eigensolver_generic.cpp

// 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/.

#include "main.h"
#include <limits>
#include <Eigen/Eigenvalues>

template<typename EigType,typename MatType>
void check_eigensolver_for_given_mat(const EigType &eig, const MatType& a)
{
  typedef typename NumTraits<typename MatType::Scalar>::Real RealScalar;
  typedef Matrix<RealScalar, MatType::RowsAtCompileTime, 1> RealVectorType;
  typedef typename std::complex<RealScalar> Complex;
  Index n = a.rows();
  VERIFY_IS_EQUAL(eig.info(), Success);
  VERIFY_IS_APPROX(a * eig.pseudoEigenvectors(), eig.pseudoEigenvectors() * eig.pseudoEigenvalueMatrix());
  VERIFY_IS_APPROX(a.template cast<Complex>() * eig.eigenvectors(),
                   eig.eigenvectors() * eig.eigenvalues().asDiagonal());
  VERIFY_IS_APPROX(eig.eigenvectors().colwise().norm(), RealVectorType::Ones(n).transpose());
  VERIFY_IS_APPROX(a.eigenvalues(), eig.eigenvalues());
}

template<typename MatrixType> void eigensolver(const MatrixType& m)
{
  /* this test covers the following files:
     EigenSolver.h
  */
  Index rows = m.rows();
  Index cols = m.cols();

  typedef typename MatrixType::Scalar Scalar;
  typedef typename NumTraits<Scalar>::Real RealScalar;
  typedef typename std::complex<RealScalar> Complex;

  MatrixType a = MatrixType::Random(rows,cols);
  MatrixType a1 = MatrixType::Random(rows,cols);
  MatrixType symmA =  a.adjoint() * a + a1.adjoint() * a1;

  EigenSolver<MatrixType> ei0(symmA);
  VERIFY_IS_EQUAL(ei0.info(), Success);
  VERIFY_IS_APPROX(symmA * ei0.pseudoEigenvectors(), ei0.pseudoEigenvectors() * ei0.pseudoEigenvalueMatrix());
  VERIFY_IS_APPROX((symmA.template cast<Complex>()) * (ei0.pseudoEigenvectors().template cast<Complex>()),
    (ei0.pseudoEigenvectors().template cast<Complex>()) * (ei0.eigenvalues().asDiagonal()));

  EigenSolver<MatrixType> ei1(a);
  CALL_SUBTEST( check_eigensolver_for_given_mat(ei1,a) );

  EigenSolver<MatrixType> ei2;
  ei2.setMaxIterations(RealSchur<MatrixType>::m_maxIterationsPerRow * rows).compute(a);
  VERIFY_IS_EQUAL(ei2.info(), Success);
  VERIFY_IS_EQUAL(ei2.eigenvectors(), ei1.eigenvectors());
  VERIFY_IS_EQUAL(ei2.eigenvalues(), ei1.eigenvalues());
  if (rows > 2) {
    ei2.setMaxIterations(1).compute(a);
    VERIFY_IS_EQUAL(ei2.info(), NoConvergence);
    VERIFY_IS_EQUAL(ei2.getMaxIterations(), 1);
  }

  EigenSolver<MatrixType> eiNoEivecs(a, false);
  VERIFY_IS_EQUAL(eiNoEivecs.info(), Success);
  VERIFY_IS_APPROX(ei1.eigenvalues(), eiNoEivecs.eigenvalues());
  VERIFY_IS_APPROX(ei1.pseudoEigenvalueMatrix(), eiNoEivecs.pseudoEigenvalueMatrix());

  MatrixType id = MatrixType::Identity(rows, cols);
  VERIFY_IS_APPROX(id.operatorNorm(), RealScalar(1));

  if (rows > 2 && rows < 20)
  {
    // Test matrix with NaN
    a(0,0) = std::numeric_limits<typename MatrixType::RealScalar>::quiet_NaN();
    EigenSolver<MatrixType> eiNaN(a);
    VERIFY_IS_NOT_EQUAL(eiNaN.info(), Success);
  }

  // regression test for bug 1098
  {
    EigenSolver<MatrixType> eig(a.adjoint() * a);
    eig.compute(a.adjoint() * a);
  }

