text: // 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) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
  //
  // 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>
  
  template<typename MatrixType> void qr()
  {
    typedef typename MatrixType::Index Index;
  
    Index rows = internal::random<Index>(20,200), cols = internal::random<int>(20,200), cols2 = internal::random<int>(20,200);
    Index rank = internal::random<Index>(1, (std::min)(rows, cols)-1);
  
    typedef typename MatrixType::Scalar Scalar;
    typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
    typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> VectorType;
    MatrixType m1;
    createRandomPIMatrixOfRank(rank,rows,cols,m1);
    FullPivHouseholderQR<MatrixType> qr(m1);
    VERIFY(rank == qr.rank());
    VERIFY(cols - qr.rank() == qr.dimensionOfKernel());
    VERIFY(!qr.isInjective());
    VERIFY(!qr.isInvertible());
    VERIFY(!qr.isSurjective());
  
    MatrixType r = qr.matrixQR();
    
    MatrixQType q = qr.matrixQ();
    VERIFY_IS_UNITARY(q);
    
    // FIXME need better way to construct trapezoid
    for(int i = 0; i < rows; i++) for(int j = 0; j < cols; j++) if(i>j) r(i,j) = Scalar(0);
  
    MatrixType c = qr.matrixQ() * r * qr.colsPermutation().inverse();
  
    VERIFY_IS_APPROX(m1, c);
  
    MatrixType m2 = MatrixType::Random(cols,cols2);
    MatrixType m3 = m1*m2;
    m2 = MatrixType::Random(cols,cols2);
    m2 = qr.solve(m3);
    VERIFY_IS_APPROX(m3, m1*m2);
  }
  
  template<typename MatrixType> void qr_invertible()
  {
    typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
    typedef typename MatrixType::Scalar Scalar;
  
    int size = internal::random<int>(10,50);
  
    MatrixType m1(size, size), m2(size, size), m3(size, size);
    m1 = MatrixType::Random(size,size);
  
    if (internal::is_same<RealScalar,float>::value)
    {
      // let's build a matrix more stable to inverse
      MatrixType a = MatrixType::Random(size,size*2);
      m1 += a * a.adjoint();
    }
  
    FullPivHouseholderQR<MatrixType> qr(m1);
    VERIFY(qr.isInjective());
    VERIFY(qr.isInvertible());
    VERIFY(qr.isSurjective());
  
    m3 = MatrixType::Random(size,size);
    m2 = qr.solve(m3);
    VERIFY_IS_APPROX(m3, m1*m2);
  
    // now construct a matrix with prescribed determinant
    m1.setZero();
    for(int i = 0; i < size; i++) m1(i,i) = internal::random<Scalar>();
    RealScalar absdet = internal::abs(m1.diagonal().prod());
    m3 = qr.matrixQ(); // get a unitary
    m1 = m3 * m1 * m3;
    qr.compute(m1);
    VERIFY_IS_APPROX(absdet, qr.absDeterminant());
    VERIFY_IS_APPROX(internal::log(absdet), qr.logAbsDeterminant());
  }
  
  template<typename MatrixType> void qr_verify_assert()
  {
    MatrixType tmp;
  
    FullPivHouseholderQR<MatrixType> qr;
    VERIFY_RAISES_ASSERT(qr.matrixQR())
    VERIFY_RAISES_ASSERT(qr.solve(tmp))
    VERIFY_RAISES_ASSERT(qr.matrixQ())
    VERIFY_RAISES_ASSERT(qr.dimensionOfKernel())
    VERIFY_RAISES_ASSERT(qr.isInjective())
    VERIFY_RAISES_ASSERT(qr.isSurjective())
    VERIFY_RAISES_ASSERT(qr.isInvertible())
    VERIFY_RAISES_ASSERT(qr.inverse())
    VERIFY_RAISES_ASSERT(qr.absDeterminant())
    VERIFY_RAISES_ASSERT(qr.logAbsDeterminant())
  }
  
  void test_qr_fullpivoting()
  {
   for(int i = 0; i < 1; i++) {
      // FIXME : very weird bug here
  //     CALL_SUBTEST(qr(Matrix2f()) );
      CALL_SUBTEST_1( qr<MatrixXf>() );
      CALL_SUBTEST_2( qr<MatrixXd>() );
      CALL_SUBTEST_3( qr<MatrixXcd>() );
    }
  
    for(int i = 0; i < g_repeat; i++) {
      CALL_SUBTEST_1( qr_invertible<MatrixXf>() );
      CALL_SUBTEST_2( qr_invertible<MatrixXd>() );
      CALL_SUBTEST_4( qr_invertible<MatrixXcf>() );
      CALL_SUBTEST_3( qr_invertible<MatrixXcd>() );
    }
  
    CALL_SUBTEST_5(qr_verify_assert<Matrix3f>());
    CALL_SUBTEST_6(qr_verify_assert<Matrix3d>());
    CALL_SUBTEST_1(qr_verify_assert<MatrixXf>());
    CALL_SUBTEST_2(qr_verify_assert<MatrixXd>());
    CALL_SUBTEST_4(qr_verify_assert<MatrixXcf>());
    CALL_SUBTEST_3(qr_verify_assert<MatrixXcd>());
  
    // Test problem size constructors
    CALL_SUBTEST_7(FullPivHouseholderQR<MatrixXf>(10, 20));
  }
  