move eigen values related stuff of the QR module to a new EigenSolver module.

- perhaps we can find a better name ?
- note that the QR module still includes the EigenSolver module for compatibility

1 files changed, 317 insertions, 0 deletions

diff --git a/Eigen/src/EigenSolver/Tridiagonalization.h b/Eigen/src/EigenSolver/Tridiagonalization.h
new file mode 100644
index 000000000..e0bff17b9
--- /dev/null
+++ b/Eigen/src/EigenSolver/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
+
+/** \ingroup EigenSolver_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