
+// typedef typename TridiagonalizationType::TridiagonalMatrixType TridiagonalMatrixType;
+
+ SelfAdjointEigenSolver()
+ : m_eivec(int(Size), int(Size)),
+ m_eivalues(int(Size))
+ {
+ ei_assert(Size!=Dynamic);
+ }
+
+ SelfAdjointEigenSolver(int size)
+ : m_eivec(size, size),
+ m_eivalues(size)
+ {}
+
+ /** Constructors computing the eigenvalues of the selfadjoint matrix \a matrix,
+ * as well as the eigenvectors if \a computeEigenvectors is true.
+ *
+ * \sa compute(MatrixType,bool), SelfAdjointEigenSolver(MatrixType,MatrixType,bool)
+ */
+ SelfAdjointEigenSolver(const MatrixType& matrix, bool computeEigenvectors = true)
+ : m_eivec(matrix.rows(), matrix.cols()),
+ m_eivalues(matrix.cols())
+ {
+ 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.
+ *
+ * \sa compute(MatrixType,MatrixType,bool), SelfAdjointEigenSolver(MatrixType,bool)
+ */
+ SelfAdjointEigenSolver(const MatrixType& matA, const MatrixType& matB, bool computeEigenvectors = true)
+ : m_eivec(matA.rows(), matA.cols()),
+ m_eivalues(matA.cols())
+ {
+ compute(matA, matB, computeEigenvectors);
+ }
+
+ SelfAdjointEigenSolver& compute(const MatrixType& matrix, bool computeEigenvectors = true);
+
+ 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
+ {
+ #ifndef NDEBUG
+ ei_assert(m_eigenvectorsOk);
+ #endif
+ return m_eivec;
+ }
+
+ /** \returns the computed eigen values */
+ RealVectorType eigenvalues(void) const { return m_eivalues; }
+
+ /** \returns the positive square root of the matrix
+ *
+ * \note the matrix itself must be positive in order for this to make sense.
+ */
+ MatrixType operatorSqrt() const
+ {
+ return m_eivec * m_eivalues.cwise().sqrt().asDiagonal() * m_eivec.adjoint();
+ }
+
+ /** \returns the positive inverse square root of the matrix
+ *
+ * \note the matrix itself must be positive definite in order for this to make sense.
+ */
+ MatrixType operatorInverseSqrt() const
+ {
+ return m_eivec * m_eivalues.cwise().inverse().cwise().sqrt().asDiagonal() * m_eivec.adjoint();
+ }
+
+
+ protected:
+ MatrixType m_eivec;
+ RealVectorType m_eivalues;
+ #ifndef NDEBUG
+ bool m_eigenvectorsOk;
+ #endif
+};
+
+#ifndef EIGEN_HIDE_HEAVY_CODE
+
+/** \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"
+ */
+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)
+{
+ #ifndef NDEBUG
+ m_eigenvectorsOk = computeEigenvectors;
+ #endif
+ assert(matrix.cols() == matrix.rows());
+ int n = matrix.cols();
+ m_eivalues.resize(n,1);
+
+ if(n==1)
+ {
+ m_eivalues.coeffRef(0,0) = ei_real(matrix.coeff(0,0));
+ m_eivec.setOnes();
+ return *this;
+ }
+
+ m_eivec = matrix;
+
+ // FIXME, should tridiag be a local variable of this function or an attribute of SelfAdjointEigenSolver ?
+ // the latter avoids multiple memory allocation when the same SelfAdjointEigenSolver is used multiple times...
+ // (same for diag and subdiag)
+ RealVectorType& diag = m_eivalues;
+ typename TridiagonalizationType::SubDiagonalType subdiag(n-1);
+ TridiagonalizationType::decomposeInPlace(m_eivec, diag, subdiag, computeEigenvectors);
+
+ int end = n-1;
+ int start = 0;
+ while (end>0)
+ {
+ for (int i = start; i<end; ++i)
+ if (ei_isMuchSmallerThan(ei_abs(subdiag[i]),(ei_abs(diag[i])+ei_abs(diag[i+1]))))
+ subdiag[i] = 0;
+
+ // find the largest unreduced block
+ while (end>0 && subdiag[end-1]==0)
+ end--;
+ if (end<=0)
+ break;
+ start = end - 1;
+ while (start>0 && subdiag[start-1]!=0)
+ start--;
+
+ ei_tridiagonal_qr_step(diag.data(), subdiag.data(), start, end, computeEigenvectors ? m_eivec.data() : (Scalar*)0, n);
+ }
+
+ // Sort eigenvalues and corresponding vectors.
