[mq]: eigensolver

Eigen/QR

@@ -42,6 +42,8 @@ namespace Eigen {
 #include "src/QR/EigenSolver.h"
 #include "src/QR/SelfAdjointEigenSolver.h"
 #include "src/QR/HessenbergDecomposition.h"
+#include "src/QR/ComplexSchur.h"
+#include "src/QR/ComplexEigenSolver.h"
 
 // declare all classes for a given matrix type
 #define EIGEN_QR_MODULE_INSTANTIATE_TYPE(MATRIXTYPE,PREFIX) \

Eigen/src/QR/ComplexEigenSolver.h

@@ -0,0 +1,138 @@
+// 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
+
+#define MAXITER 30
+
+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)
+{
+  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))
+        z.real() = 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

Eigen/src/QR/ComplexSchur.h

@@ -0,0 +1,273 @@
+// 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
+
+#define MAXITER 30
+
+/** \ingroup QR
+  *
+  * \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 plane rotation G such that G' x |p| = | c  s' |' |p| = |z|
+//                                              |q|   |-s  c' |  |q|   |0|
+// and returns z if requested. Note that the returned c is real.
+template<typename T> void ei_make_givens(const std::complex<T>& p, const std::complex<T>& q,
+                                           JacobiRotation<std::complex<T> >& rot, std::complex<T>* z=0)
+{
+  typedef std::complex<T> Complex;
+  T scale, absx, absxy;
+  if(p==Complex(0))
+  {
+    // return identity
+    rot.c() = Complex(1,0);
+    rot.s() = Complex(0,0);
+    if(z) *z = p;
+  }
+  else
+  {
+    scale = cnorm1(p);
+    absx = scale * ei_sqrt(ei_abs2(p/scale));
+    scale = ei_abs(scale) + cnorm1(q);
+    absxy = scale * ei_sqrt((absx/scale)*(absx/scale) + ei_abs2(q/scale));
+    rot.c() = Complex(absx / absxy);
+    Complex np = p/absx;
+    rot.s() = -ei_conj(np) * q / absxy;
+    if(z) *z = np * absxy;
+  }
+}
+
+template<typename MatrixType>
+void ComplexSchur<MatrixType>::compute(const MatrixType& matrix)
+{
+  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 = cnorm1(m_matT.coeffRef(iu,iu)) + cnorm1(m_matT.coeffRef(iu-1,iu-1));
+      sd = cnorm1(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 >= MAXITER)
+    {
+      // 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 = cnorm1(m_matT.coeffRef(il,il)) + cnorm1(m_matT.coeffRef(il-1,il-1));
+      sd = cnorm1(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 = csqrt(b*b - RealScalar(4)*c);
+
+    r1 = (b+disc)/RealScalar(2);
+    r2 = (b-disc)/RealScalar(2);
+
+    if(cnorm1(r1) > cnorm1(r2))
+      r2 = c/r1;
+    else
+      r1 = c/r2;
+
+    if(cnorm1(r1-t.coeff(1,1)) < cnorm1(r2-t.coeff(1,1)))
+      kappa = sf * r1;
+    else
+      kappa = sf * r2;
+
+    // perform the QR step using Givens rotations
+    JacobiRotation<Complex> rot;
+    ei_make_givens(m_matT.coeff(il,il) - kappa, m_matT.coeff(il+1,il), rot);
+
+    for(int i=il ; i<iu ; i++)
+    {
+      m_matT.block(0,i,n,n-i).applyJacobiOnTheLeft(i, i+1, rot.adjoint());
+      m_matT.block(0,0,std::min(i+2,iu)+1,n).applyJacobiOnTheRight(i, i+1, rot);
+      m_matU.applyJacobiOnTheRight(i, i+1, rot);
+
+      if(i != iu-1)
+      {
+        int i1 = i+1;
+        int i2 = i+2;
+
+        ei_make_givens(m_matT.coeffRef(i1,i), m_matT.coeffRef(i2,i), rot, &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(m_matT.coeff(i,j).real()) < eps)
+        m_matT.coeffRef(i,j).real() = 0;
+      if(ei_abs(m_matT.coeff(i,j).imag()) < eps)
+        m_matT.coeffRef(i,j).imag() = 0;
+    }
+
+  m_isInitialized = true;
+}
+
+// norm1 of complex numbers
+template<typename T>
+T cnorm1(const std::complex<T> &Z)
+{
+  return(ei_abs(Z.real()) + ei_abs(Z.imag()));
+}
+
+/**
+  * Computes the principal value of the square root of the complex \a z.
+  */
+template<typename RealScalar>
+std::complex<RealScalar> csqrt(const std::complex<RealScalar> &z)
+{
+  RealScalar t, tre, tim;
+
+  t = ei_abs(z);
+
+  if (ei_abs(z.real()) <= ei_abs(z.imag()))
+  {
+    // No cancellation in these formulas
+    tre = ei_sqrt(0.5*(t+z.real()));
+    tim = ei_sqrt(0.5*(t-z.real()));
+  }
+  else
+  {
+    // Stable computation of the above formulas
+    if (z.real() > 0)
+    {
+      tre = t + z.real();
+      tim = ei_abs(z.imag())*ei_sqrt(0.5/tre);
+      tre = ei_sqrt(0.5*tre);
+    }
+    else
+    {
+      tim = t - z.real();
+      tre = ei_abs(z.imag())*ei_sqrt(0.5/tim);
+      tim = ei_sqrt(0.5*tim);
+    }
+  }
+  if(z.imag() < 0)
+    tim = -tim;
+
+  return (std::complex<RealScalar>(tre,tim));
+
+}
+
+
+#endif // EIGEN_COMPLEX_SCHUR_H