  // regression test for bug 478
  {
    a.setZero();
    EigenSolver<MatrixType> ei3(a);
    VERIFY_IS_EQUAL(ei3.info(), Success);
    VERIFY_IS_MUCH_SMALLER_THAN(ei3.eigenvalues().norm(),RealScalar(1));
    VERIFY((ei3.eigenvectors().transpose()*ei3.eigenvectors().transpose()).eval().isIdentity());
  }
}

template<typename MatrixType> void eigensolver_verify_assert(const MatrixType& m)
{
  EigenSolver<MatrixType> eig;
  VERIFY_RAISES_ASSERT(eig.eigenvectors());
  VERIFY_RAISES_ASSERT(eig.pseudoEigenvectors());
  VERIFY_RAISES_ASSERT(eig.pseudoEigenvalueMatrix());
  VERIFY_RAISES_ASSERT(eig.eigenvalues());

  MatrixType a = MatrixType::Random(m.rows(),m.cols());
  eig.compute(a, false);
  VERIFY_RAISES_ASSERT(eig.eigenvectors());
  VERIFY_RAISES_ASSERT(eig.pseudoEigenvectors());
}


template<typename CoeffType>
Matrix<typename CoeffType::Scalar,Dynamic,Dynamic>
make_companion(const CoeffType& coeffs)
{
  Index n = coeffs.size()-1;
  Matrix<typename CoeffType::Scalar,Dynamic,Dynamic> res(n,n);
  res.setZero();
	res.row(0) = -coeffs.tail(n) / coeffs(0);
	res.diagonal(-1).setOnes();
  return res;
}

template<int>
void eigensolver_generic_extra()
{
  {
    // regression test for bug 793
    MatrixXd a(3,3);
    a << 0,  0,  1,
        1,  1, 1,
        1, 1e+200,  1;
    Eigen::EigenSolver<MatrixXd> eig(a);
    double scale = 1e-200; // scale to avoid overflow during the comparisons
    VERIFY_IS_APPROX(a * eig.pseudoEigenvectors()*scale, eig.pseudoEigenvectors() * eig.pseudoEigenvalueMatrix()*scale);
    VERIFY_IS_APPROX(a * eig.eigenvectors()*scale, eig.eigenvectors() * eig.eigenvalues().asDiagonal()*scale);
  }
  {
    // check a case where all eigenvalues are null.
    MatrixXd a(2,2);
    a << 1,  1,
        -1, -1;
    Eigen::EigenSolver<MatrixXd> eig(a);
    VERIFY_IS_APPROX(eig.pseudoEigenvectors().squaredNorm(), 2.);
    VERIFY_IS_APPROX((a * eig.pseudoEigenvectors()).norm()+1., 1.);
    VERIFY_IS_APPROX((eig.pseudoEigenvectors() * eig.pseudoEigenvalueMatrix()).norm()+1., 1.);
    VERIFY_IS_APPROX((a * eig.eigenvectors()).norm()+1., 1.);
    VERIFY_IS_APPROX((eig.eigenvectors() * eig.eigenvalues().asDiagonal()).norm()+1., 1.);
  }

  // regression test for bug 933
  {
    {
      VectorXd coeffs(5); coeffs << 1, -3, -175, -225, 2250;
      MatrixXd C = make_companion(coeffs);
      EigenSolver<MatrixXd> eig(C);
      CALL_SUBTEST( check_eigensolver_for_given_mat(eig,C) );
    }
    {
      // this test is tricky because it requires high accuracy in smallest eigenvalues
      VectorXd coeffs(5); coeffs << 6.154671e-15, -1.003870e-10, -9.819570e-01, 3.995715e+03, 2.211511e+08;
      MatrixXd C = make_companion(coeffs);
      EigenSolver<MatrixXd> eig(C);
      CALL_SUBTEST( check_eigensolver_for_given_mat(eig,C) );
      Index n = C.rows();
      for(Index i=0;i<n;++i)
      {
        typedef std::complex<double> Complex;
        MatrixXcd ac = C.cast<Complex>();
        ac.diagonal().array() -= eig.eigenvalues()(i);
        VectorXd sv = ac.jacobiSvd().singularValues();
        // comparing to sv(0) is not enough here to catch the "bug",
        // the hard-coded 1.0 is important!
        VERIFY_IS_MUCH_SMALLER_THAN(sv(n-1), 1.0);
      }
    }
  }
  // regression test for bug 1557
  {
    // this test is interesting because it contains zeros on the diagonal.
    MatrixXd A_bug1557(3,3);
    A_bug1557 << 0, 0, 0, 1, 0, 0.5887907064808635127, 0, 1, 0;
    EigenSolver<MatrixXd> eig(A_bug1557);
    CALL_SUBTEST( check_eigensolver_for_given_mat(eig,A_bug1557) );
  }