+ // TODO make the sort optional ?
+ // TODO use a better sort algorithm !!
+ for (int i = 0; i < n-1; ++i)
+ {
+ int k;
+ m_eivalues.segment(i,n-i).minCoeff(&k);
+ if (k > 0)
+ {
+ std::swap(m_eivalues[i], m_eivalues[k+i]);
+ m_eivec.col(i).swap(m_eivec.col(k+i));
+ }
+ }
+ 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)
+{
+ ei_assert(matA.cols()==matA.rows() && matB.rows()==matA.rows() && matB.cols()==matB.rows());
+
+ // Compute the cholesky decomposition of matB = L L'
+ LLT<MatrixType> cholB(matB);
+
+ // compute C = inv(L) A inv(L')
+ MatrixType matC = matA;
+ cholB.matrixL().solveInPlace(matC);
+ // FIXME since we currently do not support A * inv(L'), let's do (inv(L) A')' :
+ matC.adjointInPlace();
+ cholB.matrixL().solveInPlace(matC);
+ matC.adjointInPlace();
+ // this version works too:
+// matC = matC.transpose();
+// cholB.matrixL().conjugate().template marked<LowerTriangular>().solveTriangularInPlace(matC);
+// matC = matC.transpose();
+ // FIXME: this should work: (currently it only does for small matrices)
+// Transpose<MatrixType> trMatC(matC);
+// cholB.matrixL().conjugate().eval().template marked<LowerTriangular>().solveTriangularInPlace(trMatC);
+
+ compute(matC, computeEigenvectors);
+
+ if (computeEigenvectors)
+ {
+ // transform back the eigen vectors: evecs = inv(U) * evecs
+ cholB.matrixU().solveInPlace(m_eivec);
+ for (int i=0; i<m_eivec.cols(); ++i)
+ m_eivec.col(i) = m_eivec.col(i).normalized();
+ }
+ return *this;
+}
+
+#endif // EIGEN_HIDE_HEAVY_CODE
+
+/** \eigenvalues_module
+ *
+ * \returns a vector listing the eigenvalues of this matrix.
+ */
+template<typename Derived>
+inline Matrix<typename NumTraits<typename ei_traits<Derived>::Scalar>::Real, ei_traits<Derived>::ColsAtCompileTime, 1>
+MatrixBase<Derived>::eigenvalues() const
+{
+ ei_assert(Flags&SelfAdjointBit);
+ return SelfAdjointEigenSolver<typename Derived::PlainMatrixType>(eval(),false).eigenvalues();
+}
+
+template<typename Derived, bool IsSelfAdjoint>
+struct ei_operatorNorm_selector
+{
+ static inline typename NumTraits<typename ei_traits<Derived>::Scalar>::Real
+ operatorNorm(const MatrixBase<Derived>& m)
+ {
+ // FIXME if it is really guaranteed that the eigenvalues are already sorted,
+ // then we don't need to compute a maxCoeff() here, comparing the 1st and last ones is enough.
+ return m.eigenvalues().cwise().abs().maxCoeff();
+ }
+};
+
+template<typename Derived> struct ei_operatorNorm_selector<Derived, false>
+{
+ static inline typename NumTraits<typename ei_traits<Derived>::Scalar>::Real
+ operatorNorm(const MatrixBase<Derived>& m)
+ {
+ typename Derived::PlainMatrixType m_eval(m);
+ // FIXME if it is really guaranteed that the eigenvalues are already sorted,
+ // then we don't need to compute a maxCoeff() here, comparing the 1st and last ones is enough.
+ return ei_sqrt(
+ (m_eval*m_eval.adjoint())
+ .template marked<SelfAdjoint>()
+ .eigenvalues()
+ .maxCoeff()
+ );
+ }
+};
+
+/** \eigenvalues_module
+ *
+ * \returns the matrix norm of this matrix.