test/CMakeLists.txt

@@ -125,6 +125,7 @@ ei_add_test(qr_colpivoting)
 ei_add_test(qr_fullpivoting)
 ei_add_test(eigensolver_selfadjoint " " "${GSL_LIBRARIES}")
 ei_add_test(eigensolver_generic " " "${GSL_LIBRARIES}")
+ei_add_test(eigensolver_complex)
 ei_add_test(svd)
 ei_add_test(jacobisvd ${EI_OFLAG})
 ei_add_test(geo_orthomethods)

test/eigensolver_complex.cpp

@@ -0,0 +1,70 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra. Eigen itself is part of the KDE project.
+//
+// Copyright (C) 2008-2009 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/>.
+
+#include "main.h"
+#include <Eigen/QR>
+#include <Eigen/LU>
+
+template<typename MatrixType> void eigensolver(const MatrixType& m)
+{
+  /* this test covers the following files:
+     ComplexEigenSolver.h
+  */
+  int rows = m.rows();
+  int cols = m.cols();
+
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename NumTraits<Scalar>::Real RealScalar;
+  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;
+  typedef Matrix<RealScalar, MatrixType::RowsAtCompileTime, 1> RealVectorType;
+  typedef typename std::complex<typename NumTraits<typename MatrixType::Scalar>::Real> Complex;
+
+  // RealScalar largerEps = 10*test_precision<RealScalar>();
+
+  MatrixType a = MatrixType::Random(rows,cols);
+  MatrixType a1 = MatrixType::Random(rows,cols);
+  MatrixType symmA =  a.adjoint() * a + a1.adjoint() * a1;
+
+//   ComplexEigenSolver<MatrixType> ei0(symmA);
+
+//   VERIFY_IS_APPROX(symmA * ei0.eigenvectors(), ei0.eigenvectors() * ei0.eigenvalues().asDiagonal());
+
+//   a.imag().setZero();
+//   std::cerr << a << "\n\n";
+  ComplexEigenSolver<MatrixType> ei1(a);
+//   exit(1);
+  VERIFY_IS_APPROX(a * ei1.eigenvectors(), ei1.eigenvectors() * ei1.eigenvalues().asDiagonal());
+
+}
+
+void test_eigensolver_complex()
+{
+  for(int i = 0; i < g_repeat; i++) {
+//     CALL_SUBTEST( eigensolver(Matrix4cf()) );
+//     CALL_SUBTEST( eigensolver(MatrixXcd(4,4)) );
+    CALL_SUBTEST( eigensolver(MatrixXcd(6,6)) );
+//     CALL_SUBTEST( eigensolver(MatrixXd(14,14)) );
+  }
+}
+