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

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)
}