+ */
+template<typename Derived>
+inline typename NumTraits<typename ei_traits<Derived>::Scalar>::Real
+MatrixBase<Derived>::operatorNorm() const
+{
+ return ei_operatorNorm_selector<Derived, Flags&SelfAdjointBit>
+ ::operatorNorm(derived());
+}
+
+#ifndef EIGEN_EXTERN_INSTANTIATIONS
+template<typename RealScalar, typename Scalar>
+static void ei_tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, int start, int end, Scalar* matrixQ, int n)
+{
+ RealScalar td = (diag[end-1] - diag[end])*RealScalar(0.5);
+ RealScalar e2 = ei_abs2(subdiag[end-1]);
+ RealScalar mu = diag[end] - e2 / (td + (td>0 ? 1 : -1) * ei_sqrt(td*td + e2));
+ RealScalar x = diag[start] - mu;
+ RealScalar z = subdiag[start];
+
+ for (int k = start; k < end; ++k)
+ {
+ PlanarRotation<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> > q(matrixQ,n,n);
+ q.applyOnTheRight(k,k+1,rot);
+ }
+ }
+}
+#endif
+
+#endif // EIGEN_SELFADJOINTEIGENSOLVER_H
diff --git a/Eigen/src/Eigenvalues/Tridiagonalization.h b/Eigen/src/Eigenvalues/Tridiagonalization.h
new file mode 100644
index 000000000..5f891bfa6
--- /dev/null
+++ b/Eigen/src/Eigenvalues/Tridiagonalization.h
@@ -0,0 +1,317 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_TRIDIAGONALIZATION_H
+#define EIGEN_TRIDIAGONALIZATION_H
+
+/** \eigenvalues_module \ingroup Eigenvalues_Module
+ * \nonstableyet
+ *
+ * \class Tridiagonalization
+ *
+ * \brief Trigiagonal decomposition of a selfadjoint matrix
+ *
+ * \param MatrixType the type of the matrix of which we are performing the tridiagonalization
+ *
+ * This class performs a tridiagonal decomposition of a selfadjoint matrix \f$ A \f$ such that:
+ * \f$ A = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real symmetric tridiagonal matrix.
+ *
+ * \sa MatrixBase::tridiagonalize()
+ */
+template<typename _MatrixType> class Tridiagonalization
+{
+ public:
+
+ typedef _MatrixType MatrixType;
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ typedef typename ei_packet_traits<Scalar>::type Packet;
+
+ enum {
+ Size = MatrixType::RowsAtCompileTime,
+ SizeMinusOne = MatrixType::RowsAtCompileTime==Dynamic
+ ? Dynamic
+ : MatrixType::RowsAtCompileTime-1,
+ PacketSize = ei_packet_traits<Scalar>::size
+ };
+
+ typedef Matrix<Scalar, SizeMinusOne, 1> CoeffVectorType;
+ typedef Matrix<RealScalar, Size, 1> DiagonalType;
+ typedef Matrix<RealScalar, SizeMinusOne, 1> SubDiagonalType;
+
+ typedef typename ei_meta_if<NumTraits<Scalar>::IsComplex,
+ typename NestByValue<Diagonal<MatrixType,0> >::RealReturnType,
+ Diagonal<MatrixType,0>
+ >::ret DiagonalReturnType;
+
+ typedef typename ei_meta_if<NumTraits<Scalar>::IsComplex,
+ typename NestByValue<Diagonal<
+ NestByValue<Block<MatrixType,SizeMinusOne,SizeMinusOne> >,0 > >::RealReturnType,
+ Diagonal<
+ NestByValue<Block<MatrixType,SizeMinusOne,SizeMinusOne> >,0 >
+ >::ret SubDiagonalReturnType;
+
+ /** This constructor initializes a Tridiagonalization object for
+ * further use with Tridiagonalization::compute()
+ */
+ Tridiagonalization(int size = Size==Dynamic ? 2 : Size)
+ : m_matrix(size,size), m_hCoeffs(size-1)
+ {}
+
+ Tridiagonalization(const MatrixType& matrix)
+ : m_matrix(matrix), m_hCoeffs(matrix.cols()-1)
+ {
+ _compute(m_matrix, m_hCoeffs);
+ }
+
+ /** Computes or re-compute the tridiagonalization for the matrix \a matrix.
+ *
+ * This method allows to re-use the allocated data.
+ */
+ void compute(const MatrixType& matrix)
+ {
+ m_matrix = matrix;
+ m_hCoeffs.resize(matrix.rows()-1, 1);
+ _compute(m_matrix, m_hCoeffs);
+ }
+
+ /** \returns the householder coefficients allowing to
+ * reconstruct the matrix Q from the packed data.
+ *
+ * \sa packedMatrix()
+ */
+ inline CoeffVectorType householderCoefficients(void) const { return m_hCoeffs; }
+
+ /** \returns the internal result of the decomposition.
