text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2009 Hauke Heibel <hauke.heibel@gmail.com>
  //
  // 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 <Eigen/Geometry>
  
  #include <Eigen/LU> // required for MatrixBase::determinant
  #include <Eigen/SVD> // required for SVD
  
  using namespace Eigen;
  
  //  Constructs a random matrix from the unitary group U(size).
  template <typename T>
  Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> randMatrixUnitary(int size)
  {
    typedef T Scalar;
    typedef Eigen::Matrix<Scalar, Eigen::Dynamic, Eigen::Dynamic> MatrixType;
  
    MatrixType Q;
  
    int max_tries = 40;
    double is_unitary = false;
  
    while (!is_unitary && max_tries > 0)
    {
      // initialize random matrix
      Q = MatrixType::Random(size, size);
  
      // orthogonalize columns using the Gram-Schmidt algorithm
      for (int col = 0; col < size; ++col)
      {
        typename MatrixType::ColXpr colVec = Q.col(col);
        for (int prevCol = 0; prevCol < col; ++prevCol)
        {
          typename MatrixType::ColXpr prevColVec = Q.col(prevCol);
          colVec -= colVec.dot(prevColVec)*prevColVec;
        }
        Q.col(col) = colVec.normalized();
      }
  
      // this additional orthogonalization is not necessary in theory but should enhance
      // the numerical orthogonality of the matrix
      for (int row = 0; row < size; ++row)
      {
        typename MatrixType::RowXpr rowVec = Q.row(row);
        for (int prevRow = 0; prevRow < row; ++prevRow)
        {
          typename MatrixType::RowXpr prevRowVec = Q.row(prevRow);
          rowVec -= rowVec.dot(prevRowVec)*prevRowVec;
        }
        Q.row(row) = rowVec.normalized();
      }
  
      // final check
      is_unitary = Q.isUnitary();
      --max_tries;
    }
  
    if (max_tries == 0)
      eigen_assert(false && "randMatrixUnitary: Could not construct unitary matrix!");
  
    return Q;
  }
  
  //  Constructs a random matrix from the special unitary group SU(size).
  template <typename T>
  Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> randMatrixSpecialUnitary(int size)
  {
    typedef T Scalar;
  
    typedef Eigen::Matrix<Scalar, Eigen::Dynamic, Eigen::Dynamic> MatrixType;
  
    // initialize unitary matrix
    MatrixType Q = randMatrixUnitary<Scalar>(size);
  
    // tweak the first column to make the determinant be 1
    Q.col(0) *= numext::conj(Q.determinant());
  
    return Q;
  }
  
  template <typename MatrixType>
  void run_test(int dim, int num_elements)
  {
    using std::abs;
    typedef typename internal::traits<MatrixType>::Scalar Scalar;
    typedef Matrix<Scalar, Eigen::Dynamic, Eigen::Dynamic> MatrixX;
    typedef Matrix<Scalar, Eigen::Dynamic, 1> VectorX;
  
    // MUST be positive because in any other case det(cR_t) may become negative for
    // odd dimensions!
    const Scalar c = abs(internal::random<Scalar>());
  
    MatrixX R = randMatrixSpecialUnitary<Scalar>(dim);
    VectorX t = Scalar(50)*VectorX::Random(dim,1);
  
    MatrixX cR_t = MatrixX::Identity(dim+1,dim+1);
    cR_t.block(0,0,dim,dim) = c*R;
    cR_t.block(0,dim,dim,1) = t;
  
    MatrixX src = MatrixX::Random(dim+1, num_elements);
    src.row(dim) = Matrix<Scalar, 1, Dynamic>::Constant(num_elements, Scalar(1));
  
    MatrixX dst = cR_t*src;
  
    MatrixX cR_t_umeyama = umeyama(src.block(0,0,dim,num_elements), dst.block(0,0,dim,num_elements));
  
    const Scalar error = ( cR_t_umeyama*src - dst ).norm() / dst.norm();
    VERIFY(error < Scalar(40)*std::numeric_limits<Scalar>::epsilon());
  }
  
  template<typename Scalar, int Dimension>
  void run_fixed_size_test(int num_elements)
  {
    using std::abs;
    typedef Matrix<Scalar, Dimension+1, Dynamic> MatrixX;
    typedef Matrix<Scalar, Dimension+1, Dimension+1> HomMatrix;
    typedef Matrix<Scalar, Dimension, Dimension> FixedMatrix;
    typedef Matrix<Scalar, Dimension, 1> FixedVector;
  
    const int dim = Dimension;
  
    // MUST be positive because in any other case det(cR_t) may become negative for
    // odd dimensions!
    // Also if c is to small compared to t.norm(), problem is ill-posed (cf. Bug 744)
    const Scalar c = internal::random<Scalar>(0.5, 2.0);
  
    FixedMatrix R = randMatrixSpecialUnitary<Scalar>(dim);
    FixedVector t = Scalar(32)*FixedVector::Random(dim,1);
  
    HomMatrix cR_t = HomMatrix::Identity(dim+1,dim+1);
    cR_t.block(0,0,dim,dim) = c*R;
    cR_t.block(0,dim,dim,1) = t;
  
    MatrixX src = MatrixX::Random(dim+1, num_elements);
    src.row(dim) = Matrix<Scalar, 1, Dynamic>::Constant(num_elements, Scalar(1));
  
    MatrixX dst = cR_t*src;
  
    Block<MatrixX, Dimension, Dynamic> src_block(src,0,0,dim,num_elements);
    Block<MatrixX, Dimension, Dynamic> dst_block(dst,0,0,dim,num_elements);
  
    HomMatrix cR_t_umeyama = umeyama(src_block, dst_block);
  
    const Scalar error = ( cR_t_umeyama*src - dst ).squaredNorm();
  
    VERIFY(error < Scalar(16)*std::numeric_limits<Scalar>::epsilon());
  }
  
  void test_umeyama()
  {
    for (int i=0; i<g_repeat; ++i)
    {
      const int num_elements = internal::random<int>(40,500);
  
      // works also for dimensions bigger than 3...
      for (int dim=2; dim<8; ++dim)
      {
        CALL_SUBTEST_1(run_test<MatrixXd>(dim, num_elements));
        CALL_SUBTEST_2(run_test<MatrixXf>(dim, num_elements));
      }
  
      CALL_SUBTEST_3((run_fixed_size_test<float, 2>(num_elements)));
      CALL_SUBTEST_4((run_fixed_size_test<float, 3>(num_elements)));
      CALL_SUBTEST_5((run_fixed_size_test<float, 4>(num_elements)));
  
      CALL_SUBTEST_6((run_fixed_size_test<double, 2>(num_elements)));
      CALL_SUBTEST_7((run_fixed_size_test<double, 3>(num_elements)));
      CALL_SUBTEST_8((run_fixed_size_test<double, 4>(num_elements)));
    }
  
    // Those two calls don't compile and result in meaningful error messages!
    // umeyama(MatrixXcf(),MatrixXcf());
    // umeyama(MatrixXcd(),MatrixXcd());
  }
  