  // regression test for bug 1174
  {
    Index n = 12;
    MatrixXf A_bug1174(n,n);
    A_bug1174 <<  262144, 0, 0, 262144, 786432, 0, 0, 0, 0, 0, 0, 786432,
                  262144, 0, 0, 262144, 786432, 0, 0, 0, 0, 0, 0, 786432,
                  262144, 0, 0, 262144, 786432, 0, 0, 0, 0, 0, 0, 786432,
                  262144, 0, 0, 262144, 786432, 0, 0, 0, 0, 0, 0, 786432,
                  0, 262144, 262144, 0, 0, 262144, 262144, 262144, 262144, 262144, 262144, 0,
                  0, 262144, 262144, 0, 0, 262144, 262144, 262144, 262144, 262144, 262144, 0,
                  0, 262144, 262144, 0, 0, 262144, 262144, 262144, 262144, 262144, 262144, 0,
                  0, 262144, 262144, 0, 0, 262144, 262144, 262144, 262144, 262144, 262144, 0,
                  0, 262144, 262144, 0, 0, 262144, 262144, 262144, 262144, 262144, 262144, 0,
                  0, 262144, 262144, 0, 0, 262144, 262144, 262144, 262144, 262144, 262144, 0,
                  0, 262144, 262144, 0, 0, 262144, 262144, 262144, 262144, 262144, 262144, 0,
                  0, 262144, 262144, 0, 0, 262144, 262144, 262144, 262144, 262144, 262144, 0;
    EigenSolver<MatrixXf> eig(A_bug1174);
    CALL_SUBTEST( check_eigensolver_for_given_mat(eig,A_bug1174) );
  }
}

EIGEN_DECLARE_TEST(eigensolver_generic)
{
  int s = 0;
  for(int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1( eigensolver(Matrix4f()) );
    s = internal::random<int>(1,EIGEN_TEST_MAX_SIZE/4);
    CALL_SUBTEST_2( eigensolver(MatrixXd(s,s)) );
    TEST_SET_BUT_UNUSED_VARIABLE(s)

    // some trivial but implementation-wise tricky cases
    CALL_SUBTEST_2( eigensolver(MatrixXd(1,1)) );
    CALL_SUBTEST_2( eigensolver(MatrixXd(2,2)) );
    CALL_SUBTEST_3( eigensolver(Matrix<double,1,1>()) );
    CALL_SUBTEST_4( eigensolver(Matrix2d()) );
  }

  CALL_SUBTEST_1( eigensolver_verify_assert(Matrix4f()) );
  s = internal::random<int>(1,EIGEN_TEST_MAX_SIZE/4);
  CALL_SUBTEST_2( eigensolver_verify_assert(MatrixXd(s,s)) );
  CALL_SUBTEST_3( eigensolver_verify_assert(Matrix<double,1,1>()) );
  CALL_SUBTEST_4( eigensolver_verify_assert(Matrix2d()) );

  // Test problem size constructors
  CALL_SUBTEST_5(EigenSolver<MatrixXf> tmp(s));

  // regression test for bug 410
  CALL_SUBTEST_2(
  {
     MatrixXd A(1,1);
     A(0,0) = std::sqrt(-1.); // is Not-a-Number
     Eigen::EigenSolver<MatrixXd> solver(A);
     VERIFY_IS_EQUAL(solver.info(), NumericalIssue);
  }
  );
  
  CALL_SUBTEST_2( eigensolver_generic_extra<0>() );
  
  TEST_SET_BUT_UNUSED_VARIABLE(s)
}