+ *
+ * The returned matrix contains the following information:
+ * - the strict upper part is equal to the input matrix A
+ * - the diagonal and lower sub-diagonal represent the tridiagonal symmetric matrix (real).
+ * - the rest of the lower part contains the Householder vectors that, combined with
+ * Householder coefficients returned by householderCoefficients(),
+ * allows to reconstruct the matrix Q as follow:
+ * Q = H_{N-1} ... H_1 H_0
+ * where the matrices H are the Householder transformations:
+ * H_i = (I - h_i * v_i * v_i')
+ * where h_i == householderCoefficients()[i] and v_i is a Householder vector:
+ * v_i = [ 0, ..., 0, 1, M(i+2,i), ..., M(N-1,i) ]
+ *
+ * See LAPACK for further details on this packed storage.
+ */
+ inline const MatrixType& packedMatrix(void) const { return m_matrix; }
+
+ MatrixType matrixQ() const;
+ template<typename QDerived> void matrixQInPlace(MatrixBase<QDerived>* q) const;
+ MatrixType matrixT() const;
+ const DiagonalReturnType diagonal(void) const;
+ const SubDiagonalReturnType subDiagonal(void) const;
+
+ static void decomposeInPlace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ = true);
+
+ static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs);
+
+ protected:
+
+ static void _decomposeInPlace3x3(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ = true);
+
+ MatrixType m_matrix;
+ CoeffVectorType m_hCoeffs;
+};
+
+/** \returns an expression of the diagonal vector */
+template<typename MatrixType>
+const typename Tridiagonalization<MatrixType>::DiagonalReturnType
+Tridiagonalization<MatrixType>::diagonal(void) const
+{
+ return m_matrix.diagonal().nestByValue();
+}
+
+/** \returns an expression of the sub-diagonal vector */
+template<typename MatrixType>
+const typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
+Tridiagonalization<MatrixType>::subDiagonal(void) const
+{
+ int n = m_matrix.rows();
+ return Block<MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1)
+ .nestByValue().diagonal().nestByValue();
+}
+
+/** constructs and returns the tridiagonal matrix T.
+ * Note that the matrix T is equivalent to the diagonal and sub-diagonal of the packed matrix.
+ * Therefore, it might be often sufficient to directly use the packed matrix, or the vector
+ * expressions returned by diagonal() and subDiagonal() instead of creating a new matrix.
+ */
+template<typename MatrixType>
+typename Tridiagonalization<MatrixType>::MatrixType
+Tridiagonalization<MatrixType>::matrixT(void) const
+{
+ // FIXME should this function (and other similar ones) rather take a matrix as argument
+ // and fill it ? (to avoid temporaries)
+ int n = m_matrix.rows();
+ MatrixType matT = m_matrix;
+ matT.corner(TopRight,n-1, n-1).diagonal() = subDiagonal().template cast<Scalar>().conjugate();
+ if (n>2)
+ {
+ matT.corner(TopRight,n-2, n-2).template triangularView<UpperTriangular>().setZero();
+ matT.corner(BottomLeft,n-2, n-2).template triangularView<LowerTriangular>().setZero();
+ }
+ return matT;
+}
+
+#ifndef EIGEN_HIDE_HEAVY_CODE
+
+/** \internal
+ * Performs a tridiagonal decomposition of \a matA in place.
+ *
+ * \param matA the input selfadjoint matrix
+ * \param hCoeffs returned Householder coefficients
+ *
+ * The result is written in the lower triangular part of \a matA.
+ *
+ * Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
+ *
+ * \sa packedMatrix()
+ */
+template<typename MatrixType>
+void Tridiagonalization<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs)
+{
+ assert(matA.rows()==matA.cols());
+ int n = matA.rows();
+ Matrix<Scalar,1,Dynamic> aux(n);
+ for (int i = 0; i<n-1; ++i)
+ {
+ int remainingSize = n-i-1;
+ RealScalar beta;
+ Scalar h;
+ matA.col(i).end(remainingSize).makeHouseholderInPlace(&h, &beta);
+
+ // Apply similarity transformation to remaining columns,
+ // i.e., A = H A H' where H = I - h v v' and v = matA.col(i).end(n-i-1)
+ matA.col(i).coeffRef(i+1) = 1;
+
+ hCoeffs.end(n-i-1) = (matA.corner(BottomRight,remainingSize,remainingSize).template selfadjointView<LowerTriangular>()
+ * (ei_conj(h) * matA.col(i).end(remainingSize)));
+
+ hCoeffs.end(n-i-1) += (ei_conj(h)*Scalar(-0.5)*(hCoeffs.end(remainingSize).dot(matA.col(i).end(remainingSize)))) * matA.col(i).end(n-i-1);
+
+ matA.corner(BottomRight, remainingSize, remainingSize).template selfadjointView<LowerTriangular>()
+ .rankUpdate(matA.col(i).end(remainingSize), hCoeffs.end(remainingSize), -1);
+
+ matA.col(i).coeffRef(i+1) = beta;
+ hCoeffs.coeffRef(i) = h;
+ }
+}
+
+/** reconstructs and returns the matrix Q */
+template<typename MatrixType>
+typename Tridiagonalization<MatrixType>::MatrixType
+Tridiagonalization<MatrixType>::matrixQ(void) const
+{
+ MatrixType matQ;
+ matrixQInPlace(&matQ);
+ return matQ;
+}
+
+template<typename MatrixType>
+template<typename QDerived>
+void Tridiagonalization<MatrixType>::matrixQInPlace(MatrixBase<QDerived>* q) const
+{
+ QDerived& matQ = q->derived();
+ int n = m_matrix.rows();
+ matQ = MatrixType::Identity(n,n);
+ Matrix<Scalar,1,Dynamic> aux(n);
+ for (int i = n-2; i>=0; i--)
+ {
+ matQ.corner(BottomRight,n-i-1,n-i-1)
+ .applyHouseholderOnTheLeft(m_matrix.col(i).end(n-i-2), ei_conj(m_hCoeffs.coeff(i)), &aux.coeffRef(0,0));
+ }
+}
+
+/** Performs a full decomposition in place */
+template<typename MatrixType>
+void Tridiagonalization<MatrixType>::decomposeInPlace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
+{
+ int n = mat.rows();
+ ei_assert(mat.cols()==n && diag.size()==n && subdiag.size()==n-1);
+ if (n==3 && (!NumTraits<Scalar>::IsComplex) )
+ {
+ _decomposeInPlace3x3(mat, diag, subdiag, extractQ);
+ }
+ else
+ {
+ Tridiagonalization tridiag(mat);
+ diag = tridiag.diagonal();
+ subdiag = tridiag.subDiagonal();
+ if (extractQ)
+ tridiag.matrixQInPlace(&mat);
+ }
+}
+
+/** \internal
+ * Optimized path for 3x3 matrices.
+ * Especially useful for plane fitting.
+ */
+template<typename MatrixType>
+void Tridiagonalization<MatrixType>::_decomposeInPlace3x3(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
+{
+ diag[0] = ei_real(mat(0,0));
+ RealScalar v1norm2 = ei_abs2(mat(0,2));
+ if (ei_isMuchSmallerThan(v1norm2, RealScalar(1)))
+ {
+ diag[1] = ei_real(mat(1,1));
+ diag[2] = ei_real(mat(2,2));
+ subdiag[0] = ei_real(mat(0,1));
+ subdiag[1] = ei_real(mat(1,2));
+ if (extractQ)
+ mat.setIdentity();
+ }
+ else
+ {
+ RealScalar beta = ei_sqrt(ei_abs2(mat(0,1))+v1norm2);
+ RealScalar invBeta = RealScalar(1)/beta;
+ Scalar m01 = mat(0,1) * invBeta;
+ Scalar m02 = mat(0,2) * invBeta;
+ Scalar q = RealScalar(2)*m01*mat(1,2) + m02*(mat(2,2) - mat(1,1));
+ diag[1] = ei_real(mat(1,1) + m02*q);
+ diag[2] = ei_real(mat(2,2) - m02*q);
+ subdiag[0] = beta;
+ subdiag[1] = ei_real(mat(1,2) - m01 * q);
+ if (extractQ)
+ {
+ mat(0,0) = 1;
+ mat(0,1) = 0;
+ mat(0,2) = 0;
+ mat(1,0) = 0;
+ mat(1,1) = m01;
+ mat(1,2) = m02;
+ mat(2,0) = 0;
+ mat(2,1) = m02;
+ mat(2,2) = -m01;
+ }
+ }
+}
+
+#endif // EIGEN_HIDE_HEAVY_CODE
+
+#endif // EIGEN_TRIDIAGONALIZATION_